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MATHEMATICAL PROGRAMMING INSTRATIFIED RANDOM SAMPLING

by

Dinesh Krishna Rao

A thesis submitted in fulfillment of the requirements for the

degree of Doctor of Philosophy in Mathematics

Copyright © 2017 by Dinesh Krishna Rao

School of Computing, Information and Mathematical Sciences

Faculty of Science, Technology and Environment

The University of the South Pacific

February, 2017

Declaration of Originality

Statement by Author:I hereby declare that the work presented in this thesis, which I now submit forassessment of the award of Doctor of Philosophy is entirely my own work and hasnot been taken from the work of others; except in citations and references whichhave been acknowledged and results in this thesis has not been previously submittedfor a university degree either in whole or in part elsewhere.

Signature: ....................................................... Date: 2nd, February, 2017

Name: Dinesh Krishna Rao

Student ID No.: S99007484

Statement by Supervisor:The research in this thesis was performed under my supervision and to my bestknowledge is the sole work of Mr. Dinesh Krishna Rao.

Signature: ....................................................... Date: 2nd, February, 2017

Name: Dr. M.G.M. Khan

Designation: Associate Professor in Statistics, The University of the South Pacific,Suva, Fiji

Dedication

This thesis is dedicated to my:parents Mr. and Mrs. Narayan Rao,

wife Mrs. Moreen Rao,and son Nimish Rao.

.

Acknowledgments

I would like to thank God, the almighty for giving me patience, health, wisdom andblessing to accomplish this thesis. I wish to take this opportunity to sincerely thankall those people who have assisted or contributed towards the production of thisthesis.

My profound thanks go to my supervisor, Dr M.G.M. Khan, Associate Professor inStatistics, who has guided me through the process of research and scholarly writingof this thesis. I deeply express my sincere appreciation for his guidance, valuablesuggestions, motivation and constant advice in every phase of this meritorious taskand proper guidance till completion.

Special thanks goes to Professor Jacek Wesolowski of Warsaw University of Tech-nology, Poland for his valuable suggestions, Mr. Harsh Saini, Systems Analyst/Pro-grammer at the USP Research Office for assisting in developing computer programsand Ms. Artila Devi for helping in editing of the thesis.

My sincere gratitude goes to the University of the South Pacific for providing financialsupport towards my studies and also for allowing me to undertake my studies whilebeing a full time staff.

I am greatly indebted to my family, especially my parents, wife and son who un-dauntedly endured several late hours of my work with subsequent negligence of fam-ily duties. I also wish to thank my sister in law, brothers and their daughters forcontinuously praying and motivating me to complete my studies.

Finally, I would like to thank everyone who has supported me in one way or theother towards the completion this thesis.

iv

Abbreviations

SRSWOR : Simple Random Sampling Without ReplacementOSB : Optimum Strata BoundariesOSW : Optimum Strata WidthsOSS : Optimum Sample Size in each stratumMPP : Mathematical Programming ProblemNLPP : Nonlinear Programming ProblemLPP : Linear Programming ProblemIPP : Integer Programming ProblemQPP : Quadratic Programming ProblemCPP : Convex Programming ProblemSPP : Seperable Programming ProblemMOPP : Multi Objective Programming ProblemFPP : Fractional Programming ProblemGPP : Geometric Programming ProblemLREG : Linear RegressionGREG : Generalized RegressionCS : Chi-Squarer.v. : random variablep.d.f : probability density functionCRF : Cumulative Root FrequencyLH : Lavallée and Hidiroglouf.p.c : finite population correctionCRF : Cumulative Root FrequencyLH : Lavallée and Hidiroglouf.p.c : finite population correctionILS : Iterative Local Searcherf : error function

v

SE : Standard ErrorSS : Sum of SquareANOVA : Analysis Of Variancew.r.t : with respect tos.t : subject tomin : minimumOCW : Optimum Calibrated WeightsPRE : Percentage Relative EfficiencyMSE : Mean Square ErrorGMREG : Generalized Multiple Regression

vi

Abstract

In the first part of this thesis, we focus on the designing of stratified sampling wherebywe discuss the problems of determining optimum strata boundaries and optimumsample size. In this study, a mathematical programming approach is used to deter-mine the Optimum Strata Boundaries (OSB) under Neyman allocation for skewedpopulations when the study variable itself or its auxiliary variable is the stratifica-tion variable. The problems of determining OSB for each population which followsLog-normal or Pareto distributions, respectively are redefined as the problems ofOptimum Strata Widths (OSW) and are formulated as Mathematical ProgrammingProblems (MPPs). The formulated MPPs turn out to be multistage decision prob-lems that can be solved by dynamic programming technique. Numerical examplesare presented to illustrate the application and computational details of the proposedmethod. Comparison studies are conducted to investigate the efficiency of the pro-posed method with the available stratification methods in the literature. The studiesreveal that the proposed method is efficient and useful to obtain OSB for skewed pop-ulations.

The second part deals with the improvement of survey estimates in stratified sam-pling at the estimation stage. It is well known that the precision of the survey esti-mates can further be increased using a calibration approach. A calibration estimatoruses calibrated weights that minimize a given distance measure to the design weightswhile satisfying a set of constraints based on the known auxiliary information. Thisstudy proposes some new calibration estimators of population mean in stratifiedsampling using the known information on single or several auxiliary variables. Theproblem of determining the Optimum Calibrated Weights (OCW) is formulated asan Mathematical Programming Problem (MPP) and solved using Lagrange multi-plier technique. The calibrated weights can then be used to compute nearly designunbiased estimators of the population mean. The proposed calibration estimatorsof population mean is derived in the form of Generalized Regression (GREG) esti-

vii

mator and the estimated variances of these estimators are also provided. Numericalexamples are presented to illustrate the application and computational details of theproposed calibration estimators. Simulation studies are also carried out with somereal populations to investigate the efficiency of the proposed calibration estimatorsas compared to some well known calibration estimators available in the literature.

viii

Preface

This thesis entitled “Mathematical Programing in Stratified Random Sampling” isa collection of Mathematical Programming Problems (MPPs) and their solutions instratified sampling. Stratified random sampling is one of the most widely used sam-pling techniques which needs the solution of the following three basic problems: (1)the determination of optimum number of strata, (2) the determination of OptimumStrata Boundaries (OSB) and (3) the determination of optimum sample size fromeach strata (OSS). This thesis is an attempt to provide solutions to the problems (2)and (3) using a mathematical programing approach.

Further to increase the precision of survey estimates, calibration estimators arewidely used in sample surveys. The calibration estimators use the calibrated weightswhich are obtained by minimizing a distance measure subject to some calibration con-straints on the known information of the auxiliary variable. The calibrated weightscan then be used to form nearly unbiased estimators of population parameters. Inthis thesis, we develop some calibration estimators of population mean in strati-fied sampling using the known mean information from a single or several auxiliaryvariables. Investigations are also carried out to study whether adding constraint onvariance information and adding constraint on calibrated weights improve the esti-mates in stratified sampling. Further, a study is conducted to find whether the in-clusion of information on more auxiliary variables increase the precision of estimates.

The details of the research carried out in each chapter is presented below:

• Chapter 1 discusses the preliminary concepts which will facilitate better under-standing of the latter chapters of the thesis. In particular, the main conceptsthat were discussed in this chapter are: survey sampling, stratified randomsampling, calibration approach, distance function, mathematical programmingproblem, dynamic programming technique and Lagrange multiplier technique.The chapter concluded by stating the objectives of the study.

ix

• Chapter 2 provides the review of literature and studies in the area of stratifi-cation and calibration approach estimators. The problem of stratification wasstudied and a detailed literature review on stratification methods was carriedout. Some common stratification methods such as Cumulative Root Frequency(CRF), Geometric, Lavallée and Hidiroglou (LH), genetic algorithm, iterativesearch algorithm and dynamic programing were studied in detailed. Somestratification methods provide approximate strata boundaries while others de-termine the OSB using optimization. There are several methods available fordetermining OSB when the study variable itself is the stratification variable.We also reviewed some methods which determined the strata boundaries of thestudy variable using the auxiliary variable. The concepts of take-all, take-someand take-none were also studied which is useful in stratifying skewed popula-tions. Further, a detailed literature review of calibration approach and thecalibration estimators was carried out. The original concept of calibration es-timation was introduced in survey sampling by Deville and Särndal [19]. Sincethe focus of this thesis is on stratified sampling, the stratified sampling cal-ibration estimators were studied. Some well known calibration estimators instratified sampling that we reviewed in detailed were: Singh et al. [92], Tracyet al. [100] and Singh [83].

• In Chapter 3, the problem of determining OSB is redefined as a MPP of Op-timum Strata Widths (OSW) that seeks the minimization of variance of es-timated population mean under Neyman allocation subject to the constraintthat sum of width of all strata be equal to the range of the distribution. Adetailed solution procedure to solve the MPP of OSW using dynamic pro-gramming technique is also provided. The procedure is then used to determineOSW, OSB and OSS for a skewed population with Log-normal study variable.Finally, a numerical example is provided to illustrate the computational detailsof the solution procedure and a comparison study was performed to investigatethe efficiency of the proposed method with other stratification methods.

• Chapter 4 extended the dynamic programming method to stratify skewed pop-ulations using auxiliary variable, since constructing OSB using the study vari-able is not realistic because they are unavailable prior to survey. Using asimilar procedure discussed in Chapter 3, we determine the OSW and OSBfor a skewed population with a Pareto auxiliary variable. Then, the regressionmodel is used to determine OSB and Neyman allocation to obtain OSS for the

x

study variable. Numerical example and comparison study are again performedusing real and simulated data.

• Chapter 5 proposes some calibration estimators of population mean in strat-ified sampling using known information (such as mean and/or variance) onsingle auxiliary variable. The optimum calibrated weights are derived by mini-mizing the chi-square distance function subject to some calibration constraintsusing a Lagrange multiplier technique. The proposed estimators are derived inthe form of GREG and their estimated variance have also been provided. Anumerical example is provided to show the computational details of the pro-posed calibration estimators and a simulation study is carried out to investigatetheir efficiencies. The chapter also investigates whether adding constraints onvariance information and constraints on calibrated weights further increase theprecision of estimates.

• Chapter 6 attempts to develop some multivariate calibration estimators of pop-ulation mean in stratified sampling. The calibrated weights are determined byusing the similar methodology discussed in Chapter 5. The proposed esti-mators are derived in Generalized Multiple Regression (GMREG) form andthe estimated variance of these estimators are provided. Numerical exampleand simulation study are also provided to investigate the efficiency and otherproperties of these estimators.

• Finally, Chapter 7 reviews how the objectives of the study have been achieved.It also presents the summary of findings and the recommendations for futureresearch.

xi

Contents

Acknowledgements iv

Abbreviations v

Abstract vii

Preface ix

List of Tables xvi

List of Figures xix

1 Introduction 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Survey sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Stratified random sampling . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Study and auxiliary variable . . . . . . . . . . . . . . . . . . . 41.3.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.3 Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.4 Estimators of population mean . . . . . . . . . . . . . . . . . 61.3.5 Methods of sample allocation . . . . . . . . . . . . . . . . . . 9

1.4 Calibration approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5 The distance function . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 The Mathematical Programming Problem (MPP) . . . . . . . . . . . 131.7 The dynamic programming technique . . . . . . . . . . . . . . . . . . 141.8 The Lagrange multiplier technique . . . . . . . . . . . . . . . . . . . 161.9 Objectives of the study . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Review of Literature and Studies 192.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

xii

2.2 Optimum stratification: Review of literature and studies . . . . . . . 192.2.1 Stratification using study variable . . . . . . . . . . . . . . . . 212.2.2 Stratification using auxiliary variable . . . . . . . . . . . . . . 26

2.3 Calibration estimation: Review of literature and studies . . . . . . . . 292.3.1 Singh, Horn and Yu estimator . . . . . . . . . . . . . . . . . . 352.3.2 Tracy, Singh and Arnab estimator . . . . . . . . . . . . . . . . 382.3.3 Singh estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Optimum Stratification using Study Variable 423.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1.1 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 The general formulation of the problem . . . . . . . . . . . . . . . . 433.3 The solution procedure using dynamic programming technique . . . . 473.4 Determining OSB for skewed population with Log-normal study variable 49

3.4.1 The Log-normal distribution . . . . . . . . . . . . . . . . . . . 493.4.2 Formulation of the problem of OSW as an MPP . . . . . . . . 503.4.3 Numerical illustration . . . . . . . . . . . . . . . . . . . . . . 52

3.5 Comparison study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Optimum Stratification using Auxiliary Variable 594.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 The general formulation of the problem . . . . . . . . . . . . . . . . . 604.3 Determination of optimum sample size . . . . . . . . . . . . . . . . . 634.4 Determining OSB for skewed population with Pareto auxiliary variable 64

4.4.1 The Pareto distribution . . . . . . . . . . . . . . . . . . . . . 644.4.2 Formulation of the problem of OSW as an MPP . . . . . . . . 654.4.3 Numerical illustration . . . . . . . . . . . . . . . . . . . . . . 71

4.5 Comparison study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.5.1 Simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . 734.5.2 Real data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 New Calibration Estimators in Stratified Sampling 835.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.1.1 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

xiii

5.2 Proposed calibration estimators . . . . . . . . . . . . . . . . . . . . . 845.2.1 Estimator I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.2.2 Estimator II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.2.3 Estimator III . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.2.4 Estimator IV . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.2.5 Estimator V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2.6 Estimator VI . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.3 Numerical illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.4 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4.1 Populations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.4.2 Simulation setup and results . . . . . . . . . . . . . . . . . . . 103

5.5 Comparison study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Multivariate Calibration Estimators in Stratified Sampling 1116.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.1.1 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.1.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2 Proposed multivariate calibration estimators . . . . . . . . . . . . . . 1126.2.1 Estimator I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.2.2 Estimator II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.2.3 Estimator III . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.2.4 Estimator IV . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.3 Numerical illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.4 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326.5 Comparison study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.5.1 Comparison among the proposed multivariate calibration esti-mators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.5.2 Comparison of multivariate verses univariate calibration esti-mators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7 Conclusion and Future Research 1407.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.2 Summary of findings . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.2.1 The stratification method . . . . . . . . . . . . . . . . . . . . 1417.2.2 The calibration estimators . . . . . . . . . . . . . . . . . . . . 142

xiv

7.3 Recommendations for future research . . . . . . . . . . . . . . . . . . 142

Bibliography 144

Publications 153

Appendix A: Comparison Results for the Stratification of Log-normalPopulation 155

Appendix B: Computer Programs 166

xv

List of Tables

1 Notations in stratified random sampling. . . . . . . . . . . . . . . . . 52 OSW, OSB, OSS and the optimum value of objective function for

Log-normal study variable. . . . . . . . . . . . . . . . . . . . . . . . . 553 OSW, OSB and the optimum value of objective function for simulated

data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744 OSB and OSS of the study variable y for simulated data. . . . . . . . 755 OSB and the optimum value of objective function of different methods

for simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766 ANOVA table for Sinuosity and Latitude. . . . . . . . . . . . . . . . 787 Linear model summary. . . . . . . . . . . . . . . . . . . . . . . . . . . 788 OSW, OSB and the optimum value of objective function for cyclone

data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799 OSB and OSS of the study variable y for cyclone data. . . . . . . . . 8010 OSB and the optimum value of objective function of different methods

for cyclone data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8111 Information on tobacco population. . . . . . . . . . . . . . . . . . . . 10012 Lagrange multipliers and estimated betas for different estimators. . . 10113 OCW for different estimators. . . . . . . . . . . . . . . . . . . . . . 10114 Estimators of Y and their estimated variance for tobacco population. 10215 Information on agriculture population. . . . . . . . . . . . . . . . . . 10416 Simulation results of agriculture population. . . . . . . . . . . . . . . 10517 Simulation results of tobacco population. . . . . . . . . . . . . . . . . 10618 Information on labor population. . . . . . . . . . . . . . . . . . . . . 10619 Simulation results of labor population. . . . . . . . . . . . . . . . . . 10720 Information on ST130 population. . . . . . . . . . . . . . . . . . . . . 10721 Simulation results of ST130 population. . . . . . . . . . . . . . . . . . 10822 Information on tobacco population for multivariate estimators. . . . . 13023 Lagrange multipliers and estimated betas for different estimators. . . 131

xvi

24 OCW for different multivariate calibration estimators. . . . . . . . . 13125 Multivariate calibration estimators of Y and their estimated variance

for tobacco population. . . . . . . . . . . . . . . . . . . . . . . . . . . 13226 Information on labor population for multivariate calibration estimators.13227 Information on ST130 population for multivariate calibration estimators.13328 Simulation results of tobacco population for multivariate estimators. . 13329 Simulation results of labor force population for multivariate estimators.13330 Simulation results of ST130 population for multivariate estimators. . 13431 Comparison results for y⊕st vs y

(1)st . . . . . . . . . . . . . . . . . . . . . 137

32 Comparison results for yst vs y(2)st . . . . . . . . . . . . . . . . . . . . . 137

33 Comparison results for y⊗st vs y(3)st . . . . . . . . . . . . . . . . . . . . . 137

34 Comparison results for yst vs y(4)st . . . . . . . . . . . . . . . . . . . . . 138

35 Correlation coefficient information. . . . . . . . . . . . . . . . . . . . 13836 OSB, V (yst)

∗, Nh and nh for skewness = 0.5994, µ = 0.00009935132,

σ = 0.1975361, N = 15000, n = 1000, nclass = 50, y0 = 0.49410530

and d = 1.58890220. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15637 OSB, V (yst)

∗, Nh and nh for skewness = 1.3466, µ = −0.0132848,

σ = 0.4077880, N = 1000, n = 100, nclass = 50, y0 = 0.28757730 andd = 3.18622440. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

38 OSB, V (yst)∗, Nh and nh for skewness = 1.7274, µ = −0.004327391,

σ = 0.506233130, N = 4000, n = 400, nclass = 50, y0 = 0.14586200

and d = 6.4382790. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15839 OSB, V (yst)

∗, Nh and nh for skewness = 2.1145, µ = −0.008319588, ,

N = 2000, n = 200, nclass = 50, y0 = 0.11417550 and d = 6.83472120. 15940 OSB, V (yst)

∗, Nh and nh for skewness = 3.5009, µ = −0.00328841,

σ = 0.69666629, N = 15000, n = 1500, nclass = 50, y0 = 0.05041042

and d = 22.44861984. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16041 OSB, V (yst)

∗, Nh and nh for skewness = 4.2624, µ = 0.0008461947,

σ = 0.8006085337, N = 5000, n = 400, nclass = 50, y0 = 0.04706870

and d = 26.12477998. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16142 OSB, V (yst)

∗, Nh and nh for skewness = 3.8763, µ = 0.004467927,

σ = 0.887740363, N = 8000, n = 700, nclass = 50, y0 = 0.05568601

and d = 28.04155725. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16243 OSB, V (yst)

∗, Nh and nh for skewness = 4.3091, µ = −0.01518327,

σ = 0.99530377, N = 10000, n = 500, nclass = 20, y0 = 0.02280605

and d = 30.29211804. . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

xvii

44 OSB, V (yst)∗, Nh and nh for skewness = 5.4744, µ = −0.009671336,

σ = 1.104066811, N = 5000, n = 400, nclass = 50, y0 = 0.01439434

and d = 46.00158162. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16445 OSB, V (yst)

∗, Nh and nh for skewness = 6.6147, µ = −0.01056465,

σ = 1.2029671, N = 3000, n = 150, nclass = 50, y0 = 0.02222465 andd = 65.25438173. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

xviii

List of Figures

1 Log-normal densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 Pareto densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653 Scatter plot for Sinuosity and Latitude. . . . . . . . . . . . . . . . . . 78

xix

Chapter 1

Introduction

1.1 Introduction

This chapter discusses the preliminary concepts and the major theoretical frame-works required to conceptualize better understanding of the latter chapters of thisthesis. In particular, the concepts and frameworks discussed in the sections of thischapter are: survey sampling, stratified random sampling, calibration approach, dis-tance function, mathematical programming problem, dynamic programming tech-nique and Lagrange multiplier technique. Finally, the objectives of the study arestated at the end of this chapter.

1.2 Survey sampling

Definition. Sampling is the systematic way of monitoring and measuring the relia-bility of useful statistical information using the theory of probability, whereas surveyrefers to the means by which the data was collected to obtain information on theunits of the population. Therefore, survey sampling refers to the means of selectinga sample of units from a population to conduct a survey.

Nowadays, survey sampling are widely accepted means of gathering useful informa-tion for both administrative and research purposes. There is a definite need forsample surveys in governments, manufacturing enterprises, education sector, publichealth, social sciences, population studies, economics, psychology, sports and en-

1

vironmental science. Sample surveys are conducted by the government offices toestimate population parameters on variables such as employment and unemploy-ment rates, income and expenditure, poverty, housing, health, education, tourism,and many other important aspects that help in policy and decision making. Theyalso conduct surveys on organizations such as manufacturers, retail outlets, farms,hospitals and schools. Market analysts use sample surveys to identify the markets forproducts, to obtain customers feedback and to determine the lifespan of products.Opinion polls uses sample surveys to measure the opinion of public on a variety oftopical issues and to determine the popularity of political leaders.

Surveys are conducted employing various sampling techniques. Some specific situa-tions in which sampling techniques can be used are:

1. When minimum cost is required with a fixed precision or estimates with max-imum precision is required with a fixed budget.

2. When the units under study varies significantly for the characteristic of thepopulation under study.

3. When a total count of the population is not possible as it maybe very costlyor destructive.

4. When the scope of the study is very wide and the population partially known.

5. When money, time, and other resources are limited.

There are variety of sampling techniques that have been developed to provide efficientestimates of the population parameters. Among these the most widely used aresimple random sampling, systematic random sampling, stratified random samplingand cluster sampling. These sampling techniques have two common properties: everyunit has a definite nonzero probability of being selected in the sample and randomselection is used at some point. From the various sampling techniques, stratifiedsampling is extensively used in sample surveys and is the focus of this thesis.

1.3 Stratified random sampling

In surveys the precision of an estimator depends not only on the sample size butalso on the heterogeneity among the units in the population. If the units in the

2

population are heterogeneous, it is advantageous to use stratified random samplingto estimate the population mean (or the population total) with greater precision.

Definition. Stratified random sampling is the process of partitioning a populationof elements into non-overlapping groups, called strata, which is more homogeneousthan the entire population, and draw simple random samples of predetermined sizesfrom each of the strata.

Stratified random sampling is the most commonly used technique out of the varioussampling designs. There are many reasons for this; the principal ones being:

1. The estimates for each stratum can be obtained separately. This ensures thatall important subgroups are represented in the sample. That is, it improves thepopulation representativeness in a study by protecting us from the possibilityof obtaining a really bad sample.

2. Administrative convenience may dictate the use of stratification; it is easierto sample separately from the strata rather than from the entire population(especially if it is large). That is, from the standpoint of the agency conductingthe survey, each sub-population can be supervised separately. This can alsoallow separate analysis of each stratum, therefore, the differences among thestrata can be evaluated.

3. The total, mean, and other parameters of the entire population can be esti-mated with high precision or accuracy. If each stratum is internally homoge-nous, the measurement vary little from one unit to another, then a estimate ofany stratum mean or total can be obtained in that stratum. These estimatescan then be combined into a precise estimate for the whole population. Forexample, persons of different ages tend to have different blood pressures, so ina blood pressure study it would be helpful to stratify by age groups.

4. Each element of the sub-population would be a perfect representative of thatcharacteristic, if the stratification variable were equal to the study variable. Itwould be sufficient to take one arbitrary element out of each strata to get theactual distribution of the characteristic in the parent population. Therefore,there are often savings in time, cost and resources needed for sampling theunits. In addition, it is usually convenient to sample separately from the stratarather than from the entire population, especially if the population is large.

3

5. Stratification increases sampling efficiency if a sub-division of the populationis made so that the variability between units within a stratum is reduced ascompared to the variability within the entire population.

1.3.1 Study and auxiliary variable

Definition. A variable is an attribute or characteristic under study that assumesdifferent values for different elements. The variable which is to be investigated iscalled a survey or study variable, whereas another variable which has some kind ofrelationship with the study variable is called an auxiliary variable.

For example, the blood pressure of a person can be taken as a study variable andthe age of a person can be taken as an auxiliary variable.

Some differences between study and auxiliary variables are:

1. The cost and effort is less when an auxiliary variable is used.

2. The availability of auxiliary variable is from current or past survey, books orjournals whereas for study variable it is from current surveys or experiments.

3. The auxiliary variable has less error in measurement as compared to the studyvariable.

1.3.2 Notations

Consider a finite population Ω = 1, 2, . . . , i, . . . , N of size N divided into L non-overlapping strata Ωh = 1, 2, . . . , i, . . . , Nh withNh units in the hth (h = 1, 2, . . . , L)

stratum from which a probability sample sh(sh ⊆ Ωh) of size nh is drawn by Sim-ple Random Sampling Without Replacement (SRSWOR) such that N =

∑Lh=1Nh

and n =∑L

h=1 nh give the total population and the sample size, respectively. Letyhi and xhi denote the values of the ith unit of the study variable y and the auxil-iary variable x in the hth stratum, respectively. We define Wh = Nh/N to be thestratum weights, fh = nh/Nh to be the sampling fraction, Yh = N−1

h

∑Nh

i=1 yhi andXh = N−1

h

∑Nh

i=1 xhi to be the stratum means and S2hy = (Nh − 1)−1∑Nh

i=1

(yhi − Yh

)2

and S2hx = (Nh − 1)−1∑Nh

i=1

(xhi − Xh

)2 to be the stratum variances of the variablesin hth stratum. We also define Y =

∑Lh=1WhYh and X =

∑Lh=1WhXh to be the

overall population mean of the study variable and auxiliary variable, respectively.The table 1 gives more details on notations used in stratified random sampling.

4

Table 1: Notations in stratified random sampling.Notation Description

y Study variablex Auxiliary variableNh Population size in the hth stratum, where h = 1, 2, . . . , L and L is the no. of strataN Population size, where N =

∑Lh=1Nh

Wh Stratum weight, where Wh = Nh/N and∑Lh=1Wh = 1

yhi Value of ith unit of the study variable y in the hth stratumxhi Value of ith unit of the auxiliary variable x in the hth stratumYh Population total of y in the hth stratum, where Yh =

∑Nh

i=1 yhi

Xh Population total of x in the hth stratum, where Xh =∑Nh

i=1 xhi

Yh Population mean of y in the hth stratum, where µhy = Yh = N−1h∑Nh

i=1 yhi

Xh Population mean of x in the hth stratum, where µhx = Xh = N−1h∑Nh

i=1 xhi

S2hy Population variance of y in the hth stratum,

where S2hy = (Nh − 1)−1

∑Nh

i=1

(yhi − Yh

)2S2hx Population variance of x in the hth stratum,

where S2hx = (Nh − 1)−1

∑Nh

i=1

(xhi − Xh

)2Shxy Population covariance between x and y in the hth stratum,

where Shxy = (Nh − 1)−1∑Nh

i=1

(xhi − Xh

) (yhi − Yh

)ρhxy Population correlation coefficient between x and y in the hth stratum,

where ρhxy = Shxy/ (ShxShy)

Y Population total of y, where Y =∑Lh=1

∑Nh

i=1 yhi =∑Lh=1NhYh

X Population total of x, where X =∑Lh=1

∑Nh

i=1 xhi =∑Lh=1NhXh

Y Population mean of y, where Y= N−1∑Lh=1

∑Nh

i=1 yhi =∑Lh=1WhYh

X Population mean of x, where X= N−1∑Lh=1

∑Nh

i=1 xhi =∑Lh=1WhXh

σ2y Population variance of y, where σ2

y = N−1∑Lh=1

∑Nh

i=1

(yhi − Y

)2σ2x Population variance x, where σ2

x = N−1∑Lh=1

∑Nh

i=1

(xhi − X

)2σxy Population covariance between x and y,

where σxy = N−1∑Lh=1

∑Nh

i=1

(xhi − X

) (yhi − Y

)ρxy Population correlation coefficient between x and y, where ρxy = σxy/ (σxσy)

nh Sample size of the hth stratumfh Sampling fraction in the hth stratum, where fh = nh/Nh

n Total sample size, where n =∑Lh=1 nh

yh Sample mean of y in the hth stratum where, yh = nh−1∑nh

i=1 yhi

xh Sample mean of x in the hth stratum where, xh = nh−1∑nh

i=1 xhi

s2hy Sample variance of y in the hth stratum, where s2hy = (nh − 1)−1∑nh

i=1 (yhi − yh)2

s2hx Sample variance of x in the hth stratum, where s2hx = (nh − 1)−1∑nh

i=1 (xhi − xh)2

shxy Sample covariance between x and y in the hth stratum,where shxy = (nh − 1)−1

∑nh

i=1 (xhi − xh) (yhi − yh)

5

1.3.3 Stratification

Definition. The methodology of constructing homogeneous and non-overlapping strataof a population is termed as stratification.

Stratification can be done in two ways. The first way is, the surveyors stratify thepopulation in most convenient manner, such as the use of geographical regions (e.g.North, Central, West, etc.), administrative regions (e.g. provinces, districts, etc.),demographic or other natural characteristics (e.g. gender, age, race, income, etc.).For example, in an income and expenditure survey of a country; states, province anddistricts may be considered as strata. In survey of auditing financial transactions,the transactions may be grouped into strata on the basis of their nominal values,that is, high-value, medium-value, low-value, etc. In agricultural surveys, villagesand geographical regions may form the strata. For business surveys on employee size,production, and sales, strata are usually based on industrial classifications. While inmarketing studies, where the target consumer population is defined, strata can beformed by sex, age, income or other demographic variables.

However, the stratification by convenient manner is not a reasonable criteria asthe strata formed may not be internally homogeneous with respect to the variable ofinterest, which may end up with reduction of the precision of survey estimates. Thus,one has to look for a another way of stratification which is one of the focuses of thisthesis. In the second way, the strata are determined based on stratification variable,when its frequency distribution is known. Strata are so obtained where they are moreinternally homogeneous and the precision of the estimate of population parameters(mean or total) are optimized. In this technique, the OSB are constructed based oncertain rules. For example, if we are interested to estimate the mean, we can eitherminimize its variance for a fixed sample size or to minimize the sample size for afixed variance of the estimated mean to determine OSB. This stratification methodwill be discussed more rigorously in Chapter 2.

1.3.4 Estimators of population mean

Now we will discuss some common estimators of population mean in stratified sam-pling. The estimators discussed here are the stratified sampling estimator, the com-bined ratio estimator and the combined linear regression (LREG) estimator. Itshould be noted that the combined ratio and combined LREG estimator incorpo-rates the use the auxiliary information. For the detailed results and proofs for the

6

theorems discussed in this section, one can also refer to Singh [83].

Stratified estimator:

The stratified estimator of population mean Y is given by

yst =L∑h=1

Whyh, (1.1)

where Wh = Nh/N, yh = n−1h

∑nh

i=1 yhi denotes the sample mean in the hth stratum.

The estimator yst in (1.1) is an unbiased estimator and its variances are as stated inthe following theorems:

Theorem 1. Under SRSWOR, the variance of the estimator yst is given by

V (yst) =L∑h=1

W 2h (1− fh)nh

S2hy, (1.2)

where S2hy = (Nh − 1)−1

∑Nh

i=1

(yhi − Yh

)2.

Theorem 2. An unbiased estimator of V (yst) under SRSWOR is

v(yst) =L∑h=1

W 2h (1− fh)nh

s2hy, (1.3)

where s2hy = (nh − 1)−1∑nh

i=1 (yhi − yh)2 denotes the sample variance in the hth stra-tum.

Combined ratio estimator:

The combined ratio estimator of the population mean Y is given by

ycr = yst(X/xst

), (1.4)

where xst =L∑h=1

Whxh, yst =L∑h=1

Whyh and xh = n−1h

∑nh

i=1 xhi.

The variances of the estimator in (1.4) are as stated in the following theorems:

7

Theorem 3. The variance of the estimator ycr, to first order of approximation, is

V (ycr) =L∑h=1

W 2h (1− fh)nh

[S2hy +R2S2

hx − 2RShxy], (1.5)

where R = Y /X and Shxy = (Nh − 1)−1∑Nh

i=1

(xhi − Xh

) (yhi − Yh

).

Theorem 4. Another form of the variance of ycr, to first order of approximation, is

V (ycr) =L∑h=1

W 2h (1− fh)nh

(Nh − 1)−1Nh∑i=1

ε2hi, (1.6)

where εhi = (yhi − Yh)−R(xhi − Xh).

Theorem 5. The estimator of the variance of ycr is given by

v1(ycr) =L∑h=1

W 2h (1− fh)nh

[s2hy + r2s2

hx − 2rshxy], (1.7)

where,r = yst/xst, s

2hx = (nh − 1)−1∑nh

i=1 (xhi − xh)2 andshxy = (nh − 1)−1∑nh

i=1 (xhi − xh) (yhi − yh) .

Theorem 6. Another form of the variance of ycr is

v2 (ycr) =L∑h=1

W 2h (1− fh)nh

(nh − 1)−1nh∑i=1

e2hi, (1.8)

where ehi = (yhi − yh)− r(xhi − xh) with r = yst/xst.

Theorem 7. An improved estimator of the variance of ycr due to Wu [106] is givenby

v3 (ycr) =

(X

xst

)g L∑h=1

W 2h (1− fh)nh

(nh − 1)−1nh∑i=1

e2hi, (1.9)

where g denotes the suitable chosen constant such that the variance in (1.9) is min-imum.

Combined linear regression estimator:

The combined Linear Regression (LREG) estimator of the population mean Y isgiven by

yclr = yst + βclr(X − xst

), (1.10)

8

where

βclr =L∑h=1

W 2hn−1h (1− fh)shxy/

L∑h=1

W 2hn−1h (1− fh)s2

hx. (1.11)

The variances of the estimator in (1.10) are discussed in the following theorems:

Theorem 8. The asymptotic variance of yclr is

V (yclr) = V (yst)(1− ρ2

xy

), (1.12)

where

ρxy =

L∑h=1

W 2h (1− fh)nh

Shxy√L∑h=1

W 2h (1− fh)nh

S2hx

√L∑h=1

W 2h (1− fh)nh

S2hy

(1.13)

denotes the correlation coefficient in stratified sampling across all strata.

Theorem 9. The estimator of the variance of yclr is given by

v1(yclr) =L∑h=1

W 2h (1− fh)nh

[s2hy + β2

clrs2hx − 2βclrshxy

]. (1.14)

Theorem 10. Another form of the variance of yclr is given by

v2 (yclr) =L∑h=1

W 2h (1− fh)nh

(nh − 1)−1nh∑i=1

e2hi, (1.15)

where ehi = (yhi − yh)− βclr(xhi − xh).

Theorem 11. An improved estimator of the variance of yclr due to Wu [106] is givenby

v3 (yclr) =

(X

xst

)g L∑h=1

W 2h (1− fh)nh

(nh − 1)−1nh∑i=1

e2hi, (1.16)

where g denotes the suitable chosen constant such that v3 (yclr) is minimum.

1.3.5 Methods of sample allocation

In stratified sampling, the allocation of the sample size to different strata is done byconsideration of three factors:

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1. The total number of units in the stratum, i.e. stratum size.

2. The variability within the stratum.

3. The cost of taking observations per sampling unit in the stratum.

Then, the various methods of allocating sample to L strata are given as:

1. Equal allocation: This allocation method suggests the sample size in eachstratum are equal, i.e.,

nh = n/L. (1.17)

The variance of the estimator yst using equal allocation becomes

V (yst) =L

n

L∑h=1

W 2hS

2hy −

1

N

L∑h=1

WhS2hy. (1.18)

2. Proportional allocation: Under this allocation, the sample size in each stratumis proportional to the population stratum size, i.e., nn ∝ Nh, which implies

nh = k.Nh, (1.19)

where k is a constant. Taking summation on both sides of (1.19) we have

k = n/N. (1.20)

Hence, substituting (1.20) in (1.19), we have the proportional allocation to be

nh = n.Wh. (1.21)

The variance of the estimator yst using proportional allocation becomes

V (yst) =1

n

L∑h=1

WhS2hy −

1

N

L∑h=1

WhS2hy. (1.22)

3. Neyman allocation: The allocation of samples to different strata, which min-imizes V (yst) defined in (1.2) for fixed total sample size is referred to as theNeyman [63] allocation given by

nh = nWhShy∑Lh=1WhShy

. (1.23)

10

The variance of the estimator yst using Neyman allocation becomes

V (yst) =1

n

(L∑h=1

WhShy

)2

− 1

N

L∑h=1

WhS2hy. (1.24)

4. Optimum allocation: The allocation of samples to different strata, which min-imizes V (yst) for a fixed total cost

C = c0 +∑L

h=1chnh (1.25)

is called the optimum allocation and is given by

nh =C − c0

L∑h=1

WhShy√ch

WhShy√ch

, (1.26)

where ch be the cost of collecting information from a unit in hth stratum, c0

is the overhead cost and C is the total cost of the survey.

1.4 Calibration approach

Calibration approach to estimation are nowadays widely used in survey samplingto increase the precision of the estimators of population parameters when auxiliaryinformation is available. The method works by adjusting the design weights by theuse of known population characteristics of the auxiliary variables such as populationtotals (or means). A calibration estimator uses calibrated weights that are deter-mined by minimizing a given distance measure to the original design weights whilesatisfying a set of constraints related to the auxiliary information.

Several authors have defined the calibration approach as follows:

Definition. Ardilly [3] defines “calibration as a method of re-weighting used whenone has access to several variables, quantitative or qualitative, on which one wishesto carry out, jointly, an adjustment”.

Definition. Kott [49] defines “calibration weights as a set of weights, for units inthe sample, that satisfy a calibration to known population totals, and such that theresulting estimator is design consistent”.

11

Definition. The Statistics-Canada [94] says “calibration is a procedure that can beused to incorporate auxiliary data. This procedure adjusts the sampling weights bymultipliers known as calibration factors that make the estimates agree with the knowntotals. The resulting weights are called calibration weights or final estimation weights.These calibration weights will generally result in estimates that are design consistentand that have a smaller variance than the Horvitz-Thompson estimator”.

Definition. Särndal [75] defines calibration approach broadly as estimation for finitepopulations to consist:

1. a computation of weights that incorporate specified auxiliary information andare restrained by calibration equation(s).

2. the use of these is to weights to compute linearly weighted estimates of totalsand other finite population parameters.

3. an objective to obtain nearly design unbiased estimators as long as non-responseand other sampling errors are absent.

The calibration approach will be revisited and discussed more rigorously in Chapter2.

1.5 The distance function

A distance function in mathematics is the measure of distance between each pair ofelements in a set. A distance of zero implies that both elements are equal under thatmetric. Therefore, distance function is a way to measure how close two elementsare, where the elements can be any arbitrary object, numbers, vectors or matrices.Distance functions are frequently used as cost or error functions to be minimized inan optimization problem.

Definition. A metric on a set X is a function, called distance function

d : X ×X → [0,∞),

where [0,∞) implies a set of non-negative real numbers and for all x, y, z ∈ X, thefollowing conditions are to be satisfied:

1. d(x, y) ≥ 0.

12

2. d(x, y) = d(y, x).

3. d(x, y) = 0 if and only if x = y.

4. d(x, z) ≤ d(x, y) + d(y, z).

Remark. The condition 1 is the non-negativity property, condition 2 is the symmetryproperty, condition 3 is the coincidence property and condition 4 is the triangleinequality.

Some distance measures that have been studied in calibration estimation (see Devilleand Särndal [19]) are:

1.∑i∈s

(wi − di)2

2diqi.

2.∑i∈s

(wi − di)2

2wiqi.

3.∑i∈s

wiqi

log

(widi

)+∑i∈s

(di − wi)qi

.

4.∑i∈s

2(√

wi −√di)2

qi.

5.∑i∈s

(wi − di)qi

−∑i∈s

diqi

log

(widi

).

The distance measures (1), (4) and (5) are known as Chi-Square (CS), Hellinger andMinimum entropy, respectively. Further, wi are calibrated weights, di are designweights and qi are suitably chosen to obtain different forms of estimators.

1.6 The Mathematical Programming Problem (MPP)

The MPP is a technique used to determine the optimum value (maximum or min-imum) of a function of several decision variables which are subjected to a numberof constraints. The technique is commonly used to solve many decision makingproblems in the areas such as economics, education, public health, sociology, socialpsychology, demography, political science, and many others.

13

The essential components of an MPP model are the values of the decision variables,which describe the solutions; the objective function, which measures the qualityof solutions; and the constraints, which present the relationships between decisionvariables. Thus, the general form of an MPP can be stated as follows:

Maximize(or minimize): Z = f(x1, x2, x3, ..., xn)

subject to gi(x1, x2, x3, ..., xn)≤,=,≥ 0; i = 1, 2, 3, ...,m

and xj ≥ 0; j = 1, 2, ..., n.

(1.27)

Where in (1.27), only one sign among ≤,≡,≥ holds true for each i. Usually, unlessspecified otherwise, in an MPP all the involved functions are assumed to be contin-uously differentiable.

The variables xj; j = 1, 2, ..., n are called decision variables. If all the functions inan MPP are linear functions of the decision variables, the MPP is called a LinearProgramming Problem (LPP). Similarly, if some or all the functions are nonlinear,the MPP is called a Nonlinear Programming Problem (NLPP).

Depending on the nature of the functions involved and the restrictions on the objec-tive function and the decision variables, one customarily distinguishes the MPP intothe following branches:

1. Integer Programming Problem (IPP),

2. Quadratic Programming Problem (QPP),

3. Convex Programming Problem (CPP),

4. Separable Programming Problem (SPP),

5. Multi Objective Programming Problem (MOPP),

6. Fractional Programming Problem (FPP),

7. Geometric Programming Problem (GPP), etc.

1.7 The dynamic programming technique

Many decision-making or optimization processes for multivariate problems take placein several stages. The problems in which decisions are to be made sequentially at

14

different stages of solution are called multistage decision problems. Many multistagedecision problems can be formulated as a MPP. The dynamic programming tech-nique, developed by Bellman [7] in early 1950’s, is a computational method whichis well suited for solving MPPs that may be treated as a multistage decision prob-lem. The dynamic programming determines the optimum solution of a multistageproblem by decomposing it into stages, each stage comprising of a single stage. Theadvantage of the decomposition is that the optimization process at each stage in-volves one variable only, which simplifies the computational task by dealing with allvariables simultaneously. According to Taha [97] the dynamic programming has im-portant applications in many areas such as knapsack model, equipment replacementmodel, investment model, inventory models, etc. The technique is also successfullyused by Khan et al. [38, 36, 39, 44, 41, 37] and Nand and Khan [61] to solve manyoptimization problems, particularly in the use of stratified sampling.

The general nature of most of the MPPs that can be solved by dynamic programmingtechnique may be as follows:

1. The given MPP may be described as a multistage decision problem, where ateach stage, the value(s) of one or more decision variables are to be determined.

2. The problem must be defined for any number of stages and have the samestructure irrespective of the number of stages.

3. At each stage, there must be a specified set of parameters describing the state ofthe system, that is, the parameters on which the values of the decision variableand the objective function depends.

4. The same set of state parameters must be described as the state of the systemirrespective of the number of stages.

5. The decision at any stage, that is, the determination of the decision variable(s)at any stage must have no effect on the decisions of the remaining stagesexcept in changing the values of the parameters which describes the state ofthe system.

When the above conditions are fulfilled, the given MPP can be solved by dynamicprogramming technique. The solution of a problem is achieved in a sequential mannerstarting from one stage problem, moving onto a two stage problem, then to a three

15

stage problem and so on until finally all stages have been included. The solution forn stages is obtained by adding the nth stage to the solution of n − 1 stages. Thiscould be done by defining a relation between the solutions of the two adjacent stages.This relation is known as the Recurrence Relation of Dynamic Programming.

Definition. The basic concept of dynamic programming is contained in the principleof optimality proclaimed by Bellman [7] as: “An optimum policy has the propertythat whatever the initial state and initial decision are, the remaining decisions mustconstitute an optimal policy with regards to the state resulting from the first decision”.

The principle of optimality implies that given the initial state of a system, an optimalpolicy for the subsequent stages does not depend upon the policy adopted at thepreceding stages. That is, the effect of a current policy decision on any of the policydecisions of the preceding stages need not be taken into account at all. It is usuallyreferred to as the Markovian property of dynamic programming.

1.8 The Lagrange multiplier technique

The Lagrange multiplier technique can be used to solve an MPP in which all theconstraints have equality sign and the decision variables are assumed to be non-negative. The general form of such an MPP can be stated as follows:

Maximize (or minimize): Z = f(x1, x2, x3, ..., xn)

subject to gi(x1, x2, x3, ..., xn) = bi; i = 1, 2, 3, ...,m(1.28)

To solve (1.28), we associate a multiplier λi with the ith constraint in (1.28). Then,the Lagrangian function is given by

L = f(x1, x2, x3, ..., xn)−m∑i=1

λi [gi(x1, x2, x3, ..., xn)− bi] . (1.29)

Then, we attempt to find a point (x1, x2, x3, ..., xn, λ1, λ2, . . . , λm) that maximizes(or minimizes) L(x1, x2, . . . , xn, λ1, λ2, . . . , λm).

The necessary conditions for the solution to the problem in (1.28) are:

∂L

∂x1

=∂L

∂x2

= . . . =∂L

∂xn=

∂L

∂λ1

=∂L

∂λ2

= . . . =∂L

∂λm= 0 (1.30)

16

Suppose that (1.28) is a maximization or a minimization problem, then the theorembelow gives the conditions for an optimum solution.

Theorem 12. If the objective function f(x1, x2, x3, ..., xn) being maximized is a con-cave function or being minimized is a convex function and each constraintgi(x1, x2, x3, ..., xn) is linear function, then any point (x1, x2, x3, ..., xn, λ1, λ2, . . . , λm)

satisfying (1.30) will yield an optimum solution at (x1, x2, x3, ..., xn) to (1.28).

1.9 Objectives of the study

In order to achieve maximum precision in the survey estimates, the use of stratifiedsampling in sample surveys needs the solution of the following basic problems:

1. The determination of optimum number of strata.

2. The determination of optimum strata boundaries.

3. The determination of optimum sample size from each strata.

Further, at the estimation stage the design weights in stratified random samplingcan be adjusted by obtaining the calibrated weights that improve the precision ofsurvey estimates. Hence, other problems arise, that are:

1. The determination of the optimum calibrated weights, and

2. The computation of nearly unbiased estimators of population parameters usingthe calibrated weights.

The research carried out in this thesis deals with the determination of the optimumstrata boundaries and optimum sample sizes within each strata for skewed popu-lations using a mathematical programming approach. There is extensive researchpertaining to the problem of optimum stratification of study variable y, in the lit-erature, therefore, the problem of determining optimum stratum boundaries usingauxiliary variable, x, is also discussed. The research is also carried on the determi-nation of optimum calibrated weights and then using these weights nearly unbiasedestimators of finite population parameters such as mean (or total) in stratified sam-pling are computed.

17

The problems arising in stratification and calibration are usually nonlinear program-ming problems with nonlinear objective function and linear restriction on decisionvariables. The problems of determining optimum strata boundaries and optimum cal-ibrated weights are formulated as MPP and the dynamic programming or Lagrangemultiplier technique may be used as a tool to solve these MPPs. The research thatare carried out and presented in this thesis are some attempts to effectively developsome techniques to solve the problems indicated above.

The specific objectives of the thesis are as follows:

1. To develop a dynamic programming method to stratify skewed populationsusing the study variable.

2. To extend the dynamic programming method to stratify skewed populationsusing auxiliary variable.

3. To investigate the efficiency of the proposed stratification method.

4. To propose some calibration estimators of population mean in stratified sam-pling.

5. To propose some multivariate calibration estimators of population mean instratified sampling.

6. To investigate the efficiency of the proposed calibration estimators.

18

Chapter 2

Review of Literature and Studies

2.1 Introduction

This chapter presents the literature review and studies on optimum stratification andcalibration approach. In optimum stratification, we will discuss the problem of OSBmathematically and will provide the literature review on some methods availablefor stratification. Later, we discuss the problem of OCW and how the calibratedweights is used to obtain calibration estimators. Finally, the literature review oncalibration approach and some calibrated estimators will be provided. The sectionsof this chapter are organized as follows:

• Section 2.2 provides the literature review on optimum stratification.

• Section 2.3 provides the literature review on calibration approach.

• Section 2.4 summarizes the main concepts discussed in this chapter.

2.2 Optimum stratification: Review of literature and

studies

The construction of strata, on which the current research is conducted, has a longhistory in statistical sciences dating back to 1950. The basic consideration involvedin the formation of strata is that the strata should be as internally homogenous aspossible with respect to the characteristics under study. As discussed earlier, one

19

way strata could be formed is based on the geographical region or demographic vari-ables. The other way which is the focus of this thesis, is by cutting the range of thestudy variable at suitable points to maximize the precision of estimate.

Let the range of the study variable y be a to b, where b−a ≤ ∞, and the populationbe divided into L strata. Then, the (L− 1) intermediate points of dividing the rangeat y1, y2, . . . , yL−1 are referred as stratum boundary points. For this stratification,the following assumptions are made:

1. The study variable y is partitioned into L strata [a, y1], (y1, y2], . . . , (y, b] suchthat a = y0 ≤ y1 ≤ y2 ≤ . . . ≤ yL−1 ≤ yL = b and the population mean

(Y)is

to be estimated.

2. The study variable y has a continuous probability density function (p.d.f.),f(y) for y ∈ [a, b].

3. The first and the second order derivative of the p.d.f. f(y) exists for all y ∈(a, b).

4. The hth stratum weight (Wh), mean (µhy) and the variance (S2hy) respectively

are defined as:

Wh =

∫ yh

yh−1

f(y)dy, (2.1)

µhy =1

Wh

∫ yh

yh−1

yf(y)dy (2.2)

and S2hy =

1

Wh

∫ yh

yh−1

y2f(y)dy − µ2hy. (2.3)

5. Using assumption (3) and equations (2.1)-(2.3), the following results can beobtained:

∂Wh

∂yh= f(yh), (2.4)

∂Wi

∂yh= −f(yh), for i 6= h, (2.5)

∂S2hy

∂yh=

f(yh)

Wh

[(yh − µhy)2 − S2

hy

], (2.6)

and∂S2

iy

∂yh= −f(yh)

Wi

[(yh − µiy)2 − S2

iy

], for i 6= h. (2.7)

20

2.2.1 Stratification using study variable

When a study variable y itself is used as a stratification variable, the problem ofdetermining OSB by minimizing the variance of yst given by (1.2) was first discussedby Dalenius [14]. If the finite population correction (f.p.c.) factor is ignored, thevariance of yst in (1.24) under Neyman [63] allocation in (1.23) reduces to

V (yst) =

(L∑h=1

WhShy

)2

n. (2.8)

However, for a fixed number of strata L and total sample size n, the minimizationof (2.8) is equivalent to minimization of

L∑h=1

WhShy. (2.9)

Since the hth stratum boundary yh appears in (2.9) in the terms WhShy and WiSiy

(where i = h+ 1), the minimization of (2.9) is equivalent to minimization of

WhShy +WiSiy. (2.10)

Hence, minimizing (2.10) using the results (2.4)-(2.7), the strata boundary pointsare obtained are by solving the following minimal equations:

(yh − µhy)2 + S2hy

Shy=

(yh − µiy) 2 + S2iy

Siyfor i = h+ 1 and h = 1, 2, . . . , L− 1.

(2.11)Similarly, the strata boundary points for proportional allocation given in (1.21) isobtained by solving the following minimal equations:

yh =(µhy + µiy)

2for i = h+ 1 and h = 1, 2, . . . , L− 1. (2.12)

However, solving the equations (2.11) and (2.12) is not easy, unless the number ofstrata is small as µhy and S2

hy are dependent on yh. Therefore, researchers had toeither approximate the solutions or use iterative computational algorithms to solvethe minimal equations. Hence, attempts have been made by several authors todetermine optimum or approximately optimum stratum boundaries.

21

Given the number of strata, Dalenius and Gurney [16] suggested that the strataboundaries can be determined when WhShy remain constant. They found that anexplicit solution for OSB was not possible and they arrived at some relations whichthe optimum strata boundary points must satisfy. Starting with a convenientlychosen set, they proceeded towards the optimum set by iterative steps. This methodis very cumbersome for more than two strata (Cochran [11]).

Mahalanobis [57] and Hansen et al. [28] have suggested that the strata boundariescan be determined when Whµhy remain constant, that is, make stratum totals equal.The condition under which it gives optimum stratification is that the coefficientsof variation are equal within strata and remain approximately same on adjustingthe strata sizes. The greatest merit of this rule is its extreme simplicity and it hasbeen claimed that the conditions under which it gives optimum set of stratificationpoints are satisfied by a large number of real populations. When the coefficient ofvariation is constant in all strata, both the rules result in the same solution. Sethi[77] demonstrated that the rule suggested by Hansen et al. [28] does not necessarilylead to optimal points of stratification when applied to normal, gamma and beta dis-tributions. This rule was also tested by Raj [66] for exponential and right triangulardistributions and it was found that the stratification points were not optimum.

Aoyama [2] suggested a simple approximate rule and recommended to make strataof equal width yh − yh−1 where yh−1 and yh are the boundaries of the hth stratum.Cochran [11] reveals that, if this method is applied to highly skewed populations,the strata boundaries are not necessary optimum.

Ekman [20] determined the strata boundaries with the condition that Wh(yh− yh−1)

is constant. Although, this method is simple, there are difficulties to apply in prac-tice because

∑Lh=1 Wh(yh − yh−1) is not constant (Cochran [11]).

Sethi [77] developed an iterative method to solve the minimal equation proposed byDalenius [14] for different allocations and illustrated the method using some normaland chi-square distributions .

A more convenient approximate stratification method was first proposed by Daleniusand Hodges [17], called the Cumulative Root Frequency (CRF) method, which hasbeen the main method of stratification for decades. The CRF method depends crit-

22

ically on the assumption that the distribution of y is approximately uniform in eachstratum. In a comparison of some classical approximate methods, the Ekman [20]method and the CRF method have proved to work consistently well but the latteris more convenient and easier to apply (Cochran [11], Hess et al. [30], Murthy [60]and Nicolini [64]).

In actual practice, the following steps are used to obtain the stratum boundariesusing CRF method:

1. Sort the study variable y in ascending order.

2. Divide y values into a large number equal width classes.

3. Calculate the frequency within each class, say, fh.

4. Calculate the square root of the frequencies in each class,√fh.

5. Cumulative the square root of frequencies,∑L

h=1

√fh.

6. Divide the sum of the square root of the frequencies by the number of strata,

T =1

L

∑Lh=1

√fh.

7. The stratum boundary points are the y values corresponding to T, 2T, . . . , (L−1)T.

The problem with CRF method is that the final strata boundaries depends on theinitial choice of number of classes and there is no theory on how to choose the op-timum number of classes (Hedlin [29]). Also Cochran [11] cautions the use of CRFmethod to have substantial number of classes in the frequency distribution, other-wise, boundary points will not reach optimum.

Gunning and Horgan [27] developed a simple method, known as geometric method,to approximate stratification of skewed populations and they claimed that theirmethod is better that CRF method. The geometric stratification method is basedon the assumption that there is uniform distribution in each strata and the coefficientof variation in each strata are approximately same. Horgan [32] compared this ap-proach with Dalenius and Hodges [17], Ekman [20] and Lavallée and Hidiroglou [55]and confirmed that geometric progression method is more efficient on very highlyskewed distributions. However, Kozak and Verma [51] studied the usefulness of

23

Gunning and Horgan’s geometric method and obtained a different result that thisapproach is less efficient than Lavallée and Hidiroglou (LH) algorithm and suggestedthat the geometric algorithm may be used to obtain efficient initial boundary pointsfor some optimization methods. Er [21] has cautioned researchers to be more rigor-ous when using geometric method even as initial boundaries in optimization methods.

The following steps can be used in practice to obtain geometric stratification:

1. Sort the study variable y in ascending order.

2. Take the minimum value (say a = y0) as the first term and maximum value(yL) as the last term of geometric series with L+ 1 terms.

3. Compute the geometric ratio r, where r =(yLy0

)1/L

.

4. Determine the strata boundary points using the geometric progression that isyh = arh, where h = 1, 2, . . . , L− 1.

Some main problems encountered while implementing the geometric method are thatit does not work well in normal or symmetric distributions, it is inefficient when ex-treme outliers are present and sample sizes within strata are too small (nh < 2) orare greater than the stratum sizes (nh > Nh). Also the geometric method performedpoorly when populations contained very small y values (Kozak and Verma [51], Kozaket al. [52], Keskintürk and Er [35], Brito et al. [10], Baillargeon and Rivest [4] andHorgan [33]).

In practice, the populations to be stratified are finite and hence the optimum strataboundaries could be found by considering all the possible boundaries and selectingthe one which gives the minimum variance for a fixed sample size or the minimumsample size for a fixed variance. However, in this process the number of possiblebreaks increases rapidly given population size and number of strata, and this willtake a lot of time even the availability of super computers, so instead an optimalsolution can be obtained by some optimization methods. Thus, attempts have beenmade by several authors to develop optimization methods.

Unnithan [101] suggested an iterative method using Shannon’s modified Newtonmethod for determining the strata boundaries that leads to a local minimum of thevariance for Neyman allocation, if a suitable initial solution is chosen. Later on,

24

Unnithan and Nair [102] proposed a method of selecting an appropriate startingpoint for Unnithan [101] that may lead to a global minimum of the variance.

The best known iterative algorithm is that of Lavallée and Hidiroglou [55] to con-struct stratum boundaries for a highly skewed population that suggests a take-all topstrata and a number of take-some strata is necessary. The algorithm minimizes thesample size for a given allocation and a level of precision. Hidiroglou and Srinath[31] presented a more general form of the algorithm, which by assigning differentvalues to operating parameters yields a power allocation, a Neyman allocation, or acombination of these allocations. Detlefsen and Veum [18] investigated the LH algo-rithm for several strata and observed that the algorithm’s convergence was slow ornon-existent and there was no guarantee that the algorithm would provide global op-timum. They also found that the algorithm required an initial approximate startingpoint and different starting points lead to different OSB for the same population.

Niemiro [65] proposed a random search method for the stratification problem butthe algorithm did not guarantee a global optimum and goes wrong in a case of alarge population, as it requires too many iteration steps (Kozak [50]).

Lednicki and Wieczorkowski [56] presented a method of stratification based on Rivest[72] using the simplex method of Nelder and Mead [62] but the method was ratherslow and may not provide the best solution in the case of large number of variables.

Later Kozak [50] presented a random search algorithm as a method of the optimalstratification that minimizes the sample size for Neyman allocation under some con-straints. The weakness of Kozak’s algorithm is that it does not guarantee reachingthe global optimum and sometimes the sample size in a strata may exceed the stratasize. A nonrandom version of his algorithm was implemented by Baillargeon andRivest [5] but it is slower than the original.

A genetic algorithm was suggested by Keskintürk and Er [35] to solve the combinedproblem of finding strata boundaries and optimum allocation for finite populations.They compared the performance of their algorithm with CRF, geometric and mod-ified geometric using some real and simulated populations and concluded that thebest results are obtained using the genetic algorithm.

25

Brito et al. [10] proposed an iterative local search (ILS) metaheuristic algorithmto determine strata boundaries and is designed to work for variables with any dis-tribution. They tested their algorithm using sixteen skewed populations (real andsimulated) and concluded that their algorithm mostly works better than Kozak [50].

Brito et al. [9] suggested an exact algorithm based on the concept of minimal pathin a graph, and claims that this algorithm guarantees optimum strata boundaries.They illustrated the use of the algorithm using some real data obtained from theBrazilian Central Statistics Office, and reported the coefficient of variation and theCPU time for the algorithm.

Another optimization method of stratification that has been proposed in the liter-ature is due to Bühler and Deutler [8]. They formulated the problem of determin-ing OSB as an optimization problem and developed a computational technique tosolve the problem using dynamic programming. The technique was later extendedby Lavallée [53] and Lavallée [54] for two-way optimum stratification. Khan et al.[36, 44, 41], and Nand and Khan [61] propose a technique for determining OSB forthe study variables with different frequency functions using dynamic programming.They considered the problem of finding OSB as an equivalent problem of determin-ing OSW, which is formulated as an MPP and solved by the dynamic programmingtechnique. The advantage of this technique is that it does not require an initialsolution and it can be used even when complete data-set of the study variable is un-available. The algorithm only need the frequency distribution and some parametersof the study variable of the population.

2.2.2 Stratification using auxiliary variable

The problem of determining OSB when the study variable is itself the stratifica-tion variable is well known in the sampling literature. However, it is an unrealisticassumption that stratification can be made based on the frequency distribution ofstudy variable (y), which is unknown prior to conducting the survey. Thus, theunavailability of the study variable enforces one to use the auxiliary variable forstratification which is easily available with minimum cost and effort. If the strat-ification is made based on auxiliary variable x, it may lead to substantial gains inprecision of the survey estimates, although it will not be as efficient as the one basedon y.

26

Let y be the study variable, an unbiased estimator of the mean Y is yst =L∑h=1

Whyh.

If the finite population correction (f.p.c.) factor is ignored, the variance of yst underNeyman allocation is given by (2.8) and under proportional allocation it is given by

V (yst) =

L∑h=1

WhS2hy

n. (2.13)

Let the regression of y on x be given by

y = λ(x) + ε (2.14)

where λ(x) is a function of x, ε is an error term such that E (ε|x) = 0 and V (ε|x) =

φ(x) > 0 for all x ∈ (a, b) with (b− a) < ∞. Let f(x) be the marginal densityfunction of x. Then, we have

Wh =

∫ xh

xh−1

f(x)dx, (2.15)

S2hλ =

1

Wh

∫ xh

xh−1

λ2(x)f(x)dx− µ2hλ (2.16)

where µhλ =1

Wh

∫ xh

xh−1

λ(x)f(x)dx. (2.17)

Also we have

S2hy = S2

hλ + µhφ, (2.18)

where µhy = µhλ (2.19)

and µhφ =1

Wh

∫ xh

xh−1

φ(x)f(x)dx. (2.20)

Using these, the variance expression for Neyman and proportional allocation in (2.8)and (2.13) respectively reduces to

V (yst) =

(L∑h=1

Wh

√S2hλ + µhφ

)2

n(2.21)

and

V (yst) =

L∑h=1

Wh (S2hλ + µhφ)

n. (2.22)

27

The minimization of (2.21) is equivalent to minimization of

L∑h=1

Wh

√S2hλ + µhφ (2.23)

and minimization of (2.22) is equivalent to minimization of

L∑h=1

WhS2hλ. (2.24)

Using similar techniques as discussed previously, the minimal equations for Neymanallocation was proposed by Dalenius [15] to be:

[λ (xh)− µhλ]2 + S2hλ + φ (xh) + µhφ√

S2hλ + µhφ

=[λ (xh)− µiλ]2 + S2

iλ + φ (xh) + µiφ√S2iλ + µiφ

for i = h+ 1 and h = 1, 2, . . . , L− 1.

(2.25)The minimal equations for proportional allocation was obtained by Taga [96] as:

λ (xh) =(µhλ + µiλ)

2for i = h+ 1 and h = 1, 2, . . . , L− 1. (2.26)

Since the minimal equations cannot be solved easily, many methods for finding ap-proximate solutions were suggested.

Serfling [76] proposed the use of CRF method of Dalenius and Hodges [17] for op-timum stratification on the auxiliary variable, x when the regression of y on x islinear with uncorrelated homoscedastic errors and nearly perfect correlation. Sincethe rule was proposed to stratify study variable y, it does not take into account theregression of y on x and also the form of conditional variance V (y|x). Cochran [13]concluded that the rule works well, especially when there is strong linear correlationbetween y and x.

Singh and Sukhatme [81] and Singh [79] suggested a cum. 3√f rule which takes into

account the regression of y on x but assumes that the conditional variance V (y|x) > 0

for all x ∈ (a, b). Singh [80] suggested an improvement in his cum. 3√f rule for op-

timum stratification on y when the form of conditional variance V (y|x) = 0 forall x ∈ (a, b). Mehta et al. [58] and Rizvi et al. [73] have also suggested differentapproximation methods of determining optimum strata boundaries using auxiliaryinformation.

28

Sweet and Sigman [95] and Rivest [72] reviewed LH algorithm and proposed theirmodified versions of the algorithm that incorporate the different relationships be-tween the stratification and the study variables.

2.3 Calibration estimation: Review of literature and

studies

In the theory of estimation, statisticians are often interested in deriving estimatorthat increases the precision of survey estimates. The most commonly used estimatorto increase the precision of population mean or population total is the GeneralizedRegression (GREG) estimator. Now, we study a simple case of GREG estimatorwhen the information on a single auxiliary variable is known. The detailed resultsand proofs for the theorems discussed in this section can be found in Singh [83].

Consider a population, Ω = 1, 2, . . . , i, . . . , N, from which a probability samples (s ⊂ Ω) is drawn using a sampling design p(.). The inclusion probabilities πi =

P (i ∈ s) and πij = P (i ∈ s, j ∈ s) are assumed to be strictly positive and known.Let yi and xi be the value of the ith population unit of the study variable, y and theauxiliary variable, x, respectively. The aim of the study is to estimate the populationtotal Y =

∑i∈Ω

yi, using the known population total, X =∑i∈Ω

xi. Deville and Särndal

[19] used calibration to modify the sampling design weights di = πi−1, in the Horvitz

and Thompson [34] estimator

YHT =∑i∈s

yiπi

=∑i∈s

diyi. (2.27)

Theorem 13. The variance of the estimator YHT is given by

V1

(YHT

)=∑i∈Ω

1− πiπi

y2i +

∑i∈Ω

∑j(6=i)∈Ω

πij − πiπjπiπj

yiyj. (2.28)

Theorem 14. The Sen–Yates–Grundy form of the variance of the estimator YHT is

V2(YHT ) =1

2

∑i∈Ω

∑j(6=i)∈Ω

(πiπj − πij)(yiπi−yjπj

)2

. (2.29)

29

Theorem 15. An unbiased estimator of variance of YHT

v1

(YHT

)=∑i∈s

1− πiπ2i

y2i +

∑i∈s

∑j(6=i)∈s

(πij − πiπj

πij

)yiyjπiπj

. (2.30)

Theorem 16. An unbiased estimator of variance of YHT in the Sen–Yates–Grundyform is given by

v2

(YHT

)=

1

2

∑i∈s

∑j(6=i)∈s

(πiπj − πij

πij

)(yiπi−yjπj

)2

. (2.31)

A new estimator was proposed by Deville and Särndal [19] as

YG =∑i∈s

wiyi (2.32)

with the calibrated weights wi, obtained by minimizing a distance function subjectto the calibration constraint

∑i∈s

wixi = X. (2.33)

Theorem 17. Minimization of CS distance between the weights wi and design weightsdi subject to constraint (2.33), leads to a GREG estimator of population total, Y ,given as

YG =∑i∈s

diyi + β

(X −

∑i∈s

dixi

)(2.34)

where

β =∑i∈s

diqixiyi

(∑i∈s

diqix2i

)−1

. (2.35)

Proof. The CS type distance function D is defined as

D =∑i∈s

(wi − di)2(diqi)

−1 (2.36)

where qi are the suitably chosen constant to obtain different forms of the estimator.The Lagrange function L for the problem is defined as

L =∑i∈s

(wi − di)2(diqi)

−1 − 2λ

(∑i∈s

wixi −X

). (2.37)

30

On differentiating (2.37) w.r.t to wi and equating it to zero we have

wi = di + λdiqixi. (2.38)

Substituting (2.38) in (2.33) and solving λ we have

λ =

(∑i∈s

diqix2i

)−1(X −

∑i∈s

dixi

). (2.39)

Substituting (2.39) in (2.38) we get

wi = di + diqixi

(∑i∈s

diqix2i

)−1(X −

∑i∈s

dixi

). (2.40)

Finally, substituting wi from (2.40) in (2.32) leads to GREG estimator of total givenby (2.34).

Remark 1. If qi = 1/xi, then the estimator in (2.34) reduces to a ratio estimator ofpopulation total as

YR =∑i∈s

diyi

(X/∑i∈s

dixi

).

Remark 2. Singh et al. [92] reported that there is no choice of qi such that theestimator (2.34) reduces to the product estimator of population total discussed inCochran [12].

The calibration estimation increases the precision of the estimates further, if theinformation on more than one auxiliary variable is available. The following theoremdiscussed in Singh [83] presents the result when there are two auxiliary variablesavailable.

Theorem 18. Suppose X1 and X2 are the known totals of the two auxiliary variablesx1 and x2. The minimization of the CS distance function D given in (2.36) subjectto the two linear constraints ∑

i∈s

wix1i = X1 (2.41)

and ∑i∈s

wix2i = X2 (2.42)

31

leads to GREG estimator of population total, given by

YG =∑i∈s

diyi + β1

(X1 −

∑i∈s

dix1i

)+ β2

(X2 −

∑i∈s

dix2i

)(2.43)

where

β1 =

∑i∈sdiqix1iyi

∑i∈sdiqix

22i −

∑i∈sdiqix2iyi

∑i∈sdiqix1ix2i

∑i∈sdiqix2

1i

∑i∈sdiqix2

2i −(∑i∈sdiqix1ix2i

)2 (2.44)

and

β2 =

∑i∈sdiqix2iyi

∑i∈sdiqix

22i −

∑i∈sdiqix2iyi

∑i∈sdiqix1ix2i

∑i∈sdiqix2

1i

∑i∈sdiqix2

2i −(∑i∈sdiqix1ix2i

)2 . (2.45)

Proof. The Lagrange function L for this problem is defined as

L =∑i∈s

(wi − di)2(diqi)

−1−2λ1

(∑i∈s

wix1i −X1

)−2λ2

(∑i∈s

wix2i −X2

). (2.46)

On differentiating (2.46) w.r.t. to wi and equating it to zero we have

wi = di + λ1diqix1i + λ2diqix2i. (2.47)

Substituting (2.47) in (2.41)-(2.42) we have the following equations

λ1

∑i∈s

diqix21i + λ2

∑i∈s

diqix1ix2i = X1 −∑i∈s

dix1i, (2.48)

andλ1

∑i∈s

diqix1ix2i + λ2

∑i∈s

diqix22i = X2 −

∑i∈s

dix2i. (2.49)

The system of equations given by (2.48) and (2.49) can be written using matrices as∑diqix

21i

∑i∈sdiqix1ix2i∑

i∈sdiqix1ix2i

∑i∈sdiqix

22i

[λ1

λ2

]=

X1 −∑i∈sdix1i

X2 −∑i∈sdix2i

.

32

Solving the system for λ1 and λ2 we obtain

λ1 =

(X1 −

∑i∈sdix1i

)∑i∈sdiqix

22i +

(X2 −

∑i∈sdix2i

)∑i∈sdiqix1ix2i

∑i∈sdiqix2

1i

∑i∈sdiqix2

2i −(∑i∈sdiqix1ix2i

)2 ,

and

λ2 =

(X2 −

∑i∈sdix2i

)∑i∈sdiqix

21i −

(X1 −

∑i∈sdix1i

)∑i∈sdiqix1ix2i

∑i∈sdiqix2

1i

∑i∈sdiqix2

2i −(∑i∈sdiqix1ix2i

)2 .

Substituting the values of λ1 and λ2 in (2.47) we have

wi = di +

diqix1i

[(X1 −

∑i∈sdix1i

)∑i∈sdiqix

22i −

(X2 −

∑i∈sdix2i

)∑i∈sdiqix1ix2i

]∑i∈sdiqix2

1i

∑i∈sdiqix2

2i −(∑i∈sdiqix1ix2i

)2

+

diqix2i

[(X2 −

∑i∈sdix2i

)∑i∈sdiqix

21i −

(X1 −

∑i∈sdix1i

)∑i∈sdiqix1ix2i

]∑i∈sdiqix2

1i

∑i∈sdiqix2

2i −(∑i∈sdiqix1ix2i

)2 .

(2.50)Finally, substitution wi in (2.32) leads to

YG =∑i∈s

diyi + β1

(X1 −

∑i∈s

dix1i

)+ β2

(X2 −

∑i∈s

dix2i

)

where β1 and β2 are defined in (2.44) and (2.45).

Remark 3. There is no choice of qi such that the estimator in (2.43) reduces to amultivariate ratio type estimator with two auxiliary variables.

If there are p auxiliary variables xj, j = 1, 2, . . . , p and the population totals Xj =∑i∈Ω

xji are known, the problem of minimizing the distance function (2.36) subject to

the p calibration constraints:∑i∈s

wixji = Xj, for j = 1, 2, . . . , p (2.51)

is discussed in Singh [83] as follows:

33

Minimizing the Lagrange function, the following system is obtained:

∑i∈sdiqix

21i

∑i∈sdiqix1ix2i . . .

∑i∈sdiqix1ixpi∑

i∈sdiqix2ix1i

∑i∈sdiqix

22i . . .

∑i∈sdiqix2ixpi

...... . . . ...∑

i∈sdiqixpix1i

∑i∈sdiqixpix2i . . .

∑i∈sdiqix

2pi

λ1

λ2

...λp

=

X1 −∑i∈sdix1i

X2 −∑i∈sdix2i

...Xp −

∑i∈sdixpi

.

(2.52)The system (2.52) can be expressed as

Aλ = C

and its solution is given byλ = A−1C.

Hence, the optimum calibrated weights in case of p auxiliary variables is given by

wi = di + diqiλTxi

wherexi = (xji)p×1.

The estimator of the population total using p auxiliary variables is given by

YG =∑i∈s

diyi +∑i∈s

diqiyiλTxi. (2.53)

Theorem 19. The variance of the estimator YG defined in (2.53) is

V1(YG) = V (YHT )(

1−R2y|x1,x2,...,xp

)(2.54)

where R2y|x1,x2,...,xp denotes the multiple correlation coefficient between y and

x1, x2, . . . , xp.

Theorem 20. The Sen–Yates–Grundy form variance of the estimator YG is

V2(YG) =1

2

∑i∈Ω

∑j(6=i)∈Ω

(πiπj − πij) (diei − djej)2 (2.55)

where e = y − βx denotes the vector of error terms.

34

Remark 4. The major difficulty with the calibration approach discussed above is thatit may produce negative or extreme calibrated weights for some unlucky samples.These negative or extreme calibrated weights can lead to decrease in the precisionof the survey estimates. To avoid such unrealistic calibrated weights, the problem,when single auxiliary variable is available, can be stated as

Minimize :∑i∈s

(wi − di)2(diqi)−1

Subject to∑i∈swixi = X

and li 6 wi 6 ui for i ∈ s

(2.56)

where li and ui are the lower and upper limits for the calibrated weights. The prob-lem in (2.56) can be solved using quadratic programming.

The notion of calibration estimators was first introduced by Deville and Särndal [19]in survey sampling and since then several survey statisticians such as Singh et al.[92, 90, 91], Singh and Arnab [88, 89], Singh and Sedory [93], Singh [82, 83, 84, 85,86, 87], Farrell and Singh [26, 25], Wu and Sitter [105], Särndal [75], Estevao andSärndal [22, 23, 24], Kott [49, 48], Montanari and Ranalli [59], Rueda et al. [74], Kimet al. [47], Kim [45, 46], Théberge [98, 99], Ranalli [67], Singh and Mohl [78], Tracyet al. [100] and many others have contributed to the study of calibration estimation.

However, as this thesis deals with calibration approach in stratified random samplingwe will discuss some calibration estimators in stratified sampling, especially theestimators developed by Singh et al. [92], Tracy et al. [100] and Singh [83] in thefollowing sections.

2.3.1 Singh, Horn and Yu estimator

Singh et al. [92] introduced the calibration estimation in stratified random sampling.They suggested the combined Generalized Regression (GREG) estimator of popula-tion mean

(Y)using a single auxiliary variable x.

The stratified estimator of the Y is given by

yst =L∑h=1

Whyh. (2.57)

35

A new estimator of population mean proposed by Singh et al. [92] is given by

y⊕st =L∑h=1

W⊕h yh, (2.58)

with weights W⊕h called the calibrated weights which are chosen such that the CS

distance functionL∑h=1

(W⊕h −Wh

)2

Whqh(2.59)

is minimum subject to the calibration constraint

L∑h=1

W⊕h xh = X (2.60)

where the choice of weights qh produces different forms of estimators.

The Lagrange function is given by

L =L∑h=1

(W⊕h −Wh

)2

Whqh− 2λ

(L∑h=1

W⊕h xh − X

). (2.61)

On setting ∂L/∂W⊕h = 0,

W⊕h = Wh + λWhqhxh. (2.62)

Using (2.62) in (2.60)

λ =

(L∑h=1

Whqhx2h

)−1 (X − xst

), (2.63)

where xst =L∑h=1

Whxh. On substituting (2.63) in (2.62)

W⊕h = Wh +Whqhxh

(L∑h=1

Whqhx2h

)−1 (X − xst

). (2.64)

Substituting the calibrated weights in (2.58)

y⊕st = yst + β⊕(X − xst

), (2.65)

36

where

β⊕ =L∑h=1

Whqhxhyh

(L∑h=1

Whqhx2h

)−1

(2.66)

which leads to Singh et al. [92] Generalized Regression (GREG) estimator in stratifiedsampling.

Remark 5. There is no choice of qh such that the estimator in (2.65) reduces to thetraditional combined LREG estimator given by (1.10).

Remark 6. If qh = 1/xh then the estimator in (2.65) reduces to the traditionalcombined ratio estimator in (1.4) with estimated variance of

v (ycr) =

(X

xst

)2 L∑h=1

W 2h (1− fh)nh

s2eh, (2.67)

which is a special case of (1.9) for g = 2.

Theorem 21. An estimator of the variance of y⊕st is given by

v(y⊕st)

=L∑h=1

W 2h (1− fh)nh

s2eh, (2.68)

where s2eh = (nh − 1)−1∑nh

i=1 e2hi and ehi = (yhi − yh)− β⊕(xhi − xh).

Theorem 22. The lower level calibration approach yields another estimator of vari-ance of y⊕st as

vLL(y⊕st)

=L∑h=1

(W⊕h

)2(1− fh)nh

s2eh. (2.69)

Theorem 23. A higher level calibration approach leads to a new GREG estimatorof the variance of y⊕st as

vHL(y⊕st)

= vLL(y⊕st)

+ βHL [V (xst)− v (xst)] , (2.70)

where

βHL =

L∑h=1

(W⊗h

)2 1− fhnh

Qhs2hxs

2eh

L∑h=1

W 2h

1− fhnh

QhS4hx

and

v (xst) =L∑h=1

W 2h

1− fhnh

s2hx.

37

2.3.2 Tracy, Singh and Arnab estimator

Tracy et al. [100] suggested another GREG estimator of population mean(Y)as

y⊗st =L∑h=1

W⊗h yh, (2.71)

with the calibrated weightsW⊗h which are chosen such that the CS distance function

L∑h=1

(W⊗h −Wh

)2

Whqh(2.72)

is minimum subject to the calibration constraints

L∑h=1

W⊗h xh =

L∑h=1

WhXh (2.73)

andL∑h=1

W⊗h s

2hx =

L∑h=1

WhS2hx. (2.74)

It was shown that minimization of (2.72) subject to (2.73) and (2.74) leads to newcalibrated weights defined by

W⊗h = Wh

+

Whqhxh

[L∑h=1

Wh

(Xh − xh

) L∑h=1

Whqhs4hx −

L∑h=1

Wh (S2hx − s2

hx)L∑h=1

Whqhs2hx

]L∑h=1

Whqhx2h

L∑h=1

Whqhs4hx −

(L∑h=1

Whqhs2hx

)2

+

Whqhs2hx

[L∑h=1

Wh (S2hx − s2

hx)L∑h=1

Whqhx2h −

L∑h=1

Wh

(Xh − xh

) L∑h=1

Whqhxhs2hx

]L∑h=1

Whqhx2h

L∑h=1

Whqhs4hx −

(L∑h=1

Whqhs2hx

)2 .

(2.75)On substituting (2.75) in (2.71),

y⊗st = yst + β⊗1

L∑h=1

Wh

(Xh − xh

)+ β⊗2

L∑h=1

Wh

(S2hx − s2

hx

), (2.76)

38

where

β⊗1 =

L∑h=1

Whqhxhyh

[L∑h=1

Wh

(Xh − xh

) L∑h=1

Whqhs4hx −

L∑h=1

Wh (S2hx − s2

hx)L∑h=1

Whqhxhs2hx

]L∑h=1

Whqhx2h

L∑h=1

Whqhs4hx −

(L∑h=1

Whqhs2hx

)2

(2.77)and

β⊗2 =

L∑h=1

Whqhs2hxyh

[L∑h=1

Wh (S2hx − s2

hx)L∑h=1

Whqhx2h −

L∑h=1

Wh

(Xh − xh

) L∑h=1

Whqhxhs2hx

]L∑h=1

Whqhx2h

L∑h=1

Whqhs4hx −

(L∑h=1

Whqhs2hx

)2 .

(2.78)

Remark 7. It is claimed of Tracy et al. [100] that the estimator in (2.76) performsbetter than the estimators developed by Singh et al. [92] in stratified sampling.

2.3.3 Singh estimator

Singh [83] proposed another calibration estimator of population mean(Y)in strat-

ified sampling by adding a constraint on calibrated weights to the problem of Singhet al. [92] estimator. Then, the problem of determining the calibrated weights reducesto minimizing the CS distance measure subject to calibration constraints

L∑h=1

Wh =

L∑h=1

Wh (2.79)

andL∑h=1

Wh xh = X. (2.80)

The Lagrange function is defined as

L =L∑h=1

(Wh −Wh

)2

Whqh− 2λ1

(L∑h=1

Wh − 1

)− 2λ2

(L∑h=1

Wh xh − X

). (2.81)

Again on setting ∂L/∂Wh = 0,

Wh = Wh + λ1Whqh + λ2Whqhxh. (2.82)

39

The Lagrange multipliers λ1 and λ2 were obtained as

λ1 =

−(

L∑h=1

Whqhxh

)(X − xst

)L∑h=1

WhqhL∑h=1

Whqhx2h −

(L∑h=1

Whqhxh

)2 (2.83)

and

λ2 =

(L∑h=1

Whqh

)(X − xst

)L∑h=1

WhqhL∑h=1

Whqhx2h −

(L∑h=1

Whqhxh

)2 . (2.84)

‌Substituting (2.83) and (2.84) in (2.82), the calibrated weights W

h is given as

Wh = Wh +

WhqhxhL∑h=1

Whqh −WhqhL∑h=1

Whqhxh

∑Lh=1Whqh

L∑h=1

Whqhx2h −

(L∑h=1

Whqhxh

)2

(X − xst

)(2.85)

and the estimator in GREG form is

yst = yst + β(X − xst

), (2.86)

where

β =

∑Lh=1Whqhxhyh

∑Lh=1Whqh −

∑Lh=1Whqhyh

∑Lh=1Whqhxh∑L

h=1 Whqh∑L

h=1Whqhx2h −

(∑Lh=1 Whqhxh

)2 . (2.87)

Remark 8. According to Singh [83], if qh = 1 then the estimator in (2.86) reducesto the traditional combined LREG estimator given by (1.10) and hence it performsbetter than the estimators developed by Singh et al. [92] and Tracy et al. [100].

Theorem 24. An estimator of the variance of yst is given by

v(yst)

=L∑h=1

W 2h (1− fh)nh

s2eh, (2.88)

where s2eh = (nh − 1)−1∑nh

i=1 e2hi and ehi = (yhi − yh)− β(xhi − xh).

Theorem 25. The lower level calibration approach yields another estimator of vari-

40

ance of yst as

vLL(yst)

=L∑h=1

(Wh

)2(1− fh)nh

s2eh. (2.89)

2.4 Summary

This chapter provided detailed review of literature and studies on the methods ofoptimum stratification and calibration approach.

Some common methods discussed to stratify a population when the study variableis also the stratification variable are CRF method of Dalenius and Hodges [17], geo-metric method of Gunning and Horgan [27], a random search method of Kozak [50],a genetic algorithm method of Keskintürk and Er [35], an ILS algorithm of Britoet al. [10] and dynamic programming method of Khan et al. [36]. We have also dis-cussed in detail some methods proposed by Serfling [76], Singh and Sukhatme [81],Singh [79], Rivest [72] and Sweet and Sigman [95] to stratify a population using theauxiliary variable as the stratification variable.

A detailed literature review on the calibration approach was also outlined. In par-ticular, we discussed the calibration estimators proposed in Deville and Särndal [19]and Singh [83]. Since this thesis focuses on stratified sampling, literature reviewof calibration estimators in stratified sampling proposed by Singh et al. [92], Tracyet al. [100] and Singh [83] were also discussed in detail.

41

Chapter 3

Optimum Stratification using StudyVariable

3.1 Introduction

When a study variable y itself is used as a stratification variable, the problem of de-termining OSB was first discussed by Dalenius [14]. As discussed in Section 2.2.1, hepresented a set of minimal equations whose solution could provide the OSB. Unfortu-nately, the exact solution of these equations could not usually be obtained because oftheir implicit nature. Several attempts have been made by many authors for deter-mining OSB and the details of these methods can be found in Chapter 2 of this thesis.

In this chapter, the problem of finding the OSB and the optimum sample sizes withinthe stratum for a skewed population with Log-normal distribution is studied. Theproblem of determining the OSB is redefined as the problem of determining OSWand is formulated as a MPP that seeks minimization of the variance of the estimatedpopulation mean under Neyman allocation subject to the constraint that the sumof the widths of all the strata is equal to the range of the distribution. Section 3.2provides the general formulation of the problem of finding OSW as an MPP andthe solution procedure to solve the MPP is then discussed in Section 3.3. Section3.4 illustrates how to determine the OSB for a Log-normal study variable with anumerical example. In Section 3.5, a comparison study is carried out to investigatethe effectiveness of the proposed method with Dalenius and Hodges [17] CRF method,Geometric method by Gunning and Horgan [27], Lavallée and Hidiroglou [55] method

42

using Kozak’s algorithms that are available in the literature as in Baillargeon andRivest [4]. Finally Section 3.6 provides the summary of the chapter.

3.1.1 Publications

The results presented in this chapter have been published in Khan et al. [42], Khanet al. [43] and Rao et al. [69].

3.2 The general formulation of the problem

Let y be a study variable with probability density function f(y), a ≤ y ≤ b. Toestimate the population mean µy by a stratified sample, the range of y is partitionedinto L strata [a, y1], (y1, y2], . . . , (yL−1, b] such that

a = y0 ≤ y1 ≤ y2 ≤, ...,≤ yL−1 ≤ yL = b. (3.1)

Suppose that from stratum h (h = 1, 2, . . . , L), which contains Nh units, a simplerandom sample of size nh is selected. Let yhi denote the value of the ith unit in thehth stratum. Then the stratified sample mean yst =

∑Lh=1Whyh will be an unbiased

estimator of µy with variance

V (yst) =L∑h=1

(1

nh− 1

Nh

)W 2hS

2hy, (3.2)

where Wh = Nh/N , yh = 1nh

∑nh

i=1 yhi, S2hy = 1

Nh−1

∑Nh

i=1 (yhi − µhy)2 and µhy =1Nh

∑Nh

i=1 yhi.

When the frequency function f(y) is known, the values of Wh and S2hy in (3.2)

can be obtained by

Wh =

∫ yh

yh−1

f(y)dy, (3.3)

S2hy =

1

Wh

∫ yh

yh−1

y2f(y)dy − µ2hy, (3.4)

where µhy =1

Wh

∫ yh

yh−1

yf(y)dy (3.5)

43

is the mean and (yh−1, yh) are the boundaries of hth stratum.

Using the above values of Wh, µhy and S2hy, (3.2) can be expressed as a function of

yh and nh, that is,

V (yst) = V (yst|y1, . . . , yL−1, n1, . . . , nL).

Further, if population mean is estimated with a fixed total sample size:

n =L∑h=1

nh,

then, under Neyman allocation, nh; (h = 1, 2, ..., L) are given by:

nh = n · WhShy∑Lh=1WhShy

. (3.6)

If nh; (h = 1, 2, ..., L) are fixed under Neyman allocation, the objective of the op-timum stratification is to determine the stratum boundary points y1, . . . , yL−1 suchthat V (yst) is minimum subject to the restrictions that

2 ≤ nh ≤ Nh. (3.7)

The restrictions nh ≤ Nh are imposed to avoid the over sampling, which may bethe case, especially, when the population is highly skewed. Whereas, the restrictionsnh ≥ 2 are imposed, when the stratum variances S2

hy are needed to be estimated.

From (3.2), it can be seen that the second term does not have any influence on thesample size as it is independent of nh. Thus, omitting the term and substituting(3.6), the variance V (yst) in (3.2) is reduced to:

V (yst).=

(∑Lh=1WhShy

)2

n. (3.8)

However, for a fixed total sample size n, the minimization of (3.8) is equivalent tominimizing (Khan et al. [44]):

L∑h=1

WhShy. (3.9)

Thus, the problem of determining OSB and the optimum sample size may be stated

44

as:

Minimize

L∑h=1

WhShy|a = y0 ≤ y1 ≤ y2 ≤, ...,≤ yL−1 ≤ yL = b; 2 ≤ nh ≤ Nh

.

(3.10)Further, from (3.6), (3.7) and (3.9), it can be seen that the restrictions nh ≤ Nh aresatisfied, if the following condition holds:

Shy ≤∑L

h=1WhShyn/N

.

Similarly, the restrictions 2 ≤ nh are satisfied, if

WhShy ≥2∑L

h=1WhShyn

.

Let f(y) be the frequency function and y0 and yL the smallest and largest values ofy. If the population mean is estimated under (3.6), then the problem of determiningthe strata boundaries is equivalent to cut up the range,

d = yL − y0, (3.11)

at intermediate points y1 ≤ y2 ≤, ...,≤ yL−1 such that∑L

h=1 WhShy in (3.10) is min-imum.

If f(y) is integrable, using the expressions (3.3), (3.4) and (3.5), Wh, S2hy and µhy

are obtained as a function of the boundary points yh and yh−1. Thus, the objectivefunction in (3.10) could be expressed as a function of boundary points (yh−1, yh) thatis

φh(yh−1, yh) = WhShy.

The (3.10) can be treated as an optimization problem to find y1, y2, ..., yL−1 to:

MinimizeL∑h=1

φh(yh−1, yh),

subject to a = y0 ≤ y1 ≤ y2 ≤, ...,≤ yL−1 ≤ yL = b. (3.12)

Bühler and Deutler [8] have suggested a recursive optimization method for solving(3.12) using a dynamic programming technique (Khan et al. [41]).

Khan et al. [36, 44, 41], and Nand and Khan [61] treated the problem (3.12) as an

45

equivalent problem of determining OSW as follows:

Let lh = yh − yh−1 ≥ 0 be the width of the hth (h = 1, 2, ..., L) stratum. Using thedefinition of lh, the range of the distribution given in (3.11) may be expressed as thefunction of the stratum widths as:

L∑h=1

lh =L∑h=1

(yh − yh−1) = yL − y0 = d. (3.13)

The hth stratification point yh; (h = 1, 2, ..., L− 1) is then expressed as:

yh = y0 + l1 + l2 + ...+ lh

= yh−1 + lh,

which is a function of stratum width (lh) and lower boundary (yh−1) of hth stratum.

Adding (3.13) as a constraint, the problem (3.12) can be treated as an equivalentproblem of determining OSW as:

MinimizeL∑h=1

φh(lh, yh−1),

subject toL∑h=1

lh = d,

and lh ≥ 0;h = 1, 2, ..., L.

(3.14)

Initially, y0 is known. Therefore, the first term, that is, φ1(l1, y0) in the objectivefunction of MPP (3.12) is a function of l1 alone. Once l1 is known, the next stratifi-cation point y1 = y0 + l1 will be known and the second term in the objective functionφ2(l2, y1) will become a function of l2 alone. Thus, stating the objective function asa function of lh alone, we may rewrite the MPP (3.14) as:

Minimize∑L

h=1 φh(lh),

subject toL∑h=1

lh = d,

and lh ≥ 0;h = 1, 2, ..., L.

(3.15)

46

3.3 The solution procedure using dynamic program-

ming technique

The MPP (3.15) is a multistage decision problem in which the objective functionand the constraints are separable functions of lh, which allow us to use a dynamicprogramming technique. A solution procedure using such a dynamic programmingtechnique is discussed in Khan et al. [41], which is summarized below.

Consider a sub-problem of (3.15) of first k(< L) strata, that is:

Minimize∑k

h=1 φh(lh),

subject tok∑

h=1

lh = dk,

and lh ≥ 0;h = 1, 2, ..., k

(3.16)

where dk < d is the total width available for division into k strata or the state valueat stage k. Note that dk = d for k = L.

Also notedk = l1 + l2 + ...+ lk,

dk−1 = l1 + l2 + ...+ lk−1 = dk − lk,dk−2 = l1 + l2 + ...+ lk−1 = dk−1 − lk−1,

......

d2 = l1 + l2 = d3 − l3,d1 = l1 = d2 − l2.

Let Φk(dk) denote the minimum value of the objective function of (3.16), that is,

Φk(dk) = min

[k∑

h=1

φh(lh)

∣∣∣∣ k∑h=1

lh = dk, and lh ≥ 0; h = 1, 2, ..., k

].

Using the above definition of Φk(dk), the MPP (3.15) is equivalent to finding ΦL(d)

recursively by finding Φk(dk) for k = 1, 2, ..., L and 0 ≤ dk ≤ d. We can write:

Φk(dk) = min

[φk(lk) +

k−1∑h=1

φh(lh)

∣∣∣∣ k−1∑h=1

lh = dk − lk, and lh ≥ 0; h = 1, 2, ..., k − 1

].

For a fixed value of lk; 0 ≤ lk ≤ dk,

47

Φk(dk) = φk(lk)+min

[k−1∑h=1

φh(lh)

∣∣∣∣ k−1∑h=1

lh = dk − lk, and lh ≥ 0; and h = 1, 2, ..., k − 1

].

Using the Bellman [7] principle of optimality, we get the recursive relation of dynamicprogramming technique as:

Φk(dk) =min

0 ≤ lk ≤ dk[φk(lk) + Φk−1(dk − lk)] , k ≥ 2. (3.17)

For the first stage, that is, for k = 1:

Φ1(d1) = φ1(d1) =⇒ l∗1 = d1, (3.18)

where l∗1 = d1 is the optimum width of the first stratum. The relations (3.17) and(3.18) are solved recursively for each k = 1, 2, ..., L and 0 ≤ dk ≤ d, and ΦL(d) isobtained. From ΦL(d) the optimum SRSWOR of Lth stratum, l∗L, is obtained. FromΦL−1(d− l∗L) the optimum SRSWOR of (L− 1)th stratum, l∗L−1, is obtained and soon until l∗1 is obtained. The details of the solution procedure can be seen in Khanet al. [41].

The algorithm for the solution procedure to solve the MPP (3.15) is summarized asfollows:

Step 1: Start at k = 1 and set Φ0(d0) = 0.

Step 2: Calculate Φ1(d1), the minimum value of right hand side of (3.18) forl1 = d1, 0 ≤ l1 ≤ d1, and 0 ≤ d1 ≤ d.

Step 3: Record Φ1(d1) and the value of l1.

Step 4: For k ≥ 2,3,..., L− 1, express the state variable as dk−1 = dk − lk.

Step 5: Set Φk(dk) = 0 if lk > dk and Φk(dk) =∞ if Sky >ΦL(d)

n/Nor WkSky <

2ΦL(d)

n.

Step 6: Calculate Φk(dk), the minimum value of right hand side of (3.17) forlk; 0 ≤ lk ≤ dk.

Step 7: Record Φk(dk) and lk.

48

Step 8: At k = L, ΦL(d) is obtained and hence the optimum value l∗L of lL isobtained.

Step 9: At k = L − 1, using the backward calculation for dL−1 = d − l∗L, readthe value of ΦL−1(dL−1) and the optimum value l∗L−1 of lL−1.

Step 10: Repeat Step 9 until the optimum value l∗1 of l1 is obtained from Φ1(d1).

3.4 Determining OSB for skewed population with

Log-normal study variable

3.4.1 The Log-normal distribution

The Log-normal distribution is a positively skewed distribution, meaning that mostof the distribution is concentrated around the left end. Surveyors may use the Log-normal distribution for a positive valued study variable that might increase withoutlimit, such as the value of securities in financial problem or the value of propertiesin real estate or the failure rate of electronic parts in engineering problem.

The random variable y is Log-normally distributed if ln(y) is normally distributedwhere "ln" stands for the natural logarithm. The general formula for the probabilitydensity function of the Log-normal distribution is

f(y) =exp

[−(((ln (y)− µ) /m))2 / (2σ2)

)]yσ√

2π; y > 0, µ ∈ R, m > 0, σ > 0,

(3.19)where σ is the shape parameter, µ is the location parameter and m is the scale pa-rameter.

With m = 1, (3.19) gives the Log-normal density as

f(y) =exp

[−((ln(y)− µ)2 / (2σ2)

)]yσ√

2π; y > 0, µ ∈ R, σ > 0. (3.20)

The Figure 1 shows the plot of Log-normal densities with µ = 0 and σ = 0.25, 0.5, 1.

49

Figure 1: Log-normal densities.

3.4.2 Formulation of the problem of OSW as an MPP

When the study variable y has a Log-normal frequency distribution, the formulationof the problem of determining OSW may be expressed as an MPP as discussed inSection 3.2. Using the definitions (3.3), (3.4), (3.5) and (3.20), the terms Wh, µhyand S2

hy can be expressed as

50

Wh =1

2

[erf(

ln (lh + yh−1)− µσ√

2

)− erf

(ln (yh−1)− µ

σ√

2

)], (3.21)

µhy = exp

(σ2

2+ µ

) erf(

ln (lh + yh−1)− µ− σ2

σ√

2

)− erf

(ln (yh−1)− µ− σ2

σ√

2

)erf(

ln (lh + yh−1)− µσ√

2

)− erf

(ln (yh−1)− µ

σ√

2

) , (3.22)

S2hy =

1[erf(

ln (lh + yh−1)− µσ√

2

)− erf

(ln (yh−1)− µ

σ√

2

)]2 ×

[exp

(2σ2 + 2µ

)(erf(

ln (lh + yh−1)− 2σ2 − µσ√

2

)− erf

(ln (yh−1)− 2σ2 − µ

σ√

2

))][erf(

ln (lh + yh−1)− µσ√

2

)− erf

(ln (yh−1)− µ

σ√

2

)]−[exp

(σ2

2+ µ

)(erf(

ln (lh + yh−1)− σ2 − µσ√

2

)− erf

(ln (yh−1)− σ2 − µ

σ√

2

))]2. (3.23)

Note that an error function (erf) is used to counter the integrations with Log-normaldensity function. The error function is defined as

erf(z) =2√π

∫ z

0

e−t2

dt. (3.24)

The probability that a Log-normal variate assumes a value in the range [z1, z2] isgiven by:

1√2π

∫ z2

z1

e−t2/2 dt =

1

2[erf(z2)− erf(z1)] .

Common properties of the erfs include:erf(−z) = −erf(z), erf(0) = 0, erf(∞) = 1, erf(−∞) = −1.

Using (3.21), (3.23) and (3.24) the MPP (3.15) may be expressed as:

51

MinimizeL∑h=1

1

2×

√√√√√√√√√√√√√√√√√√√√√√√√

exp

(2σ2 + 2µ

) [erf(

ln (lh + yh−1)− 2σ2 − µσ√

2

)−erf

(ln (yh−1)− 2σ2 − µ

σ√

2

)]×[

erf(

ln (lh + yh−1)− µσ√

2

)− erf

(ln (yh−1)− µ

σ√

2

)]

−

exp

(σ2

2+ µ

)[erf(

ln (lh + yh−1)− σ2 − µσ√

2

)−erf

(ln (yh−1)− σ2 − µ

σ√

2

)]2

subject to

L∑h=1

lh = d,

and lh ≥ 0; h = 1, 2, ..., L.

(3.25)Treating (3.25) as a multistage decision problem, the MPP may be solved for de-termining the OSW using the dynamic programming technique. At each stage thevalue of the OSW and hence the OSB for a stratum as well as its optimum samplesize is worked out with a forward recursive equation as discussed in Section 3.3.

Note that upon determining the optimum boundary points (yh−1, yh) of the hthstratum, the problem of determining its optimum sample size, nh, can be solved byusing (3.3)-(3.6).

3.4.3 Numerical illustration

In this section, a numerical example is presented to illustrate the computational de-tails of the solution procedure for MPP (3.25). Assume that y follows the standardLog-normal distribution in the interval [0.00001, 13.00001], that is, a = y0 = 0.00001,b = yL = 13.00001, µ = 0 and σ = 1. This implies that d = yL − y0 = 13. Then theMPP (3.25) is expressed as:

52

MinimizeL∑h=1

1

2×

√√√√√√√√√√√√√√√√√√√√√√√√

exp (2)

[erf(

ln (lh + yh−1)− 2√2

)−erf

(ln (yh−1)− 2√

2

)]×[

erf(

ln (lh + yh−1)√2

)− erf

(ln (yh−1)√

2

)]

−

exp

(1

2

)[erf(

ln (lh + yh−1)− 1√2

)−erf

(ln (yh−1)− 1√

2

)]2

Subject to

L∑h=1

lh = 13,

and lh ≥ 0; h = 1, 2, ..., L.

(3.26)

Also

yk−1 = y0 + l1 + l2 + ...+ lk−1

= 0.00001 + l1 + l2 + ...+ lk−1

= dk−1 + 0.00001

= dk − lk + 0.00001.

Substituting this value of yk−1 in (3.26) and using (3.18) and (3.17), the recurrencerelations for solving MPP (3.26) are obtained as:

For first stage (k = 1):

Φ1(d1) =1

2×

√√√√√√√√√√√√√√√√√√√√√√√√

exp (2)

[erf(

ln (d1 + 0.00001)− 2√2

)−erf

(ln (0.00001)− 2√

2

)]×[

erf(

ln (d1 + 0.00001)√2

)− erf

(ln (0.00001)√

2

)]

−

exp

(1

2

)[erf(

ln (d1 + 0.00001)− 1√2

)−erf

(ln (0.00001)− 1√

2

)]2

(3.27)

53

at l1 = d1,

and for the stages k ≥ 2:

Φk(dk) =min

0 ≤ lk ≤ dk

1

2×

√√√√√√√√√√√√√√√√√√√√√√√√√√√√√

exp (2)

[erf(

ln (dk + 0.00001)− 2√2

)−erf

(ln (dk − lk + 0.00001)− 2√

2

)]

×

[erf(

ln (dk + 0.00001)√2

)−erf

(ln (dk − lk + 0.00001)√

2

)]

−

exp

(1

2

)[erf(

ln (dk + 0.00001)− 1√2

)−erf

(ln (dk − lk + 0.00001)− 1√

2

)]2

+Φk−1(dk − lk)

.

(3.28)Solving the recursive equations (3.27) and (3.28) by executing a C++ computer pro-gram (refer to Appendix B.1) developed for the solution procedure described inSection 3.3, the OSW are obtained. The results of optimum strata widths l∗h andhence the optimum strata boundaries y∗h = y∗h−1 + l∗h along with the optimum values

of the objective functionL∑h=1

WhShy for L = 2, 3, 4, 5 and 6 are presented in Table

2. The table also presents the OSS (nh; h = 1, 2, ..., L) using (3.3)-(3.6) for a fixedtotal sample size n = 100.

54

Table 2: OSW, OSB, OSS and the optimum value of objective function for Log-normal study variable.No. of strata OSW OSB OSS Optimum value of

L (l∗h) (y∗h = y∗h−1 + l∗h) nh = n · WhShy∑Lh=1WhShy

L∑h=1

WhShy

2 l∗1 = 2.23652 y∗1= 2.23653 51 0.8569355124l∗2 = 10.76348 49l∗1 = 1.30859 y∗1= 1.30860 35

3 l∗2 = 2.35085 y∗2= 3.65945 32 0.5773613579l∗3 = 9.34056 33l∗1 = 0.95459 y∗1= 0.95460 26

4 l∗2 = 1.25278 y∗2= 2.20738 25 0.4358095763l∗3 = 2.53417 y∗3= 4.74155 24l∗4 = 8.25846 25l∗1 = 0.76589 y∗1= 0.76590 20l∗2 = 0.84332 y∗2= 1.60922 20

5 l∗3 = 1.36367 y∗3= 2.97289 20 0.3501356776l∗4 = 2.62141 y∗4= 5.59430 20l∗5 = 7.40571 20l∗1 = 0.64767 y∗1= 0.64768 17l∗2 = 0.63431 y∗2= 1.28199 17

6 l∗3 = 0.90957 y∗3= 2.19156 16 0.2926636591l∗4 = 1.44256 y∗4= 3.63412 16l∗5 = 2.65047 y∗5= 6.28459 16l∗6 = 6.71542 18

55

3.5 Comparison study

In this section, a comparison study is carried out to compare and investigate theeffectiveness of the proposed dynamic programming method with the other meth-ods available in the literature. The study is undertaken to compare the followingmethods:

1. Dalenius and Hodges [17] CRF method.

2. Geometric method by Gunning and Horgan [27].

3. Generalized Lavallée and Hidiroglou [55] (LH) method using Kozak [50] algo-rithm.

For the purpose of comparison, ten artificial skewed populations that follow Log-normal distribution were randomly generated using the R statistical software forvarious combinations of parameters, such as the shape parameter (σ) that variesfrom 0.2 to 1.2, skewness that varies from 0.6 to 6.6 and population size (N) thatvaries from 1000 to 15000, etc.

For these populations the OSB are determined by using the proposed dynamic pro-gramming method as discussed in previous sections. For each population the strat-ification is made for 5 different number of strata, i.e. L = 2, 3, 4, 5 and 6. Thevariance

V (yst)∗ =

L∑h=1

W 2hS

2hy

(1

nh− 1

Nh

)is calculated, which is used to compare the efficiency of the different methods. Foreach method, the OSB obtained along with the result of V (yst)

∗, stratum size (Nh),optimum sample size (nh) with a fixed n that varies from 100 to 1500 are presented inTables 36-45 in the Appendix A. The minimum value (y0) and the range of the dis-tribution (d) required to determine the OSB of each population are different, whichare captioned in each table.

The results for the proposed method are obtained by solving the recursive equations(3.17) and (3.18) using a computer program coded in C++. Whereas, the resultsfor other methods are obtained by using the R package "stratification", version2.2-3, developed by Baillargeon and Rivest [4, 5] to undertake the comparison.

In comparison of the proposed method with CRF and Geometric methods, it has

56

been observed that the proposed method provides least variance of the estimate (i.e.V (yst)

∗) in almost all the cases. The study also reveals that the proposed methodperforms even better than the two methods when the skewness increases. The Geo-metric method performs very badly as compared to others and may not be useful asit violates the required restrictions on sample sizes given in (3.7), especially, when Lincreases. The results support the findings of Kozak and Verma [51] which showedthat the Geometric method is less efficient than Lavallée and Hidiroglou [55] method.However, the findings in this study contradict with that of Gunning and Horgan [27],which showed that the geometric method is more efficient than the CRF method.On the other hand, although, the CRF method performs better than the Geometricmethod, it sometimes fails to determine the OSB if L is large and nclass is small(e.g. see Table 43 for L = 6).

The comparison between Lavallée and Hidiroglou [55] method and other methodsreveals that LH method provides least variances in all cases. However, the perfor-mance of proposed method is very similar to the LH method as there is not muchsignificant difference in the variances. It should be noted that the comparison ismade using the criteria of minimum variance calculated using all the data valuesthat fall within the stratum boundaries from the data-set of a population. Exceptthe proposed method, the minimum variances are calculated for each method usingthe OSB that are obtained using the data-set which is the basis this comparison.Whereas, the minimum variances are calculated for the proposed method using theOSB that are obtained using the values that fall on the density function of theLog-normal distributions and not using the values in the data-set. Because of thedifference in the procedure used, LH method produces slight better results over theproposed method. Further, an advantage of the proposed method over LH methodis that it needs neither any initial solution nor the complete data-set. In many situa-tions, the complete data-set may not be available, in such cases the proposed methodworks as it requires only the parameters of the distribution.

3.6 Summary

This chapter deals with the problem of determining OSB and the sample allocationto strata for a skewed population that has Log-normal frequency distribution. Theproblem was formulated as an MPP, which was solved by developing a method usingdynamic programming technique.

57

A numerical example on determining OSB was presented to show the computa-tional details and the applications of proposed method using dynamic programmingtechnique. Based on the results, we concluded that the proposed method is helpful inchoosing the best boundary points for stratification if parameters of the populationare available. Furthermore, a comparison study was carried out using ten artificialpopulations to compare efficiency of the proposed method is useful and more efficientthan the other stratification methods.

The basic advantage of dynamic programming over the classical optimization tech-niques is that it can determine OSB efficiently when the density function of thepopulation is known or approximately known from the previous studies. Many otheriterative methods are also available for determining strata boundaries but these iter-ative methods require approximate initial solutions, whereas, the proposed methoddoes not. Also there is no guarantee that an iterative method will converge andgive the global minimum variance in the absence of a suitably chosen initial solution(Amini et al. [1], Khan et al. [41]).

More importantly, the proposed technique has a wide scope of application as com-pared to other methods. In practice, the complete data-set of the study variableis unknown, which diminishes the uses of many stratification techniques. In such asituation, only the proposed technique can be used as it requires only the values ofparameters of the population which can easily be available from past studies. Thus,we may conclude that the proposed method is relatively efficient and very useful fordetermining the OSB for any skewed population.

58

Chapter 4

Optimum Stratification usingAuxiliary Variable

4.1 Introduction

Unquestionably, ideal stratification could be accomplished viably when the frequencydistribution of the study variable is known, and strata could be formed by cutting therange of the distribution at reasonable points. However, constructing OSB using thestudy variable is not plausible since the study variable of interest is not known priorto the survey. As auxiliary information is promptly accessible, stratification shouldbe possible utilizing the auxiliary variable x. It is assumed that the study variabley and the auxiliary variable x should be highly correlated with a valid regressionmodel. In case the stratification is made using x, it might prompt substantial gainsin the precision of survey estimate, although it may not be as efficient as the onebased on y. The literature review on optimum stratification using auxiliary variablehas been discussed in detailed in Chapter 2 of this thesis.

In this chapter, we determine OSB of the study variable, y, using the auxiliary vari-able x. It is assumed the frequency distribution of auxiliary variable is known andthere is a valid regression model that holds between y and x. Again, the problem ofdetermining the OSB is redefined as the problem of determining OSW and is formu-lated as a MPP that seeks minimization of the variance of the estimated populationmean under Neyman allocation subject to the constraint that the sum of the widthsof all the strata is equal to the range of the distribution. Section 4.2 provides the

59

general formulation of the problem of finding OSW as an MPP. Section 4.4 illus-trates the procedure to determine the OSW, OSB and OSS for a Pareto auxiliaryvariable where the regression models considered are linear or quadratic. A compari-son study is carried out in Section 4.5, to investigate the effectiveness of the proposedmethod with Dalenius and Hodges [17] CRF method, Geometric method by Gunningand Horgan [27], Lavallée and Hidiroglou [55] LH method that are available in theliterature as in Baillargeon and Rivest [4]. In Section 4.6 the chapter is summarized.

4.2 The general formulation of the problem

Let the population be stratified into L strata based on a single auxiliary variable xand the estimation of the mean of study variable y is of interest. If a simple randomsample of size nh is to be drawn from hth stratum with sample mean yh; (h =

1, 2, ..., L), then the stratified sample mean, yst, is given by

yst =L∑h=1

Whyh,

where Wh is the proportion of the population contained in the hth stratum.

When the finite population correction factors are ignored, the variance of yst underNeyman allocation is given by

V (yst) =

(∑Lh=1WhShy

)2

n(4.1)

where Wh and S2hy are the weight and variance of the study variable y for the hth

stratum; h = 1, 2, ..., L respectively and n is the predetermined total sample size.

Consider the generalized regression model:

y = λ(x) + ε (4.2)

where λ(x) is a linear or nonlinear function of x and ε is an error term such thatE(ε|x) = 0 and V (ε|x) = φ(x) ≥ 0 for all x.

Under model (4.2), the stratum mean µhy and the stratum variance S2hy can be

60

expressed as (Singh and Sukhatme [81]):

µhy = µhλ (4.3)

S2hy = S2

hλ + µhφ (4.4)

where µhλ and µhφ are the expected values of functions λ(x) and φ(x), respectively,and S2

hλ denotes the variance of λ(x) in the hth stratum.

If λ and ε are uncorrelated, from the model (4.2) , S2hy can also be expressed as

(Dalenius and Gurney [16])

S2hy = S2

hλ + S2hε (4.5)

where S2hε is the variance of ε in the hth stratum. It can be verified that the expres-

sion (4.4) and (4.5) are equivalent.

Let f(x); a ≤ x ≤ b be the frequency function of the auxiliary variable x that is usedfor the stratification. If the population mean of the study variable y is estimatedunder Neyman allocation, then the problem of determining the strata boundaries isto cut up the range, (a, b) at (L− 1) intermediate points a = x0 ≤ x1 ≤ x2 ≤, ...,≤xL−1 ≤ xL = b such that (4.1) is minimum.

For a fixed sample size n, minimizing (4.1) is equivalent to minimizing∑L

h=1WhShy.Thus, from (4.4), we minimize

L∑h=1

Wh

√S2hλ + µhφ. (4.6)

If f(x) is known and integrable, Wh, S2hλ and µhφ can be obtained as a function of

61

the boundary points (xh−1, xh) by using the following expressions:

Wh =

∫ xh

xh−1

f(x)dx, (4.7)

S2hλ =

1

Wh

∫ xh

xh−1

λ2(x)f(x)dx− µ2hλ (4.8)

where µhλ =1

Wh

∫ xh

xh−1

λ(x)f(x)dx (4.9)

and µhφ =1

Wh

∫ xh

xh−1

φ(x)f(x)dx (4.10)

is the mean and (xh−1, xh) are the boundaries of hth stratum.

Thus, the objective function in (4.6) could be expressed as a function of boundarypoints (xh−1, xh) only.

Let

φh(xh, xh−1) = WhShy = Wh

√S2hλ + µhφ.

Then, the problem of determination of OSB can be expressed as the following opti-mization problem: Find x1, x2, ..., xL−1 that

MinimizeL∑h=1

φh(xh, xh−1)

subject to a = x0 ≤ x1 ≤ x2 ≤, ...,≤ xL−1 ≤ xL = b.

(4.11)

We further definelh = xh − xh−1; h = 1, 2, ..., L (4.12)

where lh ≥ 0 denotes the range or width of the hth stratum.

Obviously, with this definition of lh, the range of the distribution, b− a = d(say), isexpressed as a function of stratum width as:

L∑h=1

lh =L∑h=1

(xh − xh−1) = b− a = xL − x0 = d. (4.13)

62

The hth stratification point xh; h = 1, 2, ..., L is then expressed as

xh = x0 +∑h

i=1 li

= xh−1 + lh.(4.14)

Adding (4.13) as a new constraint, the problem (4.11) can be treated as an equivalentproblem of determining OSW, l1, l2, ..., lL, and is expressed as the following MPP:

MinimizeL∑h=1

φh(lh, xh−1),

subject toL∑h=1

lh = d,

and lh ≥ 0;h = 1, 2, ..., L. (4.15)

Initially, x0 is known. Therefore, the first term, that is, φ1(l1, x0) in the objectivefunction of the MPP (4.15) is a function of l1 alone. Once l1 is known, the secondterm φ2(l2, x1) will become a function of l2 alone and so on. Due to the special natureof functions, the MPP (4.15) may be treated as a function of lh alone and can beexpressed as:

MinimizeL∑h=1

φh(lh),

subject toL∑h=1

lh = d,

and lh ≥ 0; h = 1, 2, ..., L. (4.16)

The MPP given by (4.16) can be solved using the same solution procedure usingdynamic programming technique developed in Section 3.3.

4.3 Determination of optimum sample size

When the OSW and the OSB have been determined using the procedure discussedin Section 4.2, the optimum sample size (OSS) within stratum (nh, h = 1, 2, . . . , L)can be computed which minimizes the variance of the survey estimate.

63

If the study variable y holds the regression model (4.2) with the auxiliary variablex across the strata, then the OSS (nh) , under Neyman allocation for a fixed totalsample size n =

∑Lh=1 nh, can be obtained as:

nh = n ·Wh

√S2hλ + µhφ∑L

h=1Wh

√S2hλ + µhφ

, h = 1, 2, . . . ,L. (4.17)

4.4 Determining OSB for skewed population with

Pareto auxiliary variable

4.4.1 The Pareto distribution

The Pareto distribution, named after the Italian economist Vilfredo Pareto, is askewed, heavy-tailed distribution that coincides with social, scientific, geophysical,actuarial, and many other types of observable phenomena. Outside the field of eco-nomics it is at times referred to as the Bradford distribution.

Vilfredo originally used this distribution to describe the allocation of wealth amongindividuals since it seemed to show that a larger portion of the wealth of any societyis owned by a smaller percentage of the people in that society. This idea is sometimesexpressed more simply as the Pareto principle or the "80-20 rule" which says that20% of the population controls 80% of the wealth.

If the random variable x has a Pareto distribution then its probability density func-tion (p.d.f) is given by

f(x) =αβα

xα+1; x ∈ [β,∞), (4.18)

where α > 0 is the shape parameter and β > 0 is the scale parameter.

The Figure 2 shows plot of Pareto densities with β = 1 and α = 0.5, 1, 2.

64

Figure 2: Pareto densities.

4.4.2 Formulation of the problem of OSW as an MPP

When the auxiliary variable x has a Pareto distribution with the p.d.f given by(4.18) and the function λ(x) in the regression model (4.2) is known, the problem ofdetermining OSW is formulated as an MPP which can be solved using a dynamicprogramming technique discussed in Chapter 3.

For the purpose of illustration, we consider two forms of the regression model:

1. Linear regression model.

2. Quadratic regression model.

Then, the terms Wh, S2hλ and µhφ can be derived and expressed as a function of

boundary points (xh−1, xh).

65

Using (4.7) and (4.18), the stratum weight is obtained as

Wh = αβα∫ xh

xh−1

x−α−1dx

= βα(x−αh−1 − x

−αh

). (4.19)

Thus, substituting (4.12) in (4.19) gives

Wh = βα[x−αh−1 − (xh−1 + lh)

−α] . (4.20)

Linear regression Model:

If the regression model in (4.2) is linear, that is, λ(x) is in the form

λ(x) = β0 + β1x (4.21)

then using (4.9) and (4.21) the stratum mean of λ(x) = β0 + β1x, denoted as µhλ isobtained by

µhλ =1

Wh

∫ xh

xh−1

(β0 + β1x) f(x)dx

= β0 +β1

Wh

∫ xh

xh−1

xf(x)dx. (4.22)

Now using f(x) defined by (4.18) we obtain

µhλ = β0 +β1αβ

α

Wh

∫ xh

xh−1

x−αdx.

= β0 −β1αβ

α

Wh (1− α)

(x1−αh−1 − x

1−αh

). (4.23)

Substituting (4.12) and (4.20) in (4.23) gives

µhλ = β0 −β1α

[x1−αh−1 − (xh−1 + lh)

1−α](1− α)

[x−αh−1 − (xh−1 + lh)

−α] . (4.24)

Similarly, using (4.8) and (4.21), the stratum variance S2hλ is derived as

66

S2hλ =

1

Wh

∫ xh

xh−1

(β0 + β1x)2 f(x)dx− µ2hλ.

= β20 +

2β0β1

Wh

∫ xh

xh−1

xf(x)dx+β2

1

Wh

∫ xh

xh−1

x2f(x)dx− µ2hλ. (4.25)

Substituting (4.18) in (4.25), gives

S2hλ = β2

0 −2β0β1αβ

α

Wh (1− α)

(x1−αh−1 − x

1−αh

)− β2

1αβα

Wh (2− α)

(x2−αh−1 − x

2−αh

)− µ2

hλ. (4.26)

Further, substituting (4.12), (4.20) and (4.24) in (4.26) yields

S2hλ = β2

0−2β0β1α

[x1−αh−1 − (xh−1 + lh)

1−α](1− α)

[x−αh−1 − (xh−1 + lh)

−α]−

β21α[x2−αh−1 − (xh−1 + lh)

2−α](2− α)

[x−αh−1 − (xh−1 + lh)

−α]−β2

0+2β0β1α

[x1−αh−1 − (xh−1 + lh)

1−α](1− α)

[x−αh−1 − (xh−1 + lh)

−α]−

β21α

2[x1−αh−1 − (xh−1 + lh)

1−α]2(1− α)2 [x−αh−1 − (xh−1 + lh)

−α]2 . (4.27)

Finally, the cancellation of some terms in (4.27) gives

S2hλ = −

β21α[x2−αh−1 − (xh−1 + lh)

2−α](2− α)

[x−αh−1 − (xh−1 + lh)

−α]−

β21α

2[x1−αh−1 − (xh−1 + lh)

1−α]2(1− α)2 [x−αh−1 − (xh−1 + lh)

−α]2 . (4.28)

The regression model (4.2) assumes that E(ε|x) = 0 and V (ε|x) = φ(x) ≥ 0 for allx ∈ (a, b). The function φ(x) is assumed to be of the form:

φ(x) = cxg; c ≥ 0, g ≥ 0, (4.29)

67

where c and g are constants and in many populations studied 0 ≤ g ≤ 2 (Singh andSukhatme [81], Singh [79] and Rizvi et al. [73]).

Using (4.18) and (4.29), we can express µhφ as a function of boundary points (xh−1, xh)as follows:

µhφ =1

Wh

∫ xh

xh−1

cxgf(x) dx

=αβαc

Wh

∫ xh

xh−1

x−α+g−1 dx

=αβαc

Wh (g − α)

[(xh−1 + lh)

g−α − xg−αh−1

]. (4.30)

Substituting (4.20), we obtain

µhφ = −αc[xg−αh−1 − (xh−1 + lh)

g−α](g − α)

[x−αh−1 − (xh−1 + lh)

−α] . (4.31)

Using (4.6), (4.20), (4.28) and (4.31), the MPP (4.16) to determine the OSW andhence the OSB may be expressed as:

MinimizeL∑h=1

√√√√√√√√√√√√√√√√

−β

21αβ

2α

2− α[x2−αh−1 − (xh−1 + lh)

2−α]×[x−αh−1 − (xh−1 + lh)

−α]−β

21α

2β2α

(1− α)2

[x1−αh−1 − (xh−1 + lh)

1−α]2 − αcβ2α

(g − α)

[xg−αh−1 − (xh−1 + lh)

g−α]×[x−αh−1 − (xh−1 + lh)

−α]

subject to

L∑h=1

lh = d,

and lh ≥ 0; h = 1, 2, ..., L

(4.32)

where α and β are the parameters of Pareto distribution, β1 is the regression coeffi-cient, d is the range of the distribution, and c, g are constants as defined above.

68

Quadratic regression Model:

Consider that the regression model in (4.2) is non-linear with a quadratic form, thatis

λ(x) = β0 + β1x+ β2x2, (4.33)

then using (4.9) and (4.33) the stratum mean µhλ is obtained by

µhλ =1

Wh

∫ xh

xh−1

(β0 + β1x+ β2x

2)f(x)dx

= β0 +β1

Wh

∫ xh

xh−1

xf(x)dx+β2

Wh

∫ xh

xh−1

x2f(x)dx. (4.34)

Now using f(x) defined by (4.18) we obtain

µhλ = β0 +β1αβ

α

Wh

∫ xh

xh−1

x−αdx+β2αβ

α

Wh

∫ xh

xh−1

x1−αdx.

= β0 −β1αβ

α

Wh (1− α)

(x1−αh−1 − x

1−αh

)− β2αβ

α

Wh (2− α)

(x2−αh−1 − x

2−αh

). (4.35)

Substituting (4.12) and (4.20) in (4.35) gives

µhλ = β0 −β1α

[x1−αh−1 − (xh−1 + lh)

1−α](1− α)

[x−αh−1 − (xh−1 + lh)

−α]−

β2α[x2−αh−1 − (xh−1 + lh)

2−α](2− α)

[x−αh−1 − (xh−1 + lh)

−α] . (4.36)

Using (4.9) and (4.33), the stratum variance S2hλ is derived as

S2hλ =

1

Wh

∫ xh

xh−1

(β0 + β1x+ β2x

2)2f(x)dx− µ2

hλ

= β20 +

2β0β1

Wh

∫ xh

xh−1

xf(x)dx+β2

1

Wh

∫ xh

xh−1

x2f(x)dx

+2β0β2

Wh

∫ xh

xh−1

x2f(x)dx+2β1β2

Wh

∫ xh

xh−1

x3f(x)dx

+β2

2

Wh

∫ xh

xh−1

x4f(x)dx− µ2hλ. (4.37)

69

Substituting (4.18) in (4.37), gives

S2hλ = β2

0 −2β0β1αβ

α

Wh (1− α)

(x1−αh−1 − x

1−αh

)− β2

1αβα

Wh (2− α)

(x2−αh−1 − x

2−αh

)− 2β0β2αβ

α

Wh (2− α)

(x2−αh−1 − x

2−αh

)− 2β1β2αβ

α

Wh (3− α)

(x3−αh−1 − x

3−αh

)− β2

2αβα

Wh (4− α)

(x4−αh−1 − x

4−αh

)− µ2

hλ. (4.38)

Further, substituting (4.12), (4.20) and (4.36) in (4.38) yields

S2hλ = β2

0−2β0β1α

[x1−αh−1 − (xh−1 + lh)

1−α](1− α)

[x−αh−1 − (xh−1 + lh)

−α] − β21α[x2−αh−1 − (xh−1 + lh)

2−α](2− α)

[x−αh−1 − (xh−1 + lh)

−α]−

2β0β2α[x2−αh−1 − (xh−1 + lh)

2−α](2− α)

[x−αh−1 − (xh−1 + lh)

−α] − 2β1β2α[x3−αh−1 − (xh−1 + lh)

3−α](3− α)

[x−αh−1 − (xh−1 + lh)

−α]−

β22α[x4−αh−1 − (xh−1 + lh)

4−α](4− α)

[x−αh−1 − (xh−1 + lh)

−α] − β20 +

2β0β1α[x1−αh−1 − (xh−1 + lh)

1−α](1− α)

[x−αh−1 − (xh−1 + lh)

−α]−

β21α

2[x1−αh−1 − (xh−1 + lh)

1−α]2(1− α)2 [x−αh−1 − (xh−1 + lh)

−α]2 +2β0β2α

[x2−αh−1 − (xh−1 + lh)

2−α](2− α)

[x−αh−1 − (xh−1 + lh)

−α]−

2β1β2α2[x1−αh−1 − (xh−1 + lh)

1−α] [x2−αh−1 − (xh−1 + lh)

2−α](1− α) (2− α)

[x−αh−1 − (xh−1 + lh)

−α]2−

β22α[x2−αh−1 − (xh−1 + lh)

2−α]2(2− α)2 [x−αh−1 − (xh−1 + lh)

−α]2 . (4.39)

Finally, the cancellation of some terms in (4.39) gives

S2hλ = −

β21α[x2−αh−1 − (xh−1 + lh)

2−α](2− α)

[x−αh−1 − (xh−1 + lh)

−α] − 2β1β2α[x3−αh−1 − (xh−1 + lh)

3−α](3− α)

[x−αh−1 − (xh−1 + lh)

−α]−

β22α[x4−αh−1 − (xh−1 + lh)

4−α](4− α)

[x−αh−1 − (xh−1 + lh)

−α] − β21α

2[x1−αh−1 − (xh−1 + lh)

1−α]2(1− α)2 [x−αh−1 − (xh−1 + lh)

−α]2−

2β1β2α2[x1−αh−1 − (xh−1 + lh)

1−α] [x2−αh−1 − (xh−1 + lh)

2−α](1− α) (2− α)

[x−αh−1 − (xh−1 + lh)

−α]2−

β22α[x2−αh−1 − (xh−1 + lh)

2−α]2(2− α)2 [x−αh−1 − (xh−1 + lh)

−α]2 . (4.40)

70

Using (4.6), (4.20), (4.28) and (4.31), the MPP (4.16) to determine the OSW andhence the OSB may be expressed as:

MinimizeL∑h=1

√√√√√√√√√√√√√√√√√√√√√√√√√√

−β

21αβ

2α

2− α[x2−αh−1 − (xh−1 + lh)

2−α] [x−αh−1 − (xh−1 + lh)−α]

−2β1β2αβ2α

3− α[x3−αh−1 − (xh−1 + lh)

3−α] [x−αh−1 − (xh−1 + lh)−α]

−β22αβ

2α

4− α[x4−αh−1 − (xh−1 + lh)

4−α] [x−αh−1 − (xh−1 + lh)−α]

−β21α

2β2α

(1− α)2

[x1−αh−1 − (xh−1 + lh)

1−α]2− 2β1β2α

2β2α

(1− α) (2− α)

[x1−αh−1 − (xh−1 + lh)

1−α][x2−αh−1 − (xh−1 + lh)

2−α]− β22α

2β2α

(2− α)2

[x2−αh−1 − (xh−1 + lh)

2−α]2 − αcβ2α

(g − α)

[xg−αh−1 − (xh−1 + lh)

g−α] [x−αh−1 − (xh−1 + lh)−α]

subject to

L∑h=1

lh = d,

and lh ≥ 0; h = 1, 2, ..., L

(4.41)where α and β are the parameters of Pareto distribution, β1 and β2 are the regressioncoefficient, d is the range of the distribution, and c, g are constants as definedpreviously.

4.4.3 Numerical illustration

This section illustrates the computational details of the solution procedure discussedin Section 3.3 for solving the MPP (4.32) to obtain OSW of a Pareto auxiliary vari-able.

Assume that x follows a Pareto distribution in the interval [1.000527, 28.147120],that is, x0 = 1.000527, xL = 28.147120 and d = xL − x0 = 27.146593. Also assumethat α = 1.472, β = 1.000527, β0 = 2, β1 = 0.5, c = 1 and g = 0. Then, the MPP in(4.32) is expressed as:

71

MinimizeL∑h=1

√√√√√√√√√√

−0.6981

[x0.528h−1 − (xh−1 + lh)

0.528]×[x−1.472h−1 − (xh−1 + lh)

−1.472]−2.4353

[x−0.472h−1 − (xh−1 + lh)

−0.472]2 +1.0016

[x−1.472h−1 − (xh−1 + lh)

−1.472]2

subject to

L∑h=1

lh = 27.146593,

and lh ≥ 0; h = 1, 2, ..., L.

(4.42)

Also

xk−1 = x0 + l1 + l2 + ...+ lk−1

= 1.000527 + l1 + l2 + ...+ lk−1

= dk−1 + 1.000527

= dk − lk + 1.000527.

Substituting this value of xk−1 in (3.18) and (3.17), the recurrence relations for solv-ing MPP (4.42) are obtained as:

For first stage (k = 1):

Φ1(d1) =

√√√√√√√√√√

−0.6981

[1.0003− (1.000527 + d1)0.528]

×[0.9992− (1.000527 + d1)−1.472]

−2.4353[0.9998− (1.000527 + d1)−0.472]2

+1.0016[0.9992− (1.000527 + d1)−1.472]2

(4.43)

at l1 = d1,

72

and for the stages k ≥ 2:

Φk(dk) =min

0 ≤ lk ≤ dk

√√√√√√√√√√√√√√√√√√√

−0.6981

[(dk − lk + 1.000527)0.528

− (dk + 1.000527)0.528]×[(dk − lk+

(1.000527)−1.472 −(dk + 1.000527

)−1.472]−2.4353

[(dk − lk + 1.000527)−0.472

− (dk + 1.000527)−0.472]2

+1.0016[

(dk − lk + 1.000527)−1.472

− (dk + 1.000527)−1.472]2

+Φk−1(dk − lk)

.

(4.44)Solving the recursive equations (4.43) and (4.44) by executing a C++ computer pro-gram (refer to Appendix B.2) developed using the solution procedure described inSection 3.3, the OSW are obtained. The results are presented in Section 4.5.

4.5 Comparison study

In this section, a comparison study is carried out on a simulated and a real data set,to compare and investigate the effectiveness of the proposed dynamic programmingmethod with the following stratification methods available in the literature:

1. Dalenius and Hodges [17] CRF method.

2. Geometric method by Gunning and Horgan [27].

3. Generalized Lavallée and Hidiroglou [55] method using Kozak [50] algorithm.

The results for the methods mentioned above are obtained by using the R package"stratification", version 2.2-3, developed by Baillargeon and Rivest [4, 5]. Thenusing the results, the optimum value of the objective function

∑Lh=1Wh

√S2hλ + µhφ =∑L

h=1WhShy was obtained to undertake the comparison study.

4.5.1 Simulated data

To undertake the comparison study, a data set (N = 500) following a Pareto distri-bution with parameters α = 1.472, β = 1.000527 was randomly generated using theR statistical software. The values of x0 = 1.000527 and xL = 28.147120, therefore,

73

d = 27.146593. It was assumed that n = 100, β0 = 2, β1 = 0.5 and c = 1, g = 0,

that is, µhφ = 1.

For the dynamic programming method, the OSW (l∗h) and the OSB(x∗h = x∗h−1 + l∗h

)along with the optimum values of the objective function

∑Lh=1 WhShy, for L =

2, 3, 4, 5 and 6 are presented in Table 3. The OSB for the study variable y, us-ing the regression model (4.2), with λ(x) = β0 + β1 and the OSS (nh) using (4.17)are presented in Table 4. Further, for the purpose of comparison, Table 5 presentsthe OSB for the auxiliary variable x and the optimum value of objective function∑L

h=1WhShy, for all the different methods.

Table 3: OSW, OSB and the optimum value of objective function for simulated data.No. of strata OSW OSB Optimum value ofL (l∗h) (x∗h = x∗h−1 + l∗h)

∑Lh=1WhShy

2 l∗1 = 4.571173 x∗1= 5.571700 1.214161l∗2 = 22.575420l∗1 = 2.367603 x∗1= 3.368130

3 l∗2 = 6.208830 x∗2= 9.576960 1.108447l∗3 =18.570160l∗1 = 1.592433 x∗1= 2.592960

4 l∗2 = 3.329960 x∗2= 5.922920 1.064204l∗3 = 6.827550 x∗3= 12.750470l∗4 = 15.396650l∗1 = 1.198223 x∗1= 2.198750l∗2 = 2.205550 x∗2= 4.404300

5 l∗3 =3.890620 x∗3= 8.294920 1.041234l∗4 = 6.884450 x∗4= 15.179370l∗5 = 12.967750l∗1 = 0.959373 x∗1= 1.959900l∗2 = 1.619570 x∗2= 3.579470

6 l∗3 = 2.624170 x∗3= 6.203640 1.027745l∗4 = 4.168210 x∗4= 10.371850l∗5 = 6.667360 x∗5= 17.039210l∗6 = 11.107910

74

Table 4: OSB and OSS of the study variable y for simulated data.No. of strata OSB OSS

L (y∗h = β0 + β1x∗h) nh = n ·

Wh

√S2hλ + µhφ∑L

h=1Wh

√S2hλ + µhφ

2 y∗1= 4.785850 n1 = 84

n2 = 16

y∗1= 3.684065 n1 = 78

3 y∗2= 6.788480 n2 = 15

n3 = 7

y∗1= 3.296480 n1 = 71

4 y∗2=4.961460 n2 = 18

y∗3= 8.375235 n3 = 7

n4 = 4

y∗1= 3.099375 n1 = 65

y∗2= 4.202150 n2 = 21

5 y∗3= 6.147460 n3 = 8

y∗4= 9.589685 n4 = 4

n5 = 2

y∗1= 2.979950 n1 = 58

y∗2= 3.789735 n2 = 24

6 y∗3= 5.101820 n3 = 10

y∗4= 7.185925 n4 = 4

y∗5= 10.519605 n5 = 2

n6 = 2

Using the results displayed in Table 5, it was observed that all the different meth-ods compared in this study were able to successfully stratify the simulated data. Itwas observed that the OSB obtained by CRF and Geometric methods are close tothat of dynamic programming method. The OSB obtained by LH method differssignificantly from the other methods. It can be concluded that there is significantdifference in the OSB obtained by the different methods.

To compare the efficiency of the different stratification methods, it was seen thatthe proposed dynamic programing method provides the least value of the objectivefunction

∑Lh=1WhShy in all strata (L = 2, 3, 4, 5,6). Therefore, it can be concluded

that the dynamic programming method is seen to be most efficient as comparedto the other methods. It was also observed that the Geometric method and CRFperformed better than the LH method. As the number of strata increased, it wasfound that Geometric method performed very similar to CRF but the LH performedvery poorly. Thus, the study reveals that dynamic programming method is more

75

Table 5: OSB and the optimum value of objective function of different methods forsimulated dataL

CRF Geometric LH (Kozak Algo.) Dynamic Prog.OSB

∑Lh=1WhShy OSB

∑Lh=1WhShy OSB

∑Lh=1WhShy OSB

∑Lh=1WhShy

2 3.72 1.325800 5.31 1.272993 3.53 1.337196 5.57 1.214161

3 2.36 1.175644 3.04 1.142575 2.00 1.229730 3.37 1.1084476.43 9.25 4.64 9.58

4 2.36 1.088801 2.30 1.088658 1.45 1.222915 2.59 1.0642045.07 5.31 2.38 5.9213.22 12.22 4.46 12.75

5 2.36 1.076055 1.95 1.069916 1.44 1.213954 2.20 1.0412343.72 3.80 2.20 4.406.43 7.41 3.30 8.2914.57 14.44 4.64 15.18

6 2.36 1.055725 1.74 1.052719 1.34 1.212331 1.96 1.0277453.72 3.04 1.74 3.585.07 5.31 2.38 6.209.14 9.25 3.30 10.3715.93 16.14 4.64 17.04

76

efficient than the other methods in stratifying a skewed population with Paretodistribution. An added advantage of the proposed method over other stratificationmethods is that it does not require initial solution or the complete data-set. In casewhere the complete data-set is not available, the proposed method can still be usedas it requires only the parameters of the distribution.

4.5.2 Real data

To carry out another comparison study, we used a cyclone population data, N = 262

cyclones from 1970 to 2008. The data was obtained from Fiji Meteorological Service,on the variables:

1. Sinuosity meaning how straight the path of the Tropical cyclone was which iscalculated as:

Sinuosity=Distance of the cyclone

Displacement of the cyclone.

2. Start Latitude of the cyclone.

3. Start Pressure of the cyclone.

4. Duration of cyclone (in days).

5. Distance traveled by the cyclone (in km).

Suppose that a survey is carried out to estimate the average start latitude of cyclonesusing a stratified sampling and the Sinuosity is used as the auxiliary variable. Theauxiliary variable x (Sinuosity) was found to have approximately Pareto distributionand the maximum likelihood estimators of the parameters were α = 7.845 and β = 1.The scatter plot in Figure 3 shows that there is a negative linear correlation betweenthe variables Sinuosity (x) and start latitude (y). Table 6 presents the analysis ofvariance approach (ANOVA) table which depicts that there is a significant linearregression between Sinuosity (x) and start latitude (y) and the R-square value wasnoted to be 0.858. The regression between the variables was found to be linearwith β0 = 23.0357 and β1 = −7.0619 (refer to Table 7). The values of x0 = 1 andxL = 1.6902, therefore, d = 0.6902. The total sample size was taken to be n = 50 andit was assumed that c = 1, g = 0, that is, µhφ = 1. For the dynamic programmingmethod, the OSW (l∗h) , the OSB

(x∗h = x∗h−1 + l∗h

), along with the optimum values

of the objective function∑L

h=1 WhShy for L = 2, 3, 4, 5 and 6 are presented in Table8. Further, the OSB for study variable y, using the regression model (4.2), withλ(x) = β0 + β1 and the OSS using (4.17) are presented in Table 9.

77

Figure 3: Scatter plot for Sinuosity and Latitude.

Table 6: ANOVA table for Sinuosity and Latitude.Source Sum of Squares (SS) d.f Mean Squares (MS) f p−value

Regression 334.8128 1 334.8128 24.41 0.0000Error 3566.9098 260 13.7189Total 3901.7226 261

Table 7: Linear model summary.Predictors Coefficients SE t p−value

β0 23.0357 1.6540 13.927 0.0000β1 -7.0619 1.4295 -4.940 0.0000

78

Table 8: OSW, OSB and the optimum value of objective function forcyclone data.

No. of strata OSW OSB Optimum value ofL (l∗h) (x∗h = x∗h−1 + l∗h)

∑Lh=1WhShy

2 l∗1 = 0.202010 x∗1= 1.202010 1.103626l∗2 = 0.488190l∗1 = 0.125270 x∗1= 1.125270 1.042070

3 l∗2 = 0.11790 x∗2= 1.317060l∗3 = 0.373140l∗1 = 0.091050 x∗1= 1.091050 1.017834

4 l∗2 = 0.121870 x∗2= 1.212920l∗3 = 0.176740 x∗3= 1.389660l∗4 = 0.0.300540l∗1 = 0.071500 x∗1= 1.071500 1.005985l∗2 = 0.089370 x∗2= 1.160870

5 l∗3 = 0.116460 x∗3= 1.277330l∗4 = 0.161670 x∗4= 1.439000l∗5 = 0.251200l∗1 = 0.058840 x∗1= 1.058840 0.999355l∗2 = 0.070510 x∗2= 1.129350

6 l∗3 = 0.086710 x∗3= 1.216060l∗4 = 0.110430 x∗4= 1.326490l∗5 = 0.148020 x∗5= 1.474510l∗6 = 0.215690

79

Table 9: OSB and OSS of the study variable y for cyclone data.No. of strata OSB OSS

L (y∗h = β0 + β1x∗h) nh = n ·

Wh

√S2hλ + µhφ∑L

h=1Wh

√S2hλ + µhφ

2 y∗1= 14.547226 n1 = 37

n2 = 13

y∗1= 13.734754 n1 = 30

3 y∗2= 15.089156 n2 = 14

n3 = 6

y∗1= 13.222060 n1 = 23

4 y∗2=14.470180 n2 = 15

y∗3= 15.330814 n3 = 8

n4 = 4

y∗1= 12.873626 n1 = 18

y∗2= 14.015323 n2 = 16

5 y∗3= 14.837752 n3 = 8

y∗4= 15.468874 n4 = 5

n5 = 3

y∗1= 12.622858 n1 = 16

y∗2= 13.668160 n2 = 14

6 y∗3= 14.448006 n3 = 9

y∗4= 15.060343 n4 = 6

y∗5= 15.558278 n5 = 3

n6 = 2

The OSB for x and the optimum value of the objective function∑L

h=1 WhShy, for allthe different methods using the cyclone data are presented in Table 10.

Using the results presented in Table 10, similar observations were again noted for thereal data set. Comparing the values of

∑Lh=1WhShy it was found that the dynamic

programming method is seen to be most efficient in all strata (L = 2, 3, 4, 5, 6) ascompared to the other methods. It was also observed that the Geometric methodand the CRF were performing better than the LH method as L increased. Hence, itcan be concluded that the proposed dynamic programming method is indeed moreefficient than the other methods in stratifying skewed populations.

80

Table 10: OSB and the optimum value of objective function of different methods forcyclone data.

LCRF Geometric LH (Kozak Algo.) Dynamic Prog.

OSB∑Lh=1WhShy OSB

∑Lh=1WhShy OSB

∑Lh=1WhShy OSB

∑Lh=1WhShy

2 1.17 1.156473 1.30 1.165577 1.18 1.155115 1.20 1.103626

3 1.10 1.079692 1.19 1.08075 1.06 1.102021 1.13 1.0420701.31 1.42 1.21 1.32

4 1.07 1.045982 1.14 1.048707 1.05 1.060467 1.09 1.0178341.17 1.30 1.14 1.211.38 1.48 1.31 1.39

5 1.07 1.029974 1.11 1.030670 1.05 1.032054 1.07 1.0059851.14 1.23 1.12 1.161.28 1.37 1.21 1.281.45 1.52 1.38 1.44

6 1.03 1.023338 1.09 1.022183 1.04 1.028621 1.06 0.9993551.10 1.19 1.09 1.131.17 1.30 1.14 1.221.31 1.42 1.24 1.331.45 1.55 1.38 1.48

81

4.6 Summary

This chapter dealt with the problem of determining optimum strata boundaries of thestudy variable y with the aid of the auxiliary variable x. The problem of determiningthe OSB for auxiliary variable x was redefined as the problem of determining OSWand is formulated as an MPP. Later, the procedure was used to determine the OSW,OSB and OSS for a Pareto auxiliary variable using the linear regression model.Finally, a comparison study was carried out using a simulated data and a real data tocompare the efficiency of the stratification methods such as CRF, Geometric, LH anddynamic programming method. It was concluded from the study that the dynamicprogramming method is indeed the most efficient in stratifying skewed populations.Further, it was also noted that there are some advantages of the proposed methodover other stratification methods such as the proposed method does not require initialsolution, nor the complete data set and it can be used to stratify the study variableby the means of auxiliary variable.

82

Chapter 5

New Calibration Estimators inStratified Sampling

5.1 Introduction

In Section 2.3, we have reviewed the literature and studies on the calibration ap-proach, in particular, the calibration estimators proposed by Singh et al. [92], Tracyet al. [100] and Singh [83] in stratified sampling. Motivated by these estimators, anattempt has been made in this chapter to derive some new calibration estimatorsof population mean in stratified sampling, which may improve estimate, using theinformation such as mean of single auxiliary variable. Attempts have also been madeto investigate whether adding more constraints on variance information and/or con-straints on calibrated weights, improve the estimate. The problem of determiningthe Optimum Calibrated Weights (OCW) of all the the proposed estimators is tominimize the Chi-Square (CS) type distance measure, subject to some calibrationconstraints on the auxiliary variable. Then, the problem is solved using the Lagrangemultiplier technique and the calibration estimators of population mean is derived inthe form of Generalized Regression (GREG).

Section 5.2 contains the derivations of the proposed calibration estimators and italso discusses the estimated variance of these estimators. A numerical exampleto illustrate the computational details of these proposed calibration estimators ispresented in Section 5.3. A simulation study using four real populations is alsocarried out in Section 5.4 to investigate the efficiency of the proposed calibration

83

estimators. Section 5.5 discusses the results obtained from the numerical illustrationand the simulation study. Finally, the chapter concludes with a summary given inSection 5.6.

5.1.1 Publications

Some results presented in this chapter have been accepted for publication in Raoet al. [71].

5.2 Proposed calibration estimators

5.2.1 Estimator I

It can be noted that the calibration estimator given in Section 2.3 by Tracy et al.[100] were incorrectly derived, especially the calibrated weights

(W⊗h

)in (2.75) and

the estimated regression coefficients β⊗1 and β⊗2 in (2.77). Therefore, the proposed

Estimator I is an attempt to present a corrected form of Tracy et al. [100] estimator.

Theorem 26. A calibration estimator of the population mean Y in stratified sam-pling is defined in Tracy et al. [100] as

y⊗st =L∑h=1

W⊗h yh, (5.1)

with the calibrated weights W⊗h , obtained by minimizing the CS distance function

L∑h=1

(W⊗h −Wh

)2

Whqh

subject to the calibration constraints

L∑h=1

W⊗h xh =

L∑h=1

WhXh (5.2)

andL∑h=1

W⊗h s

2hx =

L∑h=1

WhS2hx. (5.3)

84

This leads to a GREG estimator of population mean Y , given by

y⊗st = yst + β⊗1

L∑h=1

Wh

(Xh − xh

)+ β⊗2

L∑h=1

Wh

(S2hx − s2

hx

), (5.4)

where

β⊗1 =

L∑h=1

WhqhxhyhL∑h=1

Whqhs4hx −

L∑h=1

Whqhs2hxyh

L∑h=1

Whqhxhs2hx

L∑h=1

Whqhx2h

L∑h=1

Whqhs4hx −

(L∑h=1

Whqhxhs2hx

)2 (5.5)

and

β⊗2 =

L∑h=1

Whqhs2hxyh

L∑h=1

Whqhx2h −

L∑h=1

WhqhxhyhL∑h=1

Whqhxhs2hx

L∑h=1

Whqhx2h

L∑h=1

Whqhs4hx −

(L∑h=1

Whqhxhs2hx

)2 . (5.6)

Proof. The Lagrange function L of the problem is defined as

L =L∑h=1

(W⊗h −Wh

)2

Whqh− 2λ1

(L∑h=1

W⊗h xh −

L∑h=1

WhXh

)

−2λ2

(L∑h=1

W⊗h s

2hx −

L∑h=1

WhS2hx

). (5.7)

Setting ∂L/∂W⊗h = 0, we obtain

W⊗h = Wh + λ1Whqhxh + λ2Whqhs

2hx. (5.8)

Substituting (5.8) in (5.2) and (5.3) and solving for λ1 and λ2, we get

λ1 =

L∑h=1

Whqhs4hx

L∑h=1

Wh

(Xh − xh

)−

L∑h=1

Whqhxhs2hx

L∑h=1

Wh (S2hx − s2

hx)

L∑h=1

Whqhx2h

L∑h=1

Whqhs4hx −

(L∑h=1

Whqhxhs2hx

)2 , (5.9)

and

85

λ2 =

L∑h=1

Whqhx2h

L∑h=1

Wh (S2hx − s2

hx)−L∑h=1

WhQhxhs2hx

L∑h=1

Wh

(Xh − xh

)L∑h=1

Whqhx2h

L∑h=1

Whqhs4hx −

(L∑h=1

Whqhxhs2hx

)2 . (5.10)

By substituting (5.9) and (5.10) in (5.8), we obtain the calibrated weights W⊗h as

W⊗h = Wh +

WhqhxhL∑h=1

Whqhs4hx −Whqhs

2hx

L∑h=1

Whqhxhs2hx

L∑h=1

Whqhx2h

L∑h=1

Whqhs4hx −

(L∑h=1

Whqhxhs2hx

)2

L∑h=1

Wh

(Xh − xh

)

+

Whqhs2hx

L∑h=1

Whqhx2h −Whqhxh

L∑h=1

Whqhxhs2hx

L∑h=1

Whqhx2h

L∑h=1

Whqhs4hx −

(L∑h=1

Whqhxhs2hx

)2

L∑h=1

Wh

(S2hx − s2

hx

).(5.11)

Finally, substitutingW⊗h in (5.1), we obtain the GREG estimator of population mean

Y , given by (5.4).

Remark 9. The Estimator I in (5.4) includes the calibration constraints on knownmean and variance.

The following theorems state the estimated variance of the estimator y⊗st .

Theorem 27. An estimator of the variance of y⊗st is given in (Tracy et al. [100]) as

v(y⊗st)

=L∑h=1

W 2h (1− fh)nh

s2eh, (5.12)

where s2eh = (nh − 3)−1∑nh

i=1 e2hi, ehi = (yhi−yh)−β⊗1 (xhi−xh)−β⊗2 (xhi − xh)2 − s∗2hx

and s∗2hx = n−1h

∑nh

i=1 (xhi − xh)2 .

Proof. see Tracy et al. [100].

Theorem 28. The lower level calibration approach yields another estimator of thevariance of y⊗st as

vLL(y⊗st)

=L∑h=1

(W⊗h

)2(1− fh)nh

s2eh. (5.13)

86

Proof. Replacing the design weights Wh with the calibrated weights W⊗h in (5.12)

yields (5.13).

5.2.2 Estimator II

In this section, an attempt has been made to modify the Estimator I given in Section5.2.1 by adding a constraint on calibrated weights suggested by Singh [83]. Hence,Estimator II, a calibration estimator of population mean in stratified sampling isproposed as follows:

Theorem 29. A new calibration estimator of the population mean Y in stratifiedsampling is defined as

yst =L∑h=1

W h yh, (5.14)

where W h are the calibrated weights, obtained by minimizing the CS distance function

L∑h=1

(W h −Wh)

2

Whqh

subject to the calibration constraints

L∑h=1

W h =

L∑h=1

Wh, (5.15)

L∑h=1

W h xh =

L∑h=1

WhXh (5.16)

andL∑h=1

W hs

2hx =

L∑h=1

WhS2hx. (5.17)

This leads to a GREG estimator of the population mean Y , given by

yst = yst + β1

L∑h=1

Wh

(Xh − xh

)+ β2

L∑h=1

Wh

(S2hx − s2

hx

), (5.18)

where

β1 =

A1

L∑h=1

Whqhyh + A3

L∑h=1

Whqhxhyh + A4

L∑h=1

Whqhs2hxyh

B, (5.19)

87

β2 =

A2

L∑h=1

Whqhyh + A4

L∑h=1

Whqhxhyh + A5

L∑h=1

Whqhs2hxyh

B, (5.20)

A1 =L∑h=1

Whqhs2hx

L∑h=1

Whqhxhs2hx −

L∑h=1

WhqhxhL∑h=1

Whqhs4hx,

A2 =L∑h=1

WhqhxhL∑h=1

Whqhxhs2hx −

L∑h=1

Whqhs2hx

L∑h=1

Whqhx2h,

A3 =L∑h=1

WhqhL∑h=1

Whqhs4hx −

(L∑h=1

Whqhs2hx

)2

,

A4 =L∑h=1

Whqhs2hx

L∑h=1

Whqhxh −L∑h=1

WhqhL∑h=1

Whqhxhs2hx,

A5 =L∑h=1

WhqhL∑h=1

Whqhx2h −

(L∑h=1

Whqhxh

)2

,

(5.21)

and

B =L∑h=1

Whqh

L∑h=1

Whqhx2h

L∑h=1

Whqhs4hx

+ 2L∑h=1

Whqhxh

L∑h=1

Whqhs2hx

L∑h=1

Whqhxhs2hx

−L∑h=1

Whqh

(L∑h=1

Whqhxhs2hx

)2

−

(L∑h=1

Whqhs2hx

)2 L∑h=1

Whqhx2h

−

(L∑h=1

Whqhxh

)2 L∑h=1

Whqhs4hx. (5.22)

Proof. The Lagrange function L is given by

L =L∑h=1

(W h −Wh)

2

Whqh− 2λ0

(L∑h=1

W h −

L∑h=1

Wh

)

−2λ1

(L∑h=1

W h xh −

L∑h=1

WhXh

)

−2λ2

(L∑h=1

W hs

2hx −

L∑h=1

WhS2hx

). (5.23)

88

Setting ∂L/∂W h = 0, we obtain

W h = Wh + λ0Whqh + λ1Whqhxh + λ2Whqhs

2hx. (5.24)

Using (5.24) in (5.15)-(5.17), we obtain the following system of linear equations

λ0

L∑h=1

Whqh + λ1

L∑h=1

Whqhxh + λ2

L∑h=1

Whqhs2hx = 0,

λ0

L∑h=1

Whqhxh + λ1

L∑h=1

Whqhx2h + λ2

L∑h=1

Whqhxhs2hx =

L∑h=1

Wh

(Xh − xh

),

λ0

L∑h=1

Whqhs2hx + λ1

L∑h=1

Whqhxhs2hx + λ2

L∑h=1

Whqhs4hx =

L∑h=1

Wh (S2hx − s2

hx) .

Now, solving the above system for λ0, λ1 and λ2, we get

λ0 =

A1

L∑h=1

Wh

(Xh − xh

)+ A2

L∑h=1

Wh (S2hx − s2

hx)

B, (5.25)

λ1 =

A3

L∑h=1

Wh

(Xh − xh

)+ A4

L∑h=1

Wh (S2hx − s2

hx)

B, (5.26)

λ2 =

A4

L∑h=1

Wh

(Xh − xh

)+ A5

L∑h=1

Wh (S2hx − s2

hx)

B, (5.27)

where the constants A1,A2,A3,A4,A5 and B are defined in (5.21)-(5.22).

Substituting λ0, λ1 and λ2 in (5.24), we have

W h = Wh +

A1Whqh + A3Whqhxh + A4Whqhs2hx

B

L∑h=1

Wh

(Xh − xh

)+A2Whqh + A4Whqhxh + A5Whqhs

2hx

B

L∑h=1

Wh

(S2hx − s2

hx

). (5.28)

Substituting (5.28) in (5.14), we obtain the GREG estimator of population mean Y ,as defined in (5.18).

Remark 10. The Estimator II in (5.18) includes the constraints on the calibratedweights, known mean and variance.

89

As discussed in Section (5.2.1), the estimated variance of the estimator yst is statedin the following theorems:

Theorem 30. An estimator of the variance of yst is given by

v (yst) =L∑h=1

W 2h (1− fh)nh

s2eh, (5.29)

where s2eh = (nh − 3)−1∑nh

i=1 e2hi, ehi = (yhi−yh)−β1(xhi−xh)−β2 (xhi − xh)2 − s∗2hx

and s∗2hx = n−1h

∑nh

i=1 (xhi − xh)2 .

Proof. Applying Theorem 27 yields (5.29).

Theorem 31. The lower level calibration approach yields another estimator of thevariance of yst as

vLL (yst) =L∑h=1

(W h )2 (1− fh)nh

s2eh. (5.30)

Proof. Replacing the design weights Wh with the calibrated weights W h in (5.29)

yields (5.30).

5.2.3 Estimator III

To propose the Estimator III, we redefine the stratified estimator yst =L∑h=1

Whyh,

where yh = n−1h

nh∑i=1

yhi as:

yst =L∑h=1

Wh

nh

nh∑i=1

yhi =L∑h=1

nh∑i=1

Vhyhi, (5.31)

where, Vh are the ratio of design weights (Wh) and sample size (nh), that is,

Vh =Wh

nh. (5.32)

Using (5.31), we obtained the Estimator III as another estimator of the populationmean Y in stratified sampling as discussed in Theorem 32.

90

Theorem 32. A new calibration estimator of the population mean in stratified sam-pling is defined as

y⊗ =L∑h=1

nh∑i=1

V ⊗hi yhi, (5.33)

where V ⊗hi is the calibrated values of Vh. Then, the calibrated weights V ⊗hi are sochosen that the CS distance function

L∑h=1

nh∑i=1

(V ⊗hi − Vh

)2

Vhqhi(5.34)

is minimum subject to the calibration constraint

L∑h=1

nh∑i=1

V ⊗hixhi = X. (5.35)

This leads to a new GREG estimator of population mean, defined as

y⊗ = yst + β(X − xst

)(5.36)

where

β =

L∑h=1

nh∑i=1

Vhxhiyhiqhi

L∑h=1

nh∑i=1

Vhx2hiqhi

. (5.37)

Proof. The Lagrange function L is given by

L =L∑h=1

nh∑i=1

(V ⊗hi − Vh

)2

Vhqhi− 2λ

(L∑h=1

nh∑i=1

V ⊗hixhi − X

). (5.38)

Setting ∂L/∂V ⊗hi = 0, we have

V ⊗hi = Vh + Vhλxhiqhi = Vh (1 + λxhiqhi) . (5.39)

Using (5.39) in (5.35), we have

λ =X − xst

L∑h=1

nh∑i=1

Vhx2hiqhi

(5.40)

91

where

xst =L∑h=1

Whxh. (5.41)

Now, substituting (5.40) in (5.39), we obtain

V ⊗hi = Vh +Vhxhiqhi

L∑h=1

nh∑i=1

Vhx2hiqhi

(X − xst

). (5.42)

Finally, substituting (5.42) in (5.33), we obtain the GREG estimator of the popula-tion mean defined in (5.36).

Remark 11. The Estimator III in (5.36) takes only a constraint on mean.

The estimated variance of y⊗ is discussed in the theorems below.

Theorem 33. An estimator of the variance of y⊗ is given by

v(y⊗)

=L∑h=1

W 2h (1− fh)nh

s2eh, (5.43)

where s2eh = (nh − 1)−1∑nh

i=1 e2hi and ehi = (yhi − yh)− β(xhi − xh) with β as defined

in (5.37).

Proof. Applying Theorem 27 yields (5.43).

Theorem 34. The lower level calibration approach yields another estimator of thevariance of y⊗as

vLL(y⊗)

=L∑h=1

nh∑i=1

(V ⊗hi)2nh (1− fh)nh − 1

e2hi. (5.44)

Proof. The estimated variance of y⊗ in (5.43) can be written as

v(y⊗)

=L∑h=1

nh∑i=1

(Vh)2 nh (1− fh)nh − 1

e2hi. (5.45)

Now, replacing the weights Vh with calibrated weights V ⊗hi in (5.45) yields (5.44).

92

5.2.4 Estimator IV

We develop this estimator by adding the following constraint on the calibratedweights V ∗hi to the problem of Estimator III,

L∑h=1

nh∑i=1

V ∗hi =L∑h=1

nh∑i=1

Vh = 1

where Vh is given by (5.32) is discussed in Theorem 35.

Theorem 35. A new calibration estimator of the population mean in stratified sam-pling is defined as

y∗ =L∑h=1

nh∑i=1

V ∗hiyhi (5.46)

where calibrated weights V ∗hi, are chosen such that the CS distance function similarto (5.34) is minimum subject to the calibration constraints

L∑h=1

nh∑i=1

V ∗hixhi = X (5.47)

andL∑h=1

nh∑i=1

V ∗hi =L∑h=1

nh∑i=1

Vh. (5.48)

This leads to a new GREG estimator of population mean, defined as

y∗st = yst + β(X − xst

), (5.49)

where

β =

L∑h=1

nh∑i=1

VhyhiqhiL∑h=1

nh∑i=1

Vhxhiqhi −L∑h=1

nh∑i=1

VhxhiyhiqhiL∑h=1

nh∑i=1

Vhqhi(L∑h=1

nh∑i=1

Vhxhiqhi

)2

−L∑h=1

nh∑i=1

VhqhiL∑h=1

nh∑i=1

Vhx2hiqhi

. (5.50)

Proof. The Lagrange function L is given by

L =L∑h=1

nh∑i=1

(V ∗hi − Vh)2

Vhqhi− 2λ1

(L∑h=1

nh∑i=1

V ∗hixhi − X

)− 2λ2

(L∑h=1

nh∑i=1

V ∗hi − 1

).

(5.51)

93

Setting ∂L/∂V ∗hi = 0, we get

V ∗hi = Vh (1 + λ1xhiqhi + λ2qhi) . (5.52)

Substituting (5.52) in (5.47)-(5.48) and solving for λ1 and λ2, we obtain

λ1 =

−L∑h=1

nh∑i=1

Vhqhi(X − xst

)(

L∑h=1

nh∑i=1

Vhxhiqhi

)2

−L∑h=1

nh∑i=1

VhqhiL∑h=1

nh∑i=1

Vhx2hiqhi

, (5.53)

and

λ2 =

L∑h=1

nh∑i=1

Vhxhiqhi(X − xst

)(

L∑h=1

nh∑i=1

Vhxhiqhi

)2

−L∑h=1

nh∑i=1

VhqhiL∑h=1

nh∑i=1

Vhx2hiqhi

. (5.54)

Then, substituting λ1 and λ2 in (5.52), we have

V ∗hi = Vh +

VhqhiL∑h=1

nh∑i=1

Vhxhiqhi − VhxhiqhiL∑h=1

nh∑i=1

Vhqhi(L∑h=1

nh∑i=1

Vhxhiqhi

)2

−L∑h=1

nh∑i=1

VhqhiL∑h=1

nh∑i=1

Vhx2hiqhi

(X − xst

). (5.55)

Finally, substituting (5.55) in (5.46), we obtain the GREG estimator defined in(5.49).

The estimated variance of y∗ is discussed in the theorems below.

Theorem 36. An estimator of the variance of y∗ is given by

v (y∗) =L∑h=1

W 2h (1− fh)nh

s2eh, (5.56)

where s2eh = (nh − 1)−1∑nh

i=1 e2hi and ehi = (yhi − yh)− β(xhi − xh) with β as defined

in (5.50).

Proof. Applying Theorem 21 yields (5.56).

Theorem 37. The lower level calibration approach yields another estimator of the

94

variance of y∗ as

vLL (y∗) =L∑h=1

nh∑i=1

(V ∗hi)2 nh (1− fh)nh − 1

e2hi. (5.57)

Proof. Using Theorem 34 we obtain (5.57).

5.2.5 Estimator V

To develop this estimator, we define the stratified estimator yst =L∑h=1

Whyh of the

population mean Y, as

yst =L∑h=1

Wh

nh

nh∑i=1

yhi =L∑h=1

Vhyh (5.58)

whereVh =

Wh

nhand yh=

nh∑i=1

yhi.

Then, using (5.58), the Estimator V is developed as discussed in Theorem 38.

Theorem 38. A new calibration estimator of the population mean Y in stratifiedsampling is defined as

y†st =L∑h=1

V †h yh, (5.59)

where V †h are the calibrated weights of the ratio of design weights and sample size,that is Vh = Wh/nh. Then, the optimum weights are obtained by minimizing the CSdistance function

L∑h=1

(V †h − Vh

)2

Vhqh

subject to the calibration constraint

L∑h=1

V †hxh = X. (5.60)

This leads to a new GREG estimator of the population mean Y , given by

y†st = yst + β†(X − xst

), (5.61)

where

95

β† =

L∑h=1

Vhqhxhyh

L∑h=1

Vhqhx2h

(5.62)

and

xh =

nh∑i=1

xhi.

Proof. The Lagrange function L is defined as

L =L∑h=1

(V †h − Vh

)2

Vhqh− 2λ

(L∑h=1

V †hxh − X

). (5.63)

Setting ∂L/∂V †h = 0, we obtain

V †h = Vh + λVhqhxh. (5.64)

Then, substituting (5.64) in (5.60) and solving for λ, we get

λ =

(L∑h=1

Vhqhx2h

)−1 (X − xst

). (5.65)

Again, substituting (5.65) in (5.64), we obtain the OCW V †h as

V †h = Vh + Vhqhxh

(L∑h=1

Vhqhx2h

)−1 (X − xst

).

Finally, substituting V †h in (5.59), we obtain the GREG estimator of population meanY , given by (5.61).

The following theorems give the estimated variance of y†st .

Theorem 39. An estimator of the variance of y†st is given by

v(y†st

)=

L∑h=1

W 2h (1− fh)nh

s2eh, (5.66)

where s2eh = (nh − 1)−1∑nh

i=1 e2hi and ehi = (yhi − yh)− β†(xhi − xh).

96

Proof. Applying Theorem 21 yields (5.66).

Theorem 40. The lower level calibration approach yields another estimator of vari-ance of y†st as

vLL

(y†st

)=

L∑h=1

(V †h

)2

nh (1− fh) s2eh. (5.67)

Proof. The estimated variance of y†st in (5.66) can be written as

v(y†st

)=

L∑h=1

(Vh)2 nh (1− fh) s2

eh. (5.68)

Now, replacing the weights Vh with calibrated weights V †h in (5.68) yields (5.67).

5.2.6 Estimator VI

Here, we extended the Estimator V given in Section 5.2.5 by adding the constrainton the calibrated weight as expressed below:

L∑h=1

V ‡h =L∑h=1

Vh.

Hence, a new estimator of population mean in stratified sampling is proposed inTheorem 41.

Theorem 41. A new calibration estimator of the population mean Y in stratifiedsampling is defined as

y‡st =L∑h=1

V ‡h yh, (5.69)

where V ‡h are the calibrated weights, obtained by minimizing the CS distance function

L∑h=1

(V ‡h − Vh

)2

Vhqh

subject to the calibration constraint

L∑h=1

V ‡h =L∑h=1

Vh (5.70)

97

andL∑h=1

V ‡hxh = X. (5.71)

This leads to a new GREG estimator of the population mean Y , given by

y‡st = yst + β‡(X − xst

), (5.72)

where

β‡ =

L∑h=1

VhqhxhyhL∑h=1

Vhqh −L∑h=1

VhqhyhL∑h=1

Vhqhxh

L∑h=1

VhqhL∑h=1

Vhqhx2h −

(L∑h=1

Vhqhxh

)2 . (5.73)

Proof. The Lagrange function L is defined as

L =L∑h=1

(V ‡h − Vh

)2

Vhqh− 2λ1

(L∑h=1

V ‡h −L∑h=1

Vh

)− 2λ2

(L∑h=1

V ‡hxh − X

). (5.74)

Setting ∂L/∂V ‡h = 0, we obtain

V ‡h = Vh + λ1Vhqh + λ2Vhqhxh. (5.75)

Substituting (5.75) in (5.70) and (5.71), we get

λ1 =

−L∑h=1

Vhqhxh(X − xst

)L∑h=1

VhqhL∑h=1

Vhqhx2h −

(L∑h=1

Vhqhxh

)2 (5.76)

and

λ2 =

L∑h=1

Vhqh(X − xst

)L∑h=1

VhqhL∑h=1

Vhqhx2h −

(L∑h=1

Vhqhxh

)2 . (5.77)

Then, substituting (5.76) and (5.77) in (5.75), we obtain the calibrated weights V †has

98

V ‡h = Vh +

VhqhxhL∑h=1

Vhqh − VhqhL∑h=1

Vhqhxh

L∑h=1

VhqhL∑h=1

Vhqhx2h −

(L∑h=1

Vhqhxh

)2

(X − xst

).

Hence, substituting V ‡h in (5.69), we obtain the GREG estimator of population meanY , given by (5.72).

The estimated variance of the estimator y‡st is given below.

Theorem 42. An estimator of the variance of y‡st is given by

v1

(y‡st

)=

L∑h=1

W 2h (1− fh)nh

s2eh. (5.78)

where s2eh = (nh − 1)−1∑nh

i=1 e2hi and ehi = (yhi − yh)− β‡(xhi − xh).

Proof. Applying Theorem 21 yields (5.78).

Theorem 43. The lower level calibration approach yields another estimator of vari-ance of y‡st as

vLL

(y‡st

)=

L∑h=1

(V ‡h

)2

nh (1− fh) s2eh. (5.79)

Proof. The result in (5.79) follows Theorem 40.

5.3 Numerical illustration

In this section, we will demonstrate the application and computational details of thecalibration estimators proposed in Section 5.2. To accomplish this, we use a tobaccopopulation data (Source: Agriculture Statistics 1999 reported in Singh [83]) of N =

106 countries with three variables: x1 = area (in hectares), x2 = yield (in metric tons)and y = production (in metric tons). The data has L = 10 strata with stratum sizesas Nh = 6, 6, 8, 10, 12, 4, 30, 17, 10, 3 . A sample of n = 40 countries was selectedusing proportional allocation nh = 3, 3, 3, 3, 4, 2, 11, 6, 3, 2 with SRSWOR. Supposethat an estimate of average production ( ˆY ) of tobacco crop is of interest using someknown information from an auxiliary variable x = area. Table 11 provides the sampleinformation (xh, s2

hx and yh) needed on the tobacco population.

99

Table 11: Information on tobacco population.h Nh Wh Xh S2

hx nh xh s2hx yh

1 6 0.05660 3194.5 10899652.7 3 3730.0 21019948.0 8110.02 6 0.05660 14660.0 584984730.0 3 245.0 77025.0 328.33 8 0.07547 18309.4 635958094.8 3 18988.3 634924108.3 52709.74 10 0.09434 14923.5 209817189.2 3 10578.3 86754908.3 20258.35 12 0.11321 5987.8 27842810.5 4 8786.5 58966062.3 18112.56 4 0.03774 3450.0 5876666.7 2 3800.0 16820000.0 4675.07 30 0.28302 11682.7 760238523.4 11 21573.2 1918981931.4 33845.08 17 0.16038 145162.3 124004506112.8 6 105938.5 27906093819.9 151425.09 10 0.09434 33976.1 8340765245.4 3 5716.7 65160833.3 7066.710 3 0.02830 1333.3 2963333.3 2 1700.0 5120000.0 4547.5Total 106 1 40

For this population, the known stratum means (Xh) and stratum variances (S2hx)

of the auxiliary variable are also presented in Table 11. Using a computer programdeveloped in MATLAB (Refer to Appendix B.3), the values of Lagrange multipliers (λ)and estimated betas (β) for different calibration estimators, including Singh et al.[92] and Singh [83] are computed and presented in Table 12. The OCW for all thecalibration estimators are shown in Table 13 except for Estimators III and IV aseach of the estimators produce 40 calibrated weights (since nh = 40) and are notreported. The values of the estimator ˆY and the estimated variance of the estimatorwere computed for all the estimators and are presented in Table 14. The true value ofaverage production was Y = 52444.6 and qh = qhi = 1 is used for the computations.

100

Table 12: Lagrange multipliers and estimated betas for different estimators.Estimator

(ˆY)

λ β

Singh et al. [92](y⊕st)

3.51× 10−6 1.46248Singh [83]

(yst)

−1.56× 10−1 1.393815.67× 10−6

Estimator I(y⊗st)

−5.44× 10−4 1.970722.23× 10−9 -0.000002

Estimator II (yst) 1.92× 101 2.12227−2.04× 10−3 -0.0000037.23× 10−9

Estimator III (y⊗) 1.11× 10−6 1.45819Estimator IV(y∗) 1.27× 10−6 1.44227

−3.48× 10−2

Estimator V(y†st

)5.61× 10−7 1.45753

Estimator VI(y‡st

)−8.62× 10−2 1.41783

7.52× 10−7

Table 13: OCW for different estimators.h W⊕h Wh W⊗h W h V †h V ‡h

1 0.05735 0.04898 -0.05550 0.72223 0.01899 0.017402 0.05665 0.04786 0.04908 1.11563 0.01888 0.017253 0.08050 0.07184 -0.59670 -1.04877 0.02596 0.024074 0.09784 0.08530 -0.42984 -0.06798 0.03201 0.029485 0.11670 0.10121 -0.41256 0.30904 0.02886 0.026616 0.03824 0.03267 -0.03879 0.47496 0.01895 0.017357 0.30446 0.27354 -1.82583 -2.79713 0.02915 0.028108 0.22003 0.23173 0.89416 0.96518 0.03626 0.037209 0.09623 0.08270 -0.18509 0.85190 0.03175 0.0291410 0.02847 0.02416 0.00247 0.47494 0.01418 0.01297Total 1.09648 1.00000 -2.59860 1.00000 0.25498 0.23957

101

Table 14: Estimators of Y and their estimated variance for tobacco population.No. Estimator

(ˆY)

Estimated value Estimated variance

1 Stratified (yst) 43253.06 190067503.912 Singh et al. [92]

(y⊕st)

53433.83 4656733.683 Singh [83]

(yst)

52955.79 4990678.804 Estimator I

(y⊗st)

24058.85 -5 Estimator II (yst) 17048.20 -6 Estimator III (y⊗) 53403.98 4653438.507 Estimator IV(y∗) 53293.16 4669341.71

8 Estimator V(y†st

)53399.36 4653215.55

9 Estimator VI(y‡st

)53123.02 4780038.90

From Tables 13 and 14, the following were observed from the results of the numericalillustrations with the tobacco population:

• The calibrated weights obtained were negative and extreme for the Estimators4 and 5 (i.e. y⊗st and yst).

• The estimators were relatively close to the true value of the population mean,except the Estimators 1, 4 and 5 (i.e. yst, y⊗st and yst).

• The estimated variance of the Estimators 4 and 5 (i.e. y⊗st and yst) were notobtained since the sample size in one of the strata was less than 3.

5.4 Simulation study

In this section, a detailed simulation study is carried out to investigate the efficiencyof the proposed calibration estimators in stratified sampling, using four real popula-tions (Agriculture, Tobacco, Labor and ST130) as discussed below. It was assumedthat qh = qhi = 1 in the simulation study for the computation of the calibrationestimators.

5.4.1 Populations

Agriculture Population:

The agriculture census data was from the 2002 and 1997 Agricultural Censuses inIowa State conducted by the National Agricultural Statistics Service, USDA, Wash-ington D.C. (source: http://www. agcensus.usda.gov/ reported in Khan et al. [40]).

102

This data was collected for N = 99 counties in Iowa State with two variables: x =

the quantity of corn harvested in 1997 (in metric ton) and y = the quantity of cornharvested in 2002 (in metric ton). The correlation coefficient between the variablesin the agriculture population was ρx,y = 0.976.

Tobacco Population:

The tobacco data (Source: Agriculture Statistics 1999 reported in Singh [83]) wasdescribed in Section 5.3 for numerical illustrations. Further, the various correlationcoefficients between the variables in the tobacco population are: ρx1,y = 0.991, ρx2,y =

0.030, ρx2,x2 = −0.008 and ρx1x2,y = 0.992.

Labor Population:

The labor force data (Source: September 1976 Current population survey in theUnited States reported in Valliant et al. [103]) of N = 478 persons with three vari-ables: x1 = hoursPerWk (Usual number of hours worked per week), x2 = age (inyears) and y = WklyWage (Usual amount of weekly wages in U.S. dollars). The cor-relation coefficients between the variables in the labor population are: ρx1,y = 0.486,

ρx2,y = 0.191, ρx2,x2 = 0.132 and ρx1x2,y = 0.502.

ST130 Population:

The ST130 data has been collected from N = 255 students enrolled in the courseST130: Basic Statistics in Semester 2, 2015 at the University of the South Pacificwith three variables: x1 = coursework (marks from continuous assessments), x2 =

final exam (marks from final exam) and y = total mark (total mark obtained in thecourse). The correlation coefficients between the variables in the ST130 populationare: ρx1,y = 0.890, ρx2,y = 0.861, ρx2,x2 = 0.535 and ρx1x2,y = 1.000.

5.4.2 Simulation setup and results

Agriculture Population

To perform the simulation study, the agriculture population was divided into 4 strata,with stratum sizes Nh = 8, 34, 45, 12 as reported in Khan et al. [40]. Suppose thatan estimate of average production of corn harvested in 2002 (Y ) is of interest usingthe auxiliary variable x = the quantity of corn harvested in 1997. Table 15 providesthe information needed such as (Nh, Wh, Xh and S2

hx) on the agriculture population.

103

Table 15: Information on agriculture population.h Nh Wh Xh S2

hx nh

1 8 0.08081 518494.5 21601503189.8 42 34 0.34343 498036.4 19734615816.7 173 45 0.45455 309203.93 27129658750.0 234 12 0.12121 430366.2 17258237358.5 6Total 99 1 50

From this population, B = 100, 000 samples of size 50 counties was selected us-ing proportional allocation nh = 4, 17, 23, 6 with SRSWOR. For the purpose ofcomparison of estimators the following measures were computed:

1. The empirical bias (Bias) given as

Bias = B−1

B∑j=1

ˆYj − Y (5.80)

where B denotes the number of samples or the number of iterations of thesimulation, ˆYj is the estimator of the population mean Y based on jth sampleand Y is the population mean.

2. The empirical mean square error (MSE) given as

MSE = B−1

B∑j=1

[ˆYj − Y

]2

. (5.81)

3. The empirical variance (Variance) given as

Variance = MSE− (Bias)2. (5.82)

4. The percentage relative efficiency (PRE) of ˆY with respect to the stratifiedestimator (yst) given as

PRE =B−1

∑Bj=1

[(yst)j − Y

]2

B−1∑B

j=1

[ˆYj − Y

]2 × 100. (5.83)

It should be noted that the PRE of an estimator(

ˆY)is calculated with respect to

104

the stratified estimator (yst). The values of Bias, Variance, MSE and PRE for all theproposed estimators discussed in Sections 5.2.1-5.2.4, considered in the simulationwere obtained using a computer program developed in MATLAB (Refer to AppendixB.3). The true value of the average production of corn harvested in 2002 used in thissimulation was Y = 474973.90. The simulation results on the agriculture populationare presented in the Table 16. The table also includes simulation results for Singhet al. [92], Singh [83] and the stratified estimators for the purpose of comparison.

Table 16: Simulation results of agriculture population.No. Estimator

(ˆY)

Bias MSE Variance PRE

1 Stratified (yst) 3.62 339392988.06 339392974.9 1002 Singh et al. [92]

(y⊕st)

-16.38 19774354.75 19774086.33 1716.333 Singh [83]

(yst)

113.37 21205857.47 21193004.06 1600.474 Estimator I

(y⊗st)

-905.21 40684299.33 39864891.95 834.215 Estimator II (yst) -1112.98 559655977.46 558417254.4 60.646 Estimator III (y⊗) -1.26 19803884.57 19803882.98 1713.777 Estimator IV(y∗) 99.91 20192711.54 20182729.9 1680.77

8 Estimator V(y†st

)-14.93 19703608.68 19703385.66 1722.49

9 Estimator VI(y‡st

)1.38 19872065.81 19872063.91 1707.89

Tobacco Population

For the tobacco population, we had 10 strata, with the stratum sizesNh = 6, 6, 8, 10, 12, 4, 30, 17, 10, 3 as reported in (Singh [83]. Suppose that an es-timate of average production ( ˆY ) of tobacco crop is of interest using some knowninformation from a auxiliary variable x = area. The required information for thispopulation is provided in Table 11 and also discussed in Section 5.4.1.

From the tobacco population, B = 100, 000 samples of size 40 countries was selectedfrom each stratum using proportional allocation nh = 3, 3, 3, 3, 4, 2, 11, 6, 3, 2 withSRSWOR. Again for the purpose of comparison of estimators, the values of Bias,Variance, MSE and PRE were computed for all the proposed estimators, includingSingh et al. [92], Singh [83] and stratified estimators, using a MATLAB computer pro-gram. The simulation results for the tobacco population are presented in the Table17. The true value of the the average tobacco production used in this simulation wasY = 52444.56.

105

Table 17: Simulation results of tobacco population.No. Estimator

(ˆY)

Bias MSE Variance PRE

1 Stratified (yst) 201.25 1070645868.95 1070605366.94 100.002 Singh et al. [92]

(y⊕st)

-2441.49 47126396.12 41165520.42 2271.863 Singh [83]

(yst)

-3103.33 60586155.02 50955498.37 1767.154 Estimator I

(y⊗st)

-20747.68 32286483974.34 31856017622.24 3.325 Estimator II (yst) -19322.41 40679964978.99 40306609452.02 2.636 Estimator III (y⊗) -3283.59 54457962.39 43675976.26 1966.007 Estimator IV(y∗) -3519.51 60361570.34 47974646.60 1773.72

8 Estimator V(y†st

)-2620.14 46231077.06 39365961.04 2315.86

9 Estimator VI(y‡st

)-3183.15 54886880.90 44754446.48 1950.64

Labor Population

The labor population had L = 3 strata, Nh = 210, 212, 56 as reported in Val-liant et al. [103]. Suppose that from this population an estimate of average weeklywages ( ˆY ) is of interest using some known information on a auxiliary variablex = hoursPerWk. Then, the information required to compute the calibration es-timators for the labor population is provided in Table 18.

Table 18: Information on labor population.h Nh Wh Xh S2

hx nh

1 210 0.43933 38.5 113.7 262 212 0.44351 38.0 132.6 273 56 0.11715 37.6 170.6 7Total 478 1 60

To carry out the simulation, B = 100, 000 samples of size 60 persons was selectedusing proportional allocation nh = 26, 27, 7 with SRSWOR. Then, for the purposeof comparison of estimators, the values of Bias, Variance, MSE and PRE were com-puted for all the estimators discussed earlier using a MATLAB computer program. Forthis population, the true value of the population mean needed for the simulationswas Y = 294.60. The simulation results for the labor population are presented inthe Table 19.

106

Table 19: Simulation results of labor population.No. Estimator

(ˆY)

Bias MSE Variance PRE

1 Stratified (yst) 0.12209 535.42 535.40 100.002 Singh et al. [92]

(y⊕st)

0.03120 410.55 410.54 130.423 Singh [83]

(yst)

0.89048 2956.70 2955.91 18.114 Estimator I

(y⊗st)

-0.89430 2967.13 2966.33 18.045 Estimator II (yst) 83.54261 125286557.67 125279578.31 0.006 Estimator III (y⊗) 0.11130 410.62 410.61 130.397 Estimator IV(y∗) 0.95043 419.12 418.22 127.75

8 Estimator V(y†st

)0.01642 410.42 410.42 130.46

9 Estimator VI(y‡st

)-0.05108 411.04 411.04 130.26

ST130 Population

The ST130 population was divided into L = 5 strata with stratum sizesNh = 60, 60, 60, 60, 15 . Suppose that from this population an estimate of averagetotal mark ( ˆY ) is of interest using some known information on the auxiliary vari-able x = coursework. For this population, information required for the calibrationestimators is provided in Table 20.

Table 20: Information on ST130 population.h Nh Wh Xh S2

hx nh

1 60 0.23529 33.6 67.3 152 60 0.23529 34.7 70.3 153 60 0.23529 35.6 58.6 154 60 0.23529 34.1 94.9 155 15 0.05882 32.6 25.4 4Total 255 1 64

For simulations, B = 100, 000 samples of size 64 persons was selected using propor-tional allocation nh = 15, 15, 15, 15, 4 with SRSWOR. Again for the purpose ofcomparison of estimators, the values of Bias, Variance, MSE and PRE were com-puted for all the estimators using a MATLAB computer program. The true value of thepopulation mean used in this simulation was Y = 53.47 and results are presented inthe Table 21.

107

Table 21: Simulation results of ST130 population.No. Estimator

(ˆY)

Bias MSE Variance PRE

1 Stratified (yst) -0.00550 2.26390 2.26387 100.002 Singh et al. [92]

(y⊕st)

-0.00080 0.47282 0.47282 478.813 Singh [83]

(yst)

-0.01196 0.75874 0.75860 298.384 Estimator I

(y⊗st)

-0.03579 1.00161 1.00033 226.035 Estimator II (yst) -0.04441 1.88015 1.87817 120.416 Estimator III (y⊗) -0.00256 0.47241 0.47241 479.227 Estimator IV(y∗) -0.02492 0.47723 0.47661 474.39

8 Estimator V(y†st

)-0.00232 0.47194 0.47194 479.70

9 Estimator VI(y‡st

)-0.00092 0.47266 0.47266 478.97

5.5 Comparison study

In this section, we compare the estimators, based on the results obtained from thesimulation study carried out in Sections 5.3 and 5.4. The comparison study ofestimators is based on the four measures: Bias, MSE, Variance and PRE as discussedin (5.80)-(5.83). The estimators under comparison are as listed below:

1. Stratified estimator, yst [see (1.1)].

2. Singh et al. [92] estimator, y⊕st [see (2.65)].

3. Singh [83] estimator, yst [see (2.86)].

4. Proposed Estimator I, y⊗st [see (5.4)].

5. Proposed Estimator II, yst [see (5.18)].

6. Proposed Estimator III, y⊗ [see (5.36)].

7. Proposed Estimator IV, y∗ [see (5.49)].

8. Proposed Estimator V, y†st [see (5.61)].

9. Proposed Estimator VI, y‡st [see (5.72)].

As noted earlier, the percentage relative efficiency (PRE) for each estimator is cal-culated with respect to the stratified estimator.

108

Based on the results presented in Tables 16, 17, 19 and 21, the following were someobservations made from the simulation study based on 100,000 samples taken fromeach of the four real populations:

• The MATLAB program developed for the simulation was very efficient as theCPU times recorded to execute the programs were 17.65, 52.40, 74.33 and 44.04minutes for the agriculture, tobacco, labor and ST130 populations, respectively.

• Most estimators have negative bias which implies that the estimator has underestimated the population mean.

• The estimators yst, y⊗st, yst, y∗ and y‡st (i.e. the estimators 3, 4, 5, 7 and 9)

produced negative and extreme calibrated weights for many samples and inalmost all the populations. Some estimators such as yst, y

⊗st and yst (i.e. the

estimators 3, 4, and 5) did not converge to the true population mean in someunlucky samples in the populations. It was also noted that these estimatorswere mainly those which used the constraint such as (2.79), (5.15), (5.48) and(5.70) on the calibrated weights or those which used the constraints such as(5.3) and (5.17), on the stratum variance information of the auxiliary variableas additional calibration constraint.

• The estimators yst, yst, y∗ and y‡st (i.e. the estimators 3, 5, 7 and 9) that in-

corporated the calibration constraint on weights given in (2.79), (5.15), (5.48),(5.70) and (5.70) always performed worse than those which did not use such aconstraint. These estimators produced estimates with larger bias, MSE, vari-ance and smaller PRE compared to other estimators.

• Similarly the estimators y⊗st and yst (i.e. the estimators 4 and 5) that incor-porated the stratum variance information of the auxiliary variable (5.3) and(5.17), always performed worse than the other estimators when comparing thevalues of bias, variance, MSE and PRE.

• The relative Bias of the calibration estimators y⊕st, y⊗ and y†st (i.e. the estima-tors 2, 6 and 8) have been found to be relatively small in all the populations.These estimators also produce estimates with smaller variance, MSE and largerPRE compared to other estimators. These are those estimators which did notuse the constraints on weights and/or variance.

Finally, amongst all the estimators it was found that the variance and MSE of theproposed Estimator V

(y†st

)given in (5.61) that is obtained by calibrating the ratio

109

Vh = Wh/nh are the smallest. Also the PRE given in Tables 16, 17, 19 and 21 showthat y†st was consistently most efficient compared to other estimators for all the realpopulations. Thus, the estimator that calibrate the ratio of the design weights andsample size (i.e. Vh = Wh/nh) is found to be the most efficient estimate.

5.6 Summary

In this chapter, we have proposed some new calibration estimators of populationmean, using the information on a single auxiliary variable in stratified sampling.The problem of determining the OCW was to minimize the CS type distance func-tion subject to some calibration constraints on the auxiliary variable and the problemwas then solved using the Lagrange multiplier technique. The calibration estimatorsof population mean was derived in the form of Generalized Regression (GREG) andtheir estimated variance was also discussed.

We demonstrated the application and computation details of the proposed estima-tors using a numerical example. Later, a simulation study was performed based on100,000 samples from each of the four real populations to investigate the efficiencyof the proposed calibration estimator to others. It was noted from the simulationstudy that the calibration constraint on mean information of the auxiliary variable issufficient to improve the calibration estimators and additional constraints on weightand/or variance are unnecessary as these constraints increase the chance of ineffi-ciency and sometimes even non convergence of the estimator. The calibration esti-mators y⊕st, y⊗ and y†st (i.e. 2, 6 and 8) were found to be the efficient estimators sincethe bias, variance and MSE were relatively small, with larger PRE in all populations.

The proposed estimator y†st which calibrates Vh = Wh/nh was found to be the bestestimator and also most consistent over the other calibration estimators in improvingthe estimate in stratified sampling. Thus, we may conclude that the calibration ofratio of design weights and sample size, instead of calibrating only design weight,provides better estimate.

110

Chapter 6

Multivariate Calibration Estimatorsin Stratified Sampling

6.1 Introduction

In surveys, when more than one auxiliary variable is available, the precision of theestimate can further be increased by adjusting the design weights based on the infor-mation on all the auxiliary variables (Singh [83]). In this chapter, we propose somemultivariate calibration estimators with the aid of known mean information on sev-eral auxiliary variables for improving the precision of the estimate of the populationmean in stratified random sampling. The focus is also to investigate whether addi-tional information, particularly the information on known variance and calibratedweights improve the multivariate calibration estimates. In Section 6.2, we derive thevarious multivariate calibration estimators of population mean in stratified samplingusing different combinations of calibration constraints. For each estimator, we for-mulate the problem of determining OCW as an MPP that seeks the minimization ofCS type distance function. Then, the MPP is solved for the OCW using a Lagrangemultiplier technique. The estimators are written in a Generalized Multiple Regres-sion (GMREG) form and the estimated variance of these estimators are also derived.A numerical example is provided in Section 6.3 to demonstrate the application andcomputational details of the proposed multivariate calibration estimators. Furtherto investigate the efficiency of the estimators, a simulation study is carried out whichis presented in Section 6.4. In Section 6.5, we provide a comparative analysis amongthe proposed multivariate estimators and also with the corresponding univariate esti-

111

mators discussed in Chapter 5. Finally, Section 6.6 summarizes the results obtainedin this chapter.

6.1.1 Publications

The results presented in this chapter have been published in Rao et al. [68] and Raoet al. [70].

6.1.2 Notations

When several auxiliary variables are used the following notations will be helpful inthe derivations of the multivariate calibration estimators.

Let xhij denote the values of the ith sample or population unit of the jth auxiliaryvariable (xj) in the hth stratum. Let the estimation of unknown population meanY be of interest using the information from p auxiliary variables xj, j = 1, 2, . . . , p,

where p is the number of auxiliary variables. Assume that the population means Xj =∑Lh=1WhXhj are accurately known, where Xhj = N−1

h

∑Nh

i=1 xhij is the populationmean of jth auxiliary variable in hth stratum. Also, if the information

∑Lh=1WhS

2hxj

are known where, S2hxj = (Nh − 1)−1∑N

i=1

(xhij − Xhj

)2 is population variance ofjth auxiliary variable in hth stratum. The purpose of this study is to estimatethe population mean Y =

∑Lh=1WhYh by using the information from p auxiliary

variables xj, j = 1, 2, . . . , p.

6.2 Proposed multivariate calibration estimators

6.2.1 Estimator I

If Xj; j = 1, 2, . . . , p are accurately known, we propose a estimator of the populationmean Y under stratified sampling defined as

y(1)st =

L∑h=1

W(1)h yh (6.1)

with the calibrated weights W (1)h , chosen such that the CS distance function

112

p∑j=1

L∑h=1

(W

(1)h −Wh

)2

Whqhj(6.2)

is minimum, subject to the calibration constraints

L∑h=1

W(1)h xhj = Xj, for j = 1, 2, . . . , p, (6.3)

where xhj = n−1h

∑nh

i=1 xhij is the sample mean of jth auxiliary variable in hth stra-tum and nh; h = 1, 2, . . . , L is the sample size in hth stratum.

By rewriting (6.2) and (6.3), the problem of determining the optimum calibratedweight W (1)

h can be expressed as an MPP as given below:

MinimizeL∑h=1

(W

(1)h −Wh

)2

WhQh

subject toL∑h=1

W(1)h xh = X,

(6.4)

where Qh =

(∑pj=1

1

qhj

)−1

in (6.4) are suitability chosen weights which determine

the different forms of the estimator (Singh et al. [92] and Singh [83]). Also thatX =

(X1, X2, . . . , Xp

)′, xh = (xh1, xh2, . . . , xhp)

′are the (p × 1) column vectors of

the population mean and the sample means in hth stratum of the auxiliary variables,respectively.

To solve the MPP in (6.4) using a Lagrange multipliers technique, we associate aLagrange multiplier −2λ, where λ= (λ1, λ2, . . . , λp)

′is a (p × 1) column vector.

Then, the Lagrange function L is defined as:

L =L∑h=1

(W(1)h −Wh)

2

WhQh

− 2λ′

(L∑h=1

W(1)h xh − X

). (6.5)

The necessary conditions for the solution of the problem are

∂L

∂W(1)h

=∂L

∂λ= 0. (6.6)

Let(W

(1)1 ,W

(1)2 , . . . ,W

(1)L ,λ

)be a solution that satisfies the equation (6.6). It is

113

verified that the objective function in (6.4) is a convex function of W (1)h and the

calibration equations are linear. Then, the following theorem gives the conditionthat is sufficient for the point

(W

(1)1 ,W

(1)2 , . . . ,W

(1)L

)to be an optimum solution to

the problem (6.4).

Theorem 44. If the objective function is convex and each constraint is linear func-tion, then any point

(W

(1)1 ,W

(1)2 , . . . ,W

(1)L ,λ

)satisfying (6.6) will yield an optimum

solution at(W

(1)1 ,W

(1)2 , . . . ,W

(1)L

)to (6.4).

Proof. See Bazaraa et al. [6] and Winston et al. [104].

The determination of the OCW W(1)h , using the Lagrange multiplier technique dis-

cussed above is illustrated in Theorem 45, when information on the auxiliary vari-ables xj, (j = 1, 2, . . . , p) is available.

Theorem 45. The optimum solution to the MPP (6.4), that is, the OCW W(1)h is

given by

W(1)h = Wh +WhQhx

′

hλ, (6.7)

where

λ = T −1(X− xst

), (6.8)

if

T =L∑h=1

WhQhxhx′

h, (6.9)

is an invertible (p× p) square matrix and xst =∑L

h=1Whxh is a (p × 1) columnvector.

Proof. Using Lagrange multiplier technique, the function to be minimized is

L =L∑h=1

(W(1)h −Wh)

2

WhQh

− 2λ′

(L∑h=1

W(1)h xh − X

).

The necessary conditions given in (6.6) are

∂L

∂W(1)h

= 2(W

(1)h −Wh)

WhQh

− 2λ′xh = 0 (6.10)

114

and∂L

∂λ= −2

(L∑h=1

W(1)h xh − X

)= 0. (6.11)

From (6.10) we have

W(1)h = Wh +WhQhx

′

hλ.

Substituting (6.7) in (6.11), we obtain λ = T −1(X− xst

), which is the (p × 1)

vector of the Lagrange multipliers, where T =∑L

h=1WhQhxhx′

h is assumed to be(p× p) invertible matrix and xst =

∑Lh=1Whxh is a (p× 1) vector which denotes the

Horvitz Thompson estimator of X.

Hence, the Estimator I in (6.1) is obtained and the Generalized Multiple Regression(GMREG) form of the estimator is derived as discussed in Theorem 46.

Theorem 46. The estimator y(1)st leads to a GMREG estimator of the population

mean defined asy

(1)st = yst + β

(X− xst

), (6.12)

where

β =L∑h=1

WhQhyhx′

hT −1 (6.13)

is a (1× p) row vector of the form(β1, β2, . . . , βp

).

Proof. Substituting (6.8) in (6.7), the calibrated weights can be written as

W(1)h = Wh +WhQhx

′

hT −1(X− xst

). (6.14)

Finally, substitution of W (1)h from (6.14) in (6.1) leads to a GMREG estimator of

the population mean given by (6.12).

Remark 12. The multivariate Estimator I in (6.12) includes the constraints on meanof the auxiliary variables.

Now we provide the estimated variance of y(1)st in Theorems 47 and 48.

Theorem 47. The estimate of the variance of y(1)st is

v(y

(1)st

)=

L∑h=1

W 2h (1− fh)nh

s2eh, (6.15)

115

where, s2eh = (nh − p)−1∑nh

i=1 e2hi and ehi = (yhi − yh)−

∑pj=1 βj(xhij − xhj).

Proof. Extending the results in Theorem 21 yields (6.15).

Theorem 48. The lower level calibration estimate of the variance of y(1)st is

vLL

(y

(1)st

)=

L∑h=1

(W

(1)h

)2

(1− fh)

nhs2eh. (6.16)

Proof. Replacing the design weights Wh with the calibrated weights W (1)h in (6.15)

yields (6.16).

The estimator proposed in (6.12) uses the information on p auxiliary variables. Thefollowing results can be obtained from the estimator (6.12):

Corollary 1. When a single auxiliary information is available, that is p = 1, it caneasily be verified that the proposed GMREG estimator in (6.12) reduces to the Singhet al. [92] estimator given by (2.65).

Corollary 2. If the information on two auxiliary variable is available, that is, whenp = 2 the calibrated weights in (6.7) is reduced to

W(1)h = Wh +WhQh (λ1xh1 + λ2xh2) (6.17)

where

λ1 =

−L∑h=1

WhQhx2h2

(X1 −

∑Lh=1Whxh1

)+

L∑h=1

WhQhxh1xh2

(X2 −

∑Lh=1Whxh2

)(

L∑h=1

WhQhxh1xh2

)2

−L∑h=1

WhQhx2h1

L∑h=1

WhQhx2h2

,

(6.18)and

λ2 =

L∑h=1

WhQhxh1xh2

(X1 −

∑Lh=1Whxh1

)−

L∑h=1

WhQhx2h1

(X2 −

∑Lh=1Whxh2

)(

L∑h=1

WhQhxh1xh2

)2

−L∑h=1

WhQhx2h1

L∑h=1

WhQhx2h2

.

(6.19)

116

The calibrated weights in (6.17) can be written as

W(1)h = Wh +

WhQhxh2

(L∑h=1

WhQhxh1xh2

)−WhQhxh1

(L∑h=1

WhQhx2h2

)(

L∑h=1

WhQhxh1xh2

)2

−L∑h=1

WhQhx2h1

L∑h=1

WhQhx2h2

(X1 −

L∑h=1

Whxh1

)

+

WhQhxh1

(L∑h=1

WhQhxh1xh2

)−WhQhxh2

(L∑h=1

WhQhx2h1

)(

L∑h=1

WhQhxh1xh2

)2

−L∑h=1

WhQhx2h1

L∑h=1

WhQhx2h2

(X2 −

L∑h=1

Whxh2

).

(6.20)

Using the calibrated weights (6.20), the estimator in (6.12) is reduced to

y(1)st = yst + β1

(X1 −

L∑h=1

Whxh1

)+ β2

(X2 −

L∑h=1

Whxh2

), (6.21)

where

β1 =

L∑h=1

WhQhxh2yhL∑h=1

WhQhxh1xh2 −L∑h=1

WhQhxh1yhL∑h=1

WhQhx2h2(

L∑h=1

WhQhxh1xh2

)2

−L∑h=1

WhQhx2h1

L∑h=1

WhQhx2h2

,

and

β2 =

L∑h=1

WhQhxh1yhL∑h=1

WhQhxh1xh2 −L∑h=1

WhQhxh2yhL∑h=1

WhQhx2h1(

L∑h=1

WhQhxh1xh2

)2

−L∑h=1

WhQhx2h1

L∑h=1

WhQhx2h2

.

6.2.2 Estimator II

In this section, we modify modify the Estimator I given in Section 6.2.1 by addingthe constraint on calibrated weights as suggested by Singh [83]. Hence, EstimatorII, a calibration estimator of population mean with p auxiliary auxiliary variablesxj, j = 1, 2, . . . , p in stratified sampling is proposed as:

117

y(2)st =

L∑h=1

W(2)h yh (6.22)

where the calibrated weights W (2)h are suitably chosen weights such that the CS

distance function is minimum, subject to the calibration constraints

L∑h=1

W(2)h =

L∑h=1

Wh (6.23)

and

L∑h=1

W(2)h xhj = Xj, for j = 1, 2, . . . , p. (6.24)

The calibration constraints (6.23) and (6.24) can be collectively written as

L∑h=1

W(2)h x∗h = X∗, (6.25)

where X∗ =(1, X1, X2, . . . , Xp

)′and x∗h = (1, xh1, xh2, . . . , xhp)

′are defined as

(p+ 1)× 1 column vectors on the mean information.

Then, the problem of determining the OCW W(2)h may be rewritten as an MPP as

given below:

MinimizeL∑h=1

(W

(2)h −Wh

)2

WhQh

subject toL∑h=1

W(2)h x∗h = X∗,

(6.26)

where Qh =

(∑pj=1

1

qhj

)−1

.

In the similar way as discussed in Section 6.2.1, to solve (6.26) we associate a La-grange multiplier −2λ∗ where λ∗= (λ0, λ1, λ2, . . . , λp)

′is a (p+1)×1 column vector.

The determination of calibrated weights W (2)h is discussed in the Theorem 49.

Theorem 49. The optimum solution to the MPP (6.26), that is the OCW W(2)h that

minimize the CS distance subject to (6.25) is given by

W(2)h = Wh +WhQh (x∗h)

′λ∗,

118

whereλ∗ = T −1

(X∗ − x∗st

),

if

T =L∑h=1

WhQhx∗h (x∗h)

′,

is an invertible (p+ 1)× (p+ 1) square matrix and x∗st =∑L

h=1Whx∗h is a (p+1)×1

column vector.

Proof. The Lagrange function L is defined as

L =L∑h=1

(W(2)h −Wh)

2

WhQh

− 2 (λ∗)′

(L∑h=1

W(2)h x∗h − X∗

). (6.27)

Setting∂L

∂W(2)h

= 2W

(2)h −Wh

WhQh

− 2 (λ∗)′x∗h = 0, we obtain

W(2)h = Wh +WhQh (x∗h)

′λ∗. (6.28)

Substituting (6.28) in (6.25), we get λ∗ as

λ∗ = T −1(X∗ − x∗st

), (6.29)

where

T =L∑h=1

WhQhx∗h (x∗h)

′(6.30)

and x∗st =∑L

h=1 Whx∗h.

Now, the Theorem 50 discusses a GMREG form of the estimator defined in (6.22).

Theorem 50. The estimator y(2)st leads to a GMREG estimator of the population

mean defined as

y(2)st = yst + β

(X∗ − x∗st

), (6.31)

where

β =L∑h=1

WhQhyh (x∗h)′T −1 (6.32)

119

is a 1× (p+ 1) row vector of the form(β0, β1, β2, . . . , βp

).

Proof. Using (6.28) and (6.29), W (2)h can be written as

W(2)h = Wh +WhQh (x∗h)

′T −1

(X∗ − x∗st

). (6.33)

Substituting (6.33) in (6.22) leads to a GMREG estimator of the population meangiven by (6.31).

Remark 13. The Estimator II in (6.31) includes constraints on the calibrated weightsand mean of auxiliary variables.

The estimated variance of the proposed estimator y(2)st is given in Theorems 51 and

52.

Theorem 51. The estimate of the variance of the proposed estimator y(2)st is

v(y

(2)st

)=

L∑h=1

W 2h (1− fh)nh

s2eh, (6.34)

where, s2eh = (nh − p)−1∑nh

i=1 e2hi and ehi = (yhi − yh)−

∑pj=1 βj(xhij − xhj).

Proof. Extending the results in Theorem 21 yields (6.34).

Theorem 52. The lower level calibration estimate of the variance of y(2)st is

vLL

(y

(2)st

)=

L∑h=1

(W

(2)h

)2

(1− fh)

nhs2eh. (6.35)

Proof. Replacing the design weights Wh with the calibrated weights W (2)h in (6.34)

yields (6.35).

The following results can be obtained from the estimator in (6.31):

Corollary 3. When a single auxiliary information is available, that is, p = 1 it caneasily be verified that the proposed GMREG estimator in (6.31) reduces to the Singh[83] estimator given by (2.86).

120

Corollary 4. If the information on two auxiliary variables is available, that is, whenp = 2, then the calibrated weights in (6.28) is reduced to

W(2)h = Wh +WhQh (λ0 + λ1xh1 + λ2xh2) (6.36)

where,

λ0 =A1

(X1 −

∑Lh=1 Whxh1

)+ A2

(X2 −

∑Lh=1Whxh2

)B

,

λ1 =A3

(X1 −

∑Lh=1Whxh1

)+ A4

(X2 −

∑Lh=1Whxh2

)B

,

λ2 =A4

(X1 −

∑Lh=1Whxh1

)+ A5

(X2 −

∑Lh=1Whxh2

)B

,

A1 =L∑h=1

WhQhxh2

L∑h=1

WhQhxh1xh2 −L∑h=1

WhQhxh1

L∑h=1

WhQhx2h2,

A2 =L∑h=1

WhQhxh1

L∑h=1

WhQhxh1xh2 −L∑h=1

WhQhxh2

L∑h=1

WhQhx2h1,

A3 =L∑h=1

WhQh

L∑h=1

WhQhx2h2 −

(L∑h=1

WhQhxh2

)2

,

A4 =L∑h=1

WhQhxh2

L∑h=1

WhQhxh1 −L∑h=1

WhQh

L∑h=1

WhQhxh1xh2,

A5 =L∑h=1

WhQh

L∑h=1

WhQhx2h1 −

(L∑h=1

WhQhxh1

)2

,

and

B =L∑h=1

WhQh

L∑h=1

WhQhx2h1

L∑h=1

WhQhx2h2 + 2

L∑h=1

WhQhxh1

L∑h=1

WhQhxh1xh2

L∑h=1

WhQhxh2 −L∑h=1

WhQh

(L∑h=1

WhQhxh1xh2

)2

−(

L∑h=1

WhQhxh2

)2

L∑h=1

WhQhx2h1 −

(L∑h=1

WhQhxh1

)2 L∑h=1

WhQhx2h2.

121

The calibrated weights in (6.36) can be written as

W(2)h = Wh +

WhQhA1 +WhQhxh1A3 +WhQhxh2A4

B

(X1 −

L∑h=1

Whxh1

)

+WhQhA2 +WhQhxh1A4 +WhQhxh2A5

B

(X2 −

L∑h=1

Whxh2

).

Hence, the estimator in (6.31) can be written as

y(2)st = yst + β1

(X1 −

L∑h=1

Whxh1

)+ β2

(X2 −

L∑h=1

Whxh2

),

where

β1 =

A1

(L∑h=1

WhQhyh

)+ A3

(L∑h=1

WhQhxh1yh

)+ A4

(L∑h=1

WhQhxh2yh

)B

,

and

β2 =

A2

(L∑h=1

WhQhyh

)+ A4

(L∑h=1

WhQhxh1yh

)+ A5

(L∑h=1

WhQhxh2yh

)B

.

6.2.3 Estimator III

The Estimator III is developed by adding a multivariate calibration constraint usingthe known variance information such as

L∑h=1

W(3)h s2

hxj =L∑h=1

WhS2hxj, for j = 1, 2, . . . , p (6.37)

to the problem of Estimator I proposed in Section 6.2.1, where S2hxj; j = 1, 2, . . . , p

are accurately known. In the presence of p auxiliary auxiliary variables xj, j =

1, 2, . . . , p, we propose a multivariate calibration estimator of the population meanY under stratified sampling as

y(3)st =

L∑h=1

W(3)h yh (6.38)

122

where the calibrated weights W (3)h are suitably chosen weights such that the CS

distance function is minimum, subject to the calibration constraints

L∑h=1

W(3)h xhj =

L∑h=1

WhXhj, for j = 1, 2, . . . , p (6.39)

andL∑h=1

W(3)h s2

hxj =L∑h=1

WhS2hxj, for j = 1, 2, . . . , p (6.40)

where s2hxj = (nh − 1)−1∑nh

i=1 (xhij − xhj)2 denotes the sample variance of jth aux-iliary variable in hth stratum.

The above calibration constraints using matrix notations can also be written as

L∑h=1

W(3)h xh =

L∑h=1

WhXh, (6.41)

and

L∑h=1

W(3)h s2

hx =L∑h=1

WhS2hx (6.42)

where, Xh =(Xh1, Xh2, . . . , Xhp

)′, xh = (xh1, xh2, . . . , xhp)

′, S2

hx =(S2hx1, S

2hx2, . . . , S

2hxp

)′,

s2hx =

(s2hx1, s

2hx2, . . . , s

2hxp

)′are the (p× 1) column vectors on the population mean,

sample mean, population variance and sample variance information of the auxiliaryvariables in the hth stratum, respectively.

Thus, the problem of determining the OCW W(3)h may be expressed as an MPP as

given below:

MinimizeL∑h=1

(W

(3)h −Wh

)2

WhQh

,

subject toL∑h=1

W(3)h xh =

L∑h=1

WhXh,

andL∑h=1

W(3)h s2

hx =L∑h=1

WhS2hx,

(6.43)

where Qh =

(∑pj=1

1

qhj

)−1

.

To solve the MPP (6.43), we introduce the Lagrange multipliers λ= (λ1, λ2, . . . , λp)′

123

and γ= (γ1, γ2, . . . , γp)′as (p × 1) column vectors. The Theorem 53 discusses the

derivation of W (3)h using the Lagrange multiplier technique.

Theorem 53. The optimum solution to the MPP (6.43), that is, the OCW W(3)h

that minimize the CS distance function subject to (6.41) and (6.42) is given by

W(3)h = Wh +WhQh

(x′

hλ+ s2hxγ)

where [λ

γ

]= T −1

L∑h=1

WhXh −L∑h=1

Whxh

L∑h=1

WhS2hx −

L∑h=1

Whs2hx

,i f

T =

[ ∑Lh=1WhQhxhx

′

h

∑Lh=1WhQhxh (s2

hx)′∑L

h=1WhQhs2hxx

′

h

∑Lh=1WhQhs

2hx (s2

hx)′

]is an invertible (2p× 2p) square matrix.

Proof. The Lagrange function L is defined by:

L =L∑h=1

(W(3)h −Wh)

2

WhQh

− 2λ′(

L∑h=1

W(3)h xh −

L∑h=1

WhXh

)−2γ

′(

L∑h=1

W(3)h s2

hx −L∑h=1

WhS2hx

).

(6.44)

Setting∂L

∂W(3)h

=W

(3)h −Wh

WhQh

− 2λ′xh − 2γ

′s2hx = 0, we obtain

W(3)h = Wh +WhQh

[x′

hλ+(s2hx

)′γ]. (6.45)

Substituting (6.45) in (6.41) and (6.42), we obtain following equations

∑Lh=1WhQhxhx

′

hλ+∑L

h=1WhQhs2hxx

′

hγ =L∑h=1

WhXh −L∑h=1

Whxh∑Lh=1WhQhxh (s2

hx)′λ+

∑Lh=1WhQhs

2hx (s2

hx)′γ =

L∑h=1

WhS2hx −

L∑h=1

Whs2hx.

(6.46)

124

Solving (6.46), we get

[λ

γ

]= T −1

L∑h=1

WhXh −L∑h=1

Whxh

L∑h=1

WhS2hx −

L∑h=1

Whs2hx

, (6.47)

where

T =

[ ∑Lh=1WhQhxhx

′

h

∑Lh=1WhQhs

2hxx

′

h∑Lh=1WhQhxh (s2

hx)′ ∑L

h=1WhQhs2hx (s2

hx)′

]. (6.48)

The GMREG form of the estimator (6.38) is provided in Theorem 54.

Theorem 54. The estimator y(3)st leads to a GMREG estimator of the population

mean defined as

y(3)st =

∑Lh=1 Whyh + β

L∑h=1

WhXh −L∑h=1

Whxh

L∑h=1

WhS2hx −

L∑h=1

Whs2hx

, (6.49)

where

β =L∑h=1

WhQhyh

[xh

s2hx

]′T −1

is a (1× 2p) row vector of the form(β1, β2, . . . , β2p

).

Proof. Substituting (6.47) in (6.45), we obtain

W(3)h = Wh +WhQh

[xh

s2hx

]′T −1

L∑h=1

WhXh −L∑h=1

Whxh

L∑h=1

WhS2hx −

L∑h=1

Whs2hx

. (6.50)

Finally, substituting (6.50) in (6.38), we obtain (6.49).

Remark 14. The Estimator III in (6.49) includes constraints on the mean and vari-ance of auxiliary variables.

The estimated variance y(3)st is stated in Theorems 55 and 56.

125

Theorem 55. The estimated variance of y(3)st is

v(y

(3)st

)=

L∑h=1

W 2h (1− fh)nh

s2eh, (6.51)

where, s2eh = (nh − 2p− 1)−1∑nh

i=1 e2hi,

ehi = (yhi − yh)−∑p

j=1 βj(xhij − xhj)−∑p

j=1 βj+p

(xhij − xhj)2 − s∗2hxj

and s∗2hxj = n−1h

∑nh

i=1 (xhij − xhj)2 .

Proof. Extending the results in Theorem 27 yields (6.51).

Theorem 56. The lower level calibration estimated variance of y(3)st is

vLL

(y

(3)st

)=

L∑h=1

(W

(3)h

)2

(1− fh)

nhs2eh. (6.52)

Proof. Replacing the design weights Wh with the calibrated weights W (3)h in (6.51)

yields (6.52).

Corollary 5. When a single auxiliary information is available, that is p = 1, itcan easily be verified that the proposed GMREG estimator in (6.49) reduces to theEstimator I given by 5.4 in Section 5.2.

6.2.4 Estimator IV

The Estimator IV is developed by adding the constraint on calibrated weights assuggested by Singh [83] to the problem of Estimator III. We propose a multivariatecalibration estimator of the population mean Y under stratified sampling defined as

y(4)st =

L∑h=1

W(4)h yh (6.53)

where the calibrated weights W (4)h are so chosen such that the CS distance function

is minimum, subject to the calibration constraints

L∑h=1

W(4)h =

L∑h=1

Wh, (6.54)

L∑h=1

W(4)h xhj =

L∑h=1

WhXhj, for j = 1, 2, . . . , p, (6.55)

126

and

L∑h=1

W(4)h s2

hxj =L∑h=1

WhS2hxj, for j = 1, 2, . . . , p. (6.56)

The calibration constraints (6.54) and (6.55) can be collectively written in matrixnotation, respectively, as

L∑h=1

W(4)h x∗h =

L∑h=1

WhX∗h (6.57)

andL∑h=1

W(4)h s2

hx =L∑h=1

WhS2hx (6.58)

where X∗h =(1, Xh1, Xh2, . . . , Xhp

)′, x∗h = (1, xh1, xh2, . . . , xhp)

′are (p + 1) × 1 col-

umn vectors of the information on population mean and sample mean, respectivelyand S2

hx =(S2hx1, S

2hx2, . . . , S

2hxp

)′, s2

hx =(s2hx1, s

2hx2, . . . , s

2hxp

)′are the p × 1 column

vectors of the information on population variance and sample variance of the auxil-iary variables in the hth stratum, respectively.

Thus, the problem of determining the OCWW(4)h may be expressed as an MPP given

below:

MinimizeL∑h=1

(W

(4)h −Wh

)2

WhQh

subject toL∑h=1

W(4)h x∗h =

L∑h=1

WhX∗h

andL∑h=1

W(4)h s2

hx =L∑h=1

WhS2hx,

(6.59)

where Qh =

(∑pj=1

1

qhj

)−1

.

To solve the MPP (6.59) using the Lagrange multiplier technique, we introduce theLagrange multipliers λ∗= (λ0, λ1, λ2, . . . , λp)

′and γ= (γ1, γ2, . . . , γp)

′. The Theorem

57 provides the solution to the MPP (6.59).

Theorem 57. The optimum solution to the MPP (6.43), that is, the OCW W(4)h

127

that minimize the CS distance subject to (6.57) and (6.58) is given by

W(4)h = Wh +WhQh

[(x∗h)

′λ+

(s2hx

)′γ]

where [λ∗

γ

]= T −1

L∑h=1

WhX∗h −

L∑h=1

Whx∗h

L∑h=1

WhS2hx −

L∑h=1

Whs2hx

,if

T =

[ ∑Lh=1WhQhx

∗h (x∗h)

′ ∑Lh=1WhQhx

∗h (s2

hx)′∑L

h=1WhQh (x∗h)′ ∑L

h=1WhQhs2hx (s2

hx)′

]is an invertible (2p+ 1)× (2p+ 1) square matrix.

Proof. The Lagrange function L for this problem is defined by:

L =L∑h=1

(W(4)h −Wh)

2

WhQh

− 2 (λ∗)′(

L∑h=1

W(4)h x∗h −

L∑h=1

WhX∗h

)−2γ

′(

L∑h=1

W(4)h s2

hx −L∑h=1

WhS2hx

).

(6.60)

Setting∂L

∂W(4)h

= 0, we obtain

W(4)h = Wh +WhQh

[(x∗h)

′λ∗ +

(s2hx

)′γ]. (6.61)

Substituting (6.61) in (6.57) and (6.58), and solving for λ∗ and γ, we

[λ∗

γ

]= T −1

L∑h=1

WhX∗h −

L∑h=1

Whx∗h

L∑h=1

WhS2hx −

L∑h=1

Whs2hx

, (6.62)

where

T =

[ ∑Lh=1WhQhx

∗h (x∗h)

′ ∑Lh=1WhQhx

∗h (s2

hx)′∑L

h=1WhQh (x∗h)′ ∑L

h=1WhQhs2hx (s2

hx)′

]. (6.63)

The Theorem 58 provides the GMREG form of the estimator (6.53).

128

Theorem 58. The estimator y(4)st leads to a GMREG estimator of the population

mean defined as

y(4)st =

∑Lh=1 Whyh + β

L∑h=1

WhX∗h −

L∑h=1

Whx∗h

L∑h=1

WhS2hx −

L∑h=1

Whs2hx

, (6.64)

where

β =L∑h=1

WhQhyh

[x∗hs2hx

]′T −1

is a 1× (2p+ 1) row vector of the form(β0, β1, β2, . . . , β2p

).

Proof. Substituting (6.62) in (6.61), we obtain

W(4)h = Wh +WhQh

[x∗hs2hx

]′T −1

L∑h=1

WhX∗h −

L∑h=1

Whx∗h

L∑h=1

WhS2hx −

L∑h=1

Whs2hx

. (6.65)

Finally, substituting (6.65) in (6.53), we obtain (6.64).

Remark 15. The Estimator IV in (6.64) includes constraints on the calibrated weights,mean and variance of auxiliary variables.

The estimated variance of y(4)st is provided in the Theorems 59 and 60.

Theorem 59. The estimate of the variance of y(4)st is

v(y

(4)st

)=

L∑h=1

W 2h (1− fh)nh

s2eh, (6.66)

where, s2eh = (nh − 2p− 1)−1∑nh

i=1 e2hi,

ehi = (yhi − yh)−∑p

j=1 βj(xhij − xhj)−∑p

j=1 βj+p

(xhij − xhj)2 − s∗2hxj

and s∗2hxj = n−1h

∑nh

i=1 (xhij − xhj)2 .

Proof. Extending the results in Theorem 27 yields (6.66).

Theorem 60. The lower level calibration estimate of the variance of y(4)st is

vLL

(y

(4)st

)=

L∑h=1

(W

(4)h

)2

(1− fh)

nhs2eh. (6.67)

129

Proof. Replacing the design weights Wh with the calibrated weights W (4)h in (6.66)

yields (6.67).

Corollary 6. When a single auxiliary information is available, that is, p = 1 itcan easily be verified that the proposed GMREG estimator in (6.64) reduces to theEstimator II given by 5.18 in Section 5.2.

6.3 Numerical illustration

This section demonstrates the application and computational details of the multi-variate calibration estimators proposed in (6.12), (6.31), (6.49) and (6.64) in Section6.2, using the tobacco population discussed in Section 5.4. Suppose that an es-timate of average production (Y ) of tobacco crop is of interest using the knowninformation from two auxiliary variables: x1 = area (in hectares) and x2 = yield (inmetric tons). To carry out the computations, we used the same setup (i.e. L = 10,

Nh = 6, 6, 8, 10, 12, 4, 30, 17, 10, 3 , n = 40 and nh = 3, 3, 3, 3, 4, 2, 11, 6, 3, 2) andsame sample units as in Section 5.3. Table 22 provides the multivariate populationand sample information needed for the tobacco population.

Table 22: Information on tobacco population for multivariate estimators.h Xh1 Xh2 S2

hx1 S2hx2 xh1 xh2 s2hx1 s2hx2

1 3194.5 1.97333 10899652.7 0.02683 3730.0 2.08000 21019948.0 0.01290

2 14660.0 1.38833 584984730.0 0.21810 245.0 1.40000 77025.0 0.06910

3 18309.4 2.55625 635958094.8 0.34700 18988.3 2.49000 634924108.3 0.26130

4 14923.5 1.54900 209817189.2 0.23457 10578.3 1.88667 86754908.3 0.33903

5 5987.8 1.83167 27842810.5 0.58214 8786.5 1.74250 58966062.3 0.39249

6 3450.0 1.47000 5876666.7 0.15313 3800.0 1.39500 16820000.0 0.09245

7 11682.7 1.11500 760238523.4 0.34385 21573.2 1.04909 1918981931.4 0.28091

8 145162.3 1.38176 124004506112.8 0.37855 105938.5 1.35000 27906093819.9 0.22076

9 33976.1 1.72100 8340765245.4 2.01828 5716.7 2.65667 65160833.3 6.99253

10 1333.3 2.08667 2963333.3 0.97463 1700.0 1.84000 5120000.0 1.58420

Using a computer program developed in MATLAB (Refer to Appendix B.3), the valuesof Lagrange multipliers and estimated betas for all the proposed four multivariatecalibration estimators are computed and presented in Table 23. The OCW obtainedfor the four proposed estimators are presented in Table 24. The estimated values ofpopulation mean by all the estimators and the estimated variance of these estimators

130

were computed and presented in Table 25. We used Y = 52444.6 and Qh = 1 in thecomputations.

Table 23: Lagrange multipliers and estimated betas for different estimators.Estimator

(ˆY)

λ β

Estimator I(y(1)st

)5.42× 10−6 1.40403

−9.79× 10−2 2993.65680

Estimator II(y(2)st

)1.43× 10−1 -2761.22010

4.84× 10−6 1.41537−1.69× 10−1 4373.30203

Estimator III y(3)st 2.13× 10−5 1.69161−4.15× 10−2 2943.17335−2.10× 10−11 -0.000001−1.96× 10−1 -1521.61613

Estimator IV y(4)st −4.98× 10−1 37212483.73116

2.98× 10−5 -2981.645801.96× 10−1 -4533902.13082−2.82× 10−1 0.01047−2.33× 10−1 -1265266.30321

Table 24: OCW for different multivariate calibration estimators.h W

(1)h W

(2)h W

(3)h W

(4)h

1 0.04622 0.04578 0.76008 0.790792 0.04892 0.05134 0.71032 1.385073 0.06484 0.06137 -0.22873 -1.360984 0.08232 0.08251 0.11230 -0.074255 0.09929 0.10080 0.26757 0.410556 0.03336 0.03491 0.27287 0.620287 0.28706 0.30273 -3.49098 -2.239138 0.23131 0.22880 0.99568 0.935509 0.07273 0.06799 0.08730 0.0269410 0.02346 0.02376 0.31107 0.50522Total 0.98951 1.00000 -0.20252 1.00000

131

Table 25: Multivariate calibration estimators of Y and their estimated variance fortobacco population.

Estimator(

ˆY)

Estimated value of Y Estimated variance of the estimator

Estimator I(y(1)st

)52792.96 4497017.89

Estimator II(y(2)st

)52764.01 4344750.43

Estimator III(y(3)st

)37387.90 -

Estimator IV(y(4)st

)12325.88 -

The following were some of the observations from the numerical example on themultivariate estimators:

• The estimators y(1)st and y(2)

st were found to be relatively closer to the true valueof the population mean

(Y)compared to the estimators y(3)

st and y(4)st .

• The estimators y(3)st and y

(4)st provide some negative and extreme calibrated

weights. The estimated values of Y using these estimators are also far awayfrom the true value.

• The estimated variance of y(3)st and y(4)

st which are calculated using (6.51) and(6.66) , are not obtained (see Table 25) as the sample size (nh) for some strataare less than 2p+ 1.

6.4 Simulation study

In this section, we present the results obtained in the simulation carried out withthe same setup as discussed in Section 5.4 to compare the efficiency of the proposedmultivariate calibration estimators in stratified sampling, using the tobacco, laborand ST130 populations. It was assumed that Qh = 1 to compute the values of themultivariate calibration estimators. Tables 22, 26 and 27 provide the informationneeded on the tobacco, labor and ST130 populations, respectively.

Table 26: Information on labor population for multivariate calibration estimators.h Xh1 Xh2 S2

hx1 S2hx2

1 38.5 35.8 113.7 182.72 38.0 177.0 132.6 180.33 37.6 37.7 170.6 191.5

132

Table 27: Information on ST130 population for multivariate calibration estimators.h Xh1 Xh2 S2

hx1 S2hx2

1 33.6 17.7 67.3 52.92 34.7 20.8 70.3 58.63 35.6 19.4 58.6 66.44 34.1 18.2 94.9 52.75 32.6 19.3 25.4 22.0

The simulation results for the estimators were obtained by executing the MATLAB

computer program for 100,000 iterations, simultaneously for all estimators alongwith the stratified estimator (Refer to Appendix B.3). Tables 28, 29 and 30 presentthe values of Bias, MSE, Variance and PRE obtained for all the estimators in thesimulation study for the three populations, respectively.

Table 28: Simulation results of tobacco population for multivariate estimators.Estimator

(ˆY)

Bias MSE Variance PRE

Stratified (yst) 201.25 1070645868.95 1070605366.94 100.00

Estimator I(y(1)st

)-3448.63 59487301.48 47594267.87 1799.79

Estimator II(y(2)st

)-2645.16 53839864.65 46842989.60 1988.57

Estimator III(y(3)st

)-8702.41 35538084486.55 35462352506.41 3.01

Estimator IV(y(4)st

)-9777.38 24195817335.52 24100220192.43 4.42

Table 29: Simulation results of labor force population for multivariate estimators.Estimator

(ˆY)

Bias MSE Variance PRE

Stratified (yst) 0.12209 535.42 535.40 100.00

Estimator I(y(1)st

)0.00260 3978.62 3978.62 13.46

Estimator II(y(2)st

)-18.25633 4378704.00 4378370.71 0.03

Estimator III(y(3)st

)15.06651 1714209.10 1713982.10 0.01

Estimator IV(y(4)st

)153.83580 1496452318.28 1496428652.82 0.00

133

Table 30: Simulation results of ST130 population for multivariate estimators.Estimator

(ˆY)

Bias MSE Variance PRE

Stratified (yst) -0.00550 2.26390 2.26387 100.00

Estimator I(y(1)st

)-0.00033 0.00183 0.00183 124049.42

Estimator II(y(2)st

)0.00210 0.00413 0.00412 54842.59

Estimator III(y(3)st

)-0.00034 0.05901 0.05901 3836.67

Estimator IV(y(4)st

)0.01730 152.60753 152.60723 1.48

6.5 Comparison study

In this section, we compare the estimators, based on the results obtained from thethe simulation study. The comparison study of these estimators is based on themeasures: Bias, MSE, Variance and PRE which is computed using (5.80)-(5.83).The purpose of this simulation is:

1. To compare the efficiency among the proposed multivariate calibration estima-tors.

2. To compare the proposed multivariate calibration estimators with the corre-sponding univariate estimators.

6.5.1 Comparison among the proposed multivariate calibra-

tion estimators

When compareing the proposed multivariate calibration estimators y(1)st , y

(2)st , y

(3)st and

y(4)st based on the results of the simulation study on the tobacco, labor and ST130populations, which are presented in Tables 28-30, the following observations weremade:

• The estimators y(1)st , y

(2)st and y(3)

st perform better as compared to the estimatory

(4)st in most populations. It can be noted that the estimator y(4)

st incorporatethe constraints on calibrated weights and variance information. This impliesthat adding both constraints on calibrated weights and variance informationdoes not help in improving the estimator. From our simulation results, wenotice that the estimator y(4)

st produces many negative and extreme calibratedweights that may be a reason for inefficiency. This estimator produces the

134

largest Bias, MSE and Variance and smallest PRE which is evident from thesimulation results.

• The estimators y(1)st and y

(2)st perform better as compared to estimators y(3)

st

and y(4)st in all the population. It can be noted that the estimators y(3)

st andy

(4)st are those that incorporated the constraints on the variance information.This implies that adding constraint on variance information does not help inimproving the estimate. The simulation results also show that the estimatorsy

(3)st and y(4)

st produce more negative and extreme calibrated weights that maybe a reason for their poor performance. These estimates produce larger Bias,MSE and Variance and smaller PRE as compared to others.

• The estimator y(1)st performs better as compared to the estimators y(2)

st , y(3)st and

y(4)st , which has been seen for the labor and ST130 populations. The simulationresults show that the estimator y(1)

st that incorporated only the auxiliary in-formation on mean is more consistent as compared to other estimators. Also,from our research discussed in Section 6.5.2 based on the three populations,it was found that the efficiency of the estimator y(1)

st increases as the partialcorrelations of the auxiliary variables with the survey y increases.

• Moreover, it can be noted that the estimator y(2)st , which incorporated the

constraint on weight is less efficient than y(1)st . This implies that adding the

constraint on weight does not help in improving the estimate, especially whenthe partial correlations between the auxiliary and study variables are low.

Therefore, the estimator y(1)st among the multivariate calibration estimators in strat-

ified sampling is consistently the most efficient estimator provided the partial cor-relation coefficient is greater than 0.5 that is ρx2,y|x1 > 0.5. This implies that themean information of the auxiliary variables is sufficient to improve the calibrationestimate and the additional constraints on other information are not much useful.

6.5.2 Comparison of multivariate verses univariate calibra-

tion estimators

We will now compare the multivariate calibration estimators with their correspondingunivariate estimators to investigate whether the information on more than one aux-iliary variable improves the estimates. Thus, in this comparison study, we considerthe following multivariate calibration estimators and their counterpart univariatecalibration estimators:

135

1. y(1)st vs y⊕st [see (6.12) and (2.65)].

2. y(2)st vs yst [see (6.31) and (2.86)].

3. y(3)st vs y⊗st [see (6.49) and (5.4)].

4. y(4)st vs yst [see (6.64) and (5.18)].

For these comparisons we use the values of Bias, MSE, variance and PRE for thethree populations which are presented in Tables 31-34 and are discussed below.

For the tobacco and labor populations, comparing the results in Tables 31-34 it wasobserved that the multivariate calibration estimators do not perform better thanthe univariate calibration estimators. Investigating further it was noticed that theunivariate calibration estimators were developed using the information on auxiliaryvariable x1, whereas, the multivariate calibration estimators were developed usingx1 and x2. To compare these estimators, we calculate the following correlation coef-ficients:

1. Correlation coefficient of x1 and y, denoted by ρx1,y.

2. Correlation coefficient of x2 and y, denoted by ρx2,y.

3. Partial correlation coefficient of x2 and y, when x1 is fixed, denoted by ρx2,y|x1 .

The results for these correlations coefficients for each population are presented inTable 35.

It was noted that, for tobacco and labor populations, ρx2,y|x1 were significantly lowand this may be a reason that the multivariate calibration estimators did not per-form well in these two populations.

On the other-hand, for the ST130 population, the proposed multivariate estimatorsy

(1)st , y

(2)st and y(3)

st were more efficient than their corresponding univariate estimatorsy⊕st, y

st and y⊗st. That is, the bias, variance, MSE are significantly smaller and the

PRE are significantly larger for the multivariate calibration estimators. It was alsonoticed that the partial correlation ρx2,y|x1 > 0.5 was higher in ST130 population,which may be a reason that the multivariate estimators perform better than thecorresponding univariate estimators.

136

Therefore, it can concluded that the multivariate calibration estimators are moreefficient as compared to the corresponding univariate calibration estimators, if theauxiliary variables to be added correlate strongly with the survey variable.

Table 31: Comparison results for y⊕st vs y(1)st .

Population Estimator Bias MSE Variance PRE

TobaccoUnivariate

(y⊕st)

-2441.49 47126396.12 41165520.42 2271.86

Multivariate(y(1)st

)-3448.63 59487301.48 47594267.87 1799.79

LaborUnivariate

(y⊕st)

0.03120 410.55 410.54 130.42

Multivariate(y(1)st

)0.00260 3978.62 3978.62 13.46

ST130Univariate

(y⊕st)

-0.00080 0.47282 0.47282 478.81

Multivariate(y(1)st

)-0.00033 0.00183 0.00183 124049.42

Table 32: Comparison results for yst vs y(2)st .

Population Estimator Bias MSE Variance PRE

TobaccoUnivariate

(yst)

-3103.33 60586155.02 50955498.37 1767.15

Multivariate(y(2)st

)-2645.16 53839864.65 46842989.60 1988.57

LaborUnivariate

(yst)

0.89048 2956.70 2955.91 18.11

Multivariate(y(2)st

)-18.25633 4378704.00 4378370.71 0.03

ST130Univariate

(yst)

-0.01196 0.75874 0.75860 298.38

Multivariate(y(2)st

)0.00210 0.00413 0.00412 54842.59

Table 33: Comparison results for y⊗st vs y(3)st .

Population Estimator Bias MSE Variance PRE

TobaccoUnivariate

(y⊗st)

-20747.68 32286483974.34 31856017622.24 3.32

Multivariate(y(3)st

)-8702.41 35538084486.55 35462352506.41 3.01

LaborUnivariate

(y⊗st)

-0.89430 2967.13 2966.33 18.04

Multivariate(y(3)st

)15.06651 1714209.10 1713982.10 0.01

ST130Univariate

(y⊗st)

-0.03579 1.00161 1.00033 226.03

Multivariate(y(3)st

)-0.00034 0.05901 0.05901 3836.67

137

Table 34: Comparison results for yst vs y(4)st .

Population Estimator Bias MSE Variance PRE

TobaccoUnivariate (yst) -19322.41 40679964978.99 40306609452.02 2.63

Multivariate(y(4)st

)-9777.38 24195817335.52 24100220192.43 4.42

LaborUnivariate (yst) 83.54261 125286557.67 125279578.31 0.00

Multivariate(y(4)st

)153.83580 1496452318.28 1496428652.82 0.00

ST130Univariate (yst) -0.04441 1.88015 1.87817 120.41

Multivariate(y(4)st

)0.01730 152.60753 152.60723 1.48

Table 35: Correlation coefficient information.Population ρx1,y ρx2,y ρx2,y|x1

Tobacco 0.991 0.030 0.287Labor 0.486 0.191 0.146ST130 0.890 0.861 0.505

6.6 Summary

In this chapter, we study the multivariate calibration estimators of population meanwith known information from several auxiliary variables in stratified sampling. Theproblem of determining OCW was to seek the minimization of the CS type distancefunction subject to some calibration constraints on the known information from theauxiliary variables. The Lagrange multiplier technique was used to solve the prob-lem of OCW. The multivariate calibration estimators were expressed in the GMREGforms and the estimated variance of these estimators were also obtained.

A numerical illustration is presented to demonstrate the application and compu-tational details of the proposed multivariate calibration estimators. Further, themultivariate calibration estimators were included in the simulation study to inves-tigate their efficiencies. It was found that the estimator y(1)

st was consistently mostefficient multivariate calibration estimator in stratified sampling and that the meaninformation of the auxiliary variables is sufficient to improve calibration estimatorsand additional constraints are unnecessary. The results also show that multivariate

138

calibration estimators are more efficient than the corresponding univariate estima-tors, if the auxiliary variables to be added correlate strongly with the survey variable.That is, information on more than one auxiliary variable improves the estimates.

139

Chapter 7

Conclusion and Future Research

7.1 Introduction

This chapter provides how the objectives stated in Chapter 1, have been achieved.Section 7.2 presents the summary of findings and the some recommendations aregiven for future research in Section 7.3.

The specific objectives of this study were to:

1. Develop a dynamic programming method to stratify skewed populations usingthe study variable.

2. Extend the dynamic programming method to stratify skewed populations usingthe auxiliary variable.

3. Investigate the efficiency of the proposed stratification method.

4. Propose some calibration estimators of population mean in stratified sampling.

5. Propose some multivariate calibration estimators of population mean in strat-ified sampling.

6. Investigate the efficiency of the proposed calibration estimators.

In our study, the results of which are presented in this thesis, we have met all theobjectives as discussed below:

140

A dynamic programing method was developed in Chapter 3 to stratify skewed pop-ulations using the study variable. When the study variable has Log-normal distribu-tion, the problem of determining the OSB is formulated as an MPP which is solvedby developing a dynamic programming technique. The dynamic programming tech-nique was also extended to stratify the populations using the auxiliary variable asthe stratification based on the study variable is not always realistic. In Chapter 4,we developed a technique of determining OSB when the auxiliary variable has Paretodistribution. Numerical examples were provided to demonstrate the computationaldetails of the solution procedure. A detailed comparison study was also carried outin sections 3.5 and 4.6 to investigate the efficiency of the proposed dynamic program-ming technique with other stratification methods.

In Chapter 5, we have developed some calibration estimators of population meanin stratified sampling using the known information on a single auxiliary variable.The efficiencies of the proposed calibration estimators was also discussed in thischapter. Some multivariate calibration estimators of population mean in stratifiedsampling and their efficiencies were discussed in Chapter 6. Numerical illustrationsare provided to illustrate the computation and application details of the calibrationestimators. Further, the simulation study was carried to investigate the efficiency ofthe proposed calibration estimators.

7.2 Summary of findings

7.2.1 The stratification method

The proposed dynamic programing technique is found to be efficient and very usefulthan the other stratification methods for determining the optimum strata bound-aries. The study also reveals that the proposed method performs even better whenthe skewness increases. The proposed method has an advantage that it does notneed initial solutions nor complete data set to stratify the populations. Among themethods Geometric method performs very badly and may not be useful as it violatesthe required restrictions on sample sizes, especially, when L increases. However, thisfinding in this study contradicts with that of Gunning and Horgan [27], which showedthat the geometric method is more efficient than the CRF method. On the otherhand, although, the CRF method performs better than the Geometric method, itsometimes fails to determine the OSB, if L is large and nclass is small. Further,

141

it was found that the LH method performs very similar to the proposed method,particularly for stratification of the Log-normal distribution. However, an advantageof the proposed method over LH method is that it needs neither any initial solutionnor the complete data-set. In many situations, the complete data-set may not beavailable, in such cases the proposed method works as it requires only the parametersof the distribution.

In another comparison study where the stratification was carried out using the auxil-iary variable, it was found that the dynamic programming method was more efficientthan the other methods in stratifying skewed populations. The OSBs obtained byCRF and Geometric method was closer to that of dynamic programming methodbut the OSB’s obtained by the LH differs significantly from the other methods. Itwas also observed that the Geometric and CRF methods perform better than theLH method. As the number of strata increases, it was found that Geometric methodperforms very similar to CRF but the LH performs very poorly.

7.2.2 The calibration estimators

It was found that the mean information is sufficient to improve the calibration esti-mators and there is no need for employing the constraints such as “sum of calibratedweights be equal to sum of design weights” and the constraints that use the varianceinformation. The calibration estimator y†st developed by calibrating Vh = Wh/nn

instead of Wh in stratified sampling was found to be the most efficient estimator ofpopulation mean.

When information on several auxiliary variables is available, the precision of the pop-ulation mean can further be increased. It was found that the multivariate estimatorsperform better than those estimators with single auxiliary variable if the correlationbetween the auxiliary variables and the study variable is strong.

7.3 Recommendations for future research

Future research that may be carried out in the areas of optimum stratification andcalibration estimations are:

1. To develop a multivariate stratification method when more than one studyvariables are to be stratified.

142

2. To develop procedures of avoiding negative and extreme calibrated weightsthat may improve the calibration estimate.

3. To investigate the appropriate values of qh in GREG/GMREG, which mayimprove the calibration estimation.

4. To develop some efficient calibration estimators in stratified sampling usingdifferent distance functions other than chi-square type distance function.

143

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152

Publications

The following papers have been published from this thesis:

1. Khan, M. G. M., Rao, D. K., Ansari, A. H., and Ahsan, M. J. (2011). Deter-mining optimum strata boundaries for skewed population with Log-normal dis-tribution. In Proceedings of International Conference on Operations Researchand Statistics, pages 7–10. Global Science and Technology Forum. ISBN:978-981-08-8407-9.

2. Rao, D. K., Khan, M. G., and Khan, S. (2012). Mathematical programmingon multivariate calibration estimation in stratified sampling. InternationalJournal of Mathematical, Computational, Physical, Electrical and ComputerEngineering, 6(12):58–62. ISBN: 2070-3740.

3. Khan, M. G. M., Rao, D. K., Ansari, A. H., and Ahsan, M. J. (2014). De-termining optimum strata boundaries and sample sizes for skewed populationusing log-normal distribution. Communications in Statistics: Simulation andComputation, 44(5):1364–1387.

4. Rao, D. K., Khan, M. G. M., and Reddy, K. G. (2014). Optimum stratifica-tion of a skewed population. International Journal of Mathematical, Compu-tational, Physical and Quantum Engineering, 8(3):497–500.

5. Rao, D. K., Khan, M. G. M., and Reddy, K. G. (2015). Stratified calibrationestimator of population mean using multi-auxiliary information. In Proceed-ings of Asia- Pacific World Congress on Computer Science and Engineering,pages 1–5. IEEE.

6. Rao, D. K., Khan, M. G. M., and Reddy, K. G. (2015). Calibration Estimatorswith new distance functions. Presented at 8th International Conference of theERCIM WG on Computational and Methodological Statistics, University ofLondon, December 12-14, 2015.

153

The following paper has been accepted for publication from this thesis:

1. Rao, D. K., Tekabu, T., and Khan, M. G. M. (2016). New calibration esti-mators in stratified sampling. Accepted in Proceedings of Asia- Pacific WorldCongress on Computer Science and Engineering. IEEE.

154

Appendix A: Comparison Results forthe Stratification of Log-normalPopulation

155

Table36:OSB

,V

(yst

)∗,Nhan

dnhforskew

ness

=0.

5994,µ

=0.

0000

9935

132,σ

=0.

1975

361,N

=15

000,n

=10

00,nclass

=50,y 0

=0.

4941

0530

andd

=1.

5889

0220

.

LCRF

Geometric

LHDyn

amic

Prog.

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

21.03

.000014

8493

483

1.01

.000014

7924

431

1.04

.000014

8566

490

1.04

.000

014

8657

498

6507

517

7076

569

6434

510

6343

502

30.91

.000007

4660

280

0.80

.000013

1896

640.94

.000007

5517

358

0.94

.000

007

5506

357

1.16

7013

383

1.29

11601

834

1.16

6155

302

1.16

6176

304

3327

337

1503

102

3328

340

3318

339

40.84

.000004

2905

191

0.71

.000008

576

190.87

.000004

3620

259

0.87

.000

004

3716

268

1.03

5588

299

1.01

7348

428

1.03

4807

221

1.04

4937

233

1.26

4670

288

1.45

6618

518

1.23

4302

234

1.24

4231

235

1837

222

458

352271

286

2116

264

50.81

.000003

2193

164

0.66

.000006

267

90.84

.000003

2734

220

0.84

.000

003

2730

219

0.97

4386

240

0.88

3560

171

0.97

3877

184

0.97

3909

187

1.13

4433

243

1.17

8027

555

1.11

3897

187

1.11

3931

191

1.32

2782

181

1.56

2977

250

1.29

2961

183

1.29

2961

187

1206

172

169

151531

226

1469

216

60.78

.000002

1555

128

0.63

.000004

148

50.81

.000002

2107

187

0.81

.000

002

2103

187

0.91

3105

163

0.80

1748

770.93

3085

151

0.93

3165

158

1.03

3833

201

1.01

6028

368

1.03

3306

151

1.04

3384

159

1.16

3180

167

1.29

5573

419

1.16

3107

160

1.16

3083

160

1.35

2373

184

1.64

1412

122

1.33

2259

159

1.34

2186

154

954

157

919

1136

192

1079

182

156

Table37:OSB

,V(yst

)∗,Nhan

dnhforskew

ness

=1.

3466,µ

=−

0.01

3284

8,σ

=0.

4077

880,N

=10

00,n

=10

0,nclass

=50,

y 0=

0.28

7577

30an

dd

=3.

1862

2440

.

LCRF

Geometric

LHDyn

amic

Prog.

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

21.12

.000653

633

461.00

.000704

514

301.19

.000641

689

541.17

.000

642

682

53367

54486

70311

4631

847

30.92

.000317

440

350.66

.000453

160

60.88

.000315

378

270.88

.000

316

381

281.43

384

291.51

692

681.36

399

271.39

421

31176

36148

26223

4619

841

40.80

.000187

293

240.54

.000299

652

0.82

.000183

323

280.82

.000

183

319

271.12

340

211.00

449

321.19

366

261.17

362

251.63

253

251.86

422

521.69

216

211.66

220

21114

3064

1495

2599

275

0.73

.000117

229

200.47

.000192

351

0.73

.000116

227

200.75

.000

117

244

220.99

274

170.78

242

140.99

271

161.02

287

181.31

256

191.28

472

441.29

253

181.33

235

171.82

165

192.11

221

331.74

163

171.79

154

1676

2530

886

2980

276

0.67

.000087

169

150.44

.000137

231

0.73

.000083

227

240.70

.000

087

198

190.92

271

200.66

137

70.97

261

180.93

242

161.18

245

181.00

354

271.22

221

151.17

239

171.50

163

151.51

338

391.55

152

141.47

160

141.94

104

132.29

130

212.05

105

151.94

112

1548

1918

534

1449

19

157

Table38:OSB

,V

(yst

)∗,Nhan

dnhforskew

ness

=1.

7274,µ

=−

0.00

4327

391,σ

=0.

5062

3313

0,N

=40

00,n

=40

0,nclass

=50,y 0

=0.

1458

6200

andd

=6.

4382

790.

LCRF

Geometric

LHDyn

amic

Prog.

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

21.30

.000277

2829

211

0.98

.000318

1934

921.28

.000276

2757

199

1.28

.000

276

2767

201

1171

189

2066

308

1243

201

1233

199

30.92

.000134

1726

116

0.52

.000262

387

90.97

.000133

1910

139

0.97

.000

133

1916

140

1.69

1689

146

1.85

3164

314

1.68

1489

119

1.69

1500

122

585

138

449

77601

142

584

138

40.79

.000079

1293

910.38

.000171

108

20.79

.000077

1306

930.82

.000

078

1416

107

1.30

1536

118

0.98

1826

106

1.22

1305

821.28

1350

922.08

878

982.54

1938

262

1.88

964

931.99

882

90293

93128

30425

2132

352

111

50.66

.000051

867

590.31

.000114

501

0.73

.000050

1061

820.73

.000

050

1083

851.05

1288

900.67

836

331.08

1175

761.08

1170

761.43

904

631.43

2174

200

1.51

952

771.52

936

752.21

719

100

3.07

891

151

2.22

595

772.23

596

77222

8849

15217

8821

587

60.66

.000037

867

690.28

.000085

251

0.67

.000035

883

720.67

.000

036

896

740.92

859

480.52

362

120.93

898

520.96

999

641.30

1103

940.98

1547

104

1.22

830

521.29

895

651.69

586

491.85

1617

194

1.61

691

581.72

661

642.46

434

673.49

423

802.27

495

702.46

396

58151

7326

9203

9615

375

158

Table39:OSB

,V(yst

)∗,Nhan

dnhforskew

ness

=2.

1145,µ

=−

0.00

8319

588,,N

=20

00,n

=20

0,nclass

=50,y 0

=0.

1141

7550

andd

=6.

8347

2120

.L

CRF

Geometric

LHDyn

amic

Prog.

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

21.34

.000892

1387

900.89

.001181

855

291.39

.000890

1418

951.41

.000

891

1428

96613

110

1145

171

582

105

572

104

30.93

.000418

909

540.45

.000736

185

30.98

.000416

978

621.00

.000

417

1010

661.89

801

661.77

1484

128

1.89

732

581.96

727

61290

80331

69290

8026

373

40.80

.000243

727

460.32

.000477

591

0.81

.000242

740

480.82

.000

243

751

491.34

660

430.89

796

371.36

655

431.39

667

452.30

445

482.49

1024

128

2.24

424

422.37

428

48168

63121

34181

6715

458

50.66

.000156

515

320.26

.000316

291

0.72

.000151

591

410.73

.000

151

608

431.07

592

360.59

359

111.15

594

381.16

583

371.62

471

361.34

998

801.73

464

401.73

458

392.57

315

453.06

557

892.72

259

362.72

259

36107

5157

1992

4592

456

0.66

.000108

515

390.23

.000214

181

0.65

.000106

498

370.64

.000

106

474

341.07

592

430.45

167

40.98

482

280.97

498

301.48

388

280.89

670

381.37

422

301.37

435

322.03

265

251.77

814

901.90

312

281.95

324

322.98

177

293.50

291

522.90

217

372.96

205

3563

3640

1569

4064

37

159

Table40:OSB

,V

(yst

)∗,Nhan

dnhforskew

ness

=3.

5009,µ

=−

0.00

3288

41,σ

=0.

6966

6629,N

=15

000,n

=15

00,nclass

=50,y 0

=0.

0504

1042

andd

=22.4

4861

984.

LCRF

Geometric

LHDyn

amic

Prog.

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

21.40

.000197

10293

560

1.06

.000234

8025

295

1.60

.000193

1127

371

51.62

.000

193

1135

372

94707

940

6975

1205

3727

785

3647

771

30.95

.000091

7036

354

0.39

.000222

1289

141.05

.000090

7938

456

1.09

.000

090

8269

497

2.30

6220

550

2.94

12826

1256

2.29

5318

443

2.40

5156

448

1744

596

885

230

1744

601

1575

555

40.95

.000053

7036

463

0.23

.000137

271

20.86

.000051

6283

373

0.87

.000

051

6324

378

1.85

5153

414

1.06

7754

353

1.60

4973

329

1.62

5026

341

3.64

2346

351

4.90

6803

1065

2.95

2863

343

3.01

2815

345

465

272

172

80881

455

835

436

50.50

.000037

2438

850.17

.000095

691

0.72

.000032

4798

280

0.74

.000

032

5004

303

0.95

4598

218

0.58

3203

831.22

4378

248

1.28

4552

279

1.85

5153

478

1.96

9234

837

1.93

3231

254

2.05

3187

277

3.19

2124

290

6.64

2445

544

3.28

1950

283

3.53

1742

274

687

429

4935

643

435

515

367

60.50

.000023

2438

105

0.14

.000070

301

0.62

.000021

3740

214

0.66

.000

022

4135

257

0.95

4598

269

0.39

1259

221.00

3717

187

1.08

4005

230

1.40

3257

192

1.06

6736

366

1.47

3208

202

1.62

3208

233

2.30

2963

340

2.94

6090

841

2.17

2349

217

2.43

2124

229

4.09

1433

311

8.14

860

247

3.50

1455

241

3.98

1186

236

311

283

2523

531

439

342

315

160

Table41:OSB

,V

(yst

)∗,Nhan

dnhforskew

ness

=4.

2624,µ

=0.

0008

4619

47,σ

=0.

8006

0853

37,N

=50

00,n

=40

0,nclass

=50,y 0

=0.

0470

6870

andd

=26.1

2477

998.

LCRF

Geometric

LHDyn

amic

Prog.

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

21.61

.001221

3612

147

1.11

.001510

2787

691.89

.001193

3928

191

1.89

.001

193

3932

192

1388

253

2213

331

1072

209

1068

208

31.09

.000543

2748

109

0.39

.001139

581

51.19

.000536

2959

130

1.20

.000

536

2976

132

2.66

1701

114

3.18

4034

299

2.89

1567

111

2.97

1574

115

551

177

385

96474

159

450

153

41.09

.000314

2748

143

0.23

.000705

162

10.92

.000307

2312

990.92

.000

307

2306

992.14

1404

851.11

2625

881.93

1660

981.89

1623

924.23

668

765.39

2121

272

3.95

817

933.86

843

93180

9692

39211

110

228

116

50.57

.000203

1190

350.17

.000496

501

0.75

.000186

1843

790.77

.000

187

1882

821.09

1558

570.59

1207

231.38

1442

651.44

1478

712.14

1404

102

2.09

2858

196

2.36

1003

692.48

991

734.23

668

917.39

851

159

4.35

543

734.62

515

78180

115

3421

169

114

134

966

0.57

.000131

1190

420.13

.000349

211

0.67

.000126

1571

700.67

.000

128

1558

691.09

1558

700.39

560

71.18

1364

601.18

1388

621.61

864

381.11

2206

891.82

919

501.89

982

602.66

837

743.18

1828

194

2.80

647

543.00

630

594.75

426

719.13

372

984.82

380

615.29

346

63125

105

1311

122

105

9687

161

Table42:OSB

,V(yst

)∗,Nhan

dnhforskew

ness

=3.

8763,µ

=0.

0044

6792

7,σ

=0.

8877

4036

3,N

=80

00,n

=70

0,nclass

=50,

y 0=

0.05

5686

01an

dd

=28.0

4155

725.

LCRF

Geometric

LHDyn

amic

Prog.

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

22.30

.000895

6587

374

1.25

.001106

4787

134

2.10

.000888

6369

333

2.17

.000

888

6440

346

1413

326

3213

566

1631

367

1560

354

31.18

.000395

4581

197

0.44

.000739

1438

161.27

.000392

4829

225

1.31

.000

393

4933

238

3.42

2757

247

3.53

5942

502

3.40

2501

216

3.58

2462

224

662

256

620

182

670

259

605

238

41.18

.000230

4581

258

0.26

.000453

518

40.95

.000215

3807

173

0.98

.000

217

3919

184

2.30

2006

124

1.25

4269

166

2.05

2501

154

2.16

2518

166

5.10

1142

164

5.93

3023

448

4.32

1296

156

4.77

1238

166

271

154

190

82396

217

325

184

50.62

.000140

2326

780.19

.000308

274

20.73

.000135

2901

121

0.80

.000

140

3189

148

1.18

2255

850.67

2307

521.42

2328

111

1.60

2401

134

2.30

2006

150

2.33

4043

315

2.49

1533

111

2.92

1491

136

4.54

1052

148

8.09

1296

286

4.69

903

128

5.78

711

130

361

239

8045

335

229

208

152

60.62

.000102

2326

950.16

.000221

145

10.65

.000091

2490

108

0.68

.000

095

2627

122

1.18

2255

104

0.44

1293

211.20

2169

991.28

2233

112

2.30

2006

182

1.25

3349

153

1.98

1558

992.15

1561

112

3.98

942

127

3.53

2593

317

3.16

1004

973.60

981

115

7.35

367

969.96

581

180

5.42

544

986.71

465

117

104

9639

28235

199

133

122

162

Table43:OSB

,V(yst

)∗,Nhan

dnhforskew

ness

=4.

3091,µ

=−

0.01

5183

27,σ

=0.

9953

0377,N

=10

000,n

=50

0,nclass

=20,

y 0=

0.02

2806

05an

dd

=30.2

9211

804.

LCRF

Geometric

LHDyn

amic

Prog.

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

23.05

.002003

8700

300

0.83

.003618

4353

322.47

.001945

8202

238

2.54

.001

946

8277

247

1300

200

5647

468

1798

262

1723

253

31.54

.000862

6713

185

0.25

.001686

838

21.46

.000860

6550

173

1.47

.000

860

6562

174

4.57

2694

156

2.76

7640

252

4.49

2835

164

4.50

2828

164

593

159

1522

246

615

163

610

162

41.54

.000540

6713

236

0.14

.001110

261

11.00

.000466

5081

122

1.05

.000

467

5277

134

3.05

1987

760.83

4092

502.40

3043

113

2.56

3026

122

7.60

1105

116

5.02

5160

330

5.72

1508

126

6.07

1379

122

195

72487

119

368

139

318

122

51.54

.000474

6713

254

0.10

.000709

105

10.77

.000298

4054

920.85

.000

302

4458

113

3.05

1987

820.41

1748

121.62

2860

821.87

2930

100

6.08

982

781.71

5242

149

3.06

1792

853.60

1651

9718

.20

301

807.19

2682

267

6.36

998

101

7.38

746

8617

6223

71296

140

215

104

60.08

.000503

491

0.65

.000202

3358

750.71

.000

209

3666

900.25

789

41.32

2823

771.44

2819

840.83

3515

552.32

1869

742.55

1806

802.76

4125

201

3.89

1118

704.47

1088

849.14

1383

186

7.22

609

768.68

464

72139

53223

128

157

90

163

Table44:OSB

,V

(yst

)∗,Nhan

dnhforskew

ness

=5.

4744,µ

=−

0.00

9671

336,σ

=1.

1040

6681

1,N

=50

00,n

=40

0,nclass

=50,y 0

=0.

0143

9434

andd

=46.0

0158

162.

LCRF

Geometric

LHDyn

amic

Prog.

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

22.77

.003678

4130

169

0.81

.007997

2129

193.17

.003627

4295

201

3.17

.003

627

4295

201

870

231

2871

381

705

199

705

199

31.85

.001535

3559

152

0.21

.003152

422

11.54

.001501

3256

117

1.69

.001

518

3420

136

5.53

1146

100

3.12

3859

185

5.36

1437

129

5.86

1311

126

295

148

719

214

306

154

269

138

40.93

.000800

2362

680.11

.002030

113

11.08

.000785

2641

891.17

.000

795

2798

104

2.77

1768

106

0.81

2016

312.77

1486

823.16

1492

997.37

699

101

6.12

2625

255

7.07

689

968.23

574

93171

125

246

113

184

133

136

104

50.93

.000476

2362

840.07

.001318

481

0.80

.000465

2114

640.91

.000

500

2321

811.85

1197

460.36

876

71.79

1397

572.17

1504

803.69

867

651.82

2616

103

3.56

885

634.55

757

748.29

438

799.16

1344

216

7.67

442

7010

.32

318

68136

126

116

73162

146

100

976

0.93

.000362

2362

100

0.06

.000942

261

0.64

.000297

1725

500.75

.000

343

1991

711.85

1197

560.21

396

21.35

1304

461.63

1364

632.77

571

270.81

1707

332.39

923

483.04

896

675.53

575

783.12

2152

153

4.19

572

515.64

457

5911

.05

214

5811

.99

657

161

8.11

337

6612

.18

230

7881

8162

50139

139

6262

164

Table45:OSB

,V(yst

)∗,Nhan

dnhforskew

ness

=6.

6147,µ

=−

0.01

0564

65,σ

=1.

2029

671,N

=30

00,n

=15

0,nclass

=50,

y 0=

0.02

2224

65an

dd

=65.2

5438

173.

LCRF

Geometric

LHDyn

amic

Prog.

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

OSB

V(yst)∗

Nh

nh

23.94

.014646

2608

761.20

.078002

1701

143.55

.014453

2562

703.88

.014

520

2602

76392

741299

136

438

8039

874

31.33

.006523

1786

290.32

.011590

519

21.77

.006265

2043

441.95

.006

361

2143

515.24

963

454.56

2180

8156.60

784

477.59

722

50251

76301

67173

5913

549

41.33

.003452

1786

390.16

.007350

204

11.25

.003393

1731

361.32

.003

483

1778

383.94

822

351.20

1497

193.50

825

323.87

823

359.16

295

268.87

1199

949.23

347

3210

.94

333

3997

50100

3697

5066

385

1.33

.002287

1786

460.11

.004999

108

10.91

.002068

1387

250.99

.002

089

1506

312.63

580

150.54

802

52.35

903

282.51

848

295.24

383

212.68

1470

455.20

458

285.58

428

2910

.46

176

1813.22

584

7911.68

196

2713

.95

187

3375

5036

2056

4231

286

1.33

.001741

1786

520.08

.003261

661

0.76

.001344

1235

230.81

.001

389

1298

272.63

580

180.32

453

21.75

800

211.90

811

253.94

242

71.20

1182

193.48

519

253.77

484

256.55

216

144.56

998

606.75

277

247.40

264

2711

.77

120

1417.25

282

5314.18

140

2816

.64

122

2556

4519

1529

2921

21

165

Appendix B: Computer Programs

B.1 C++ Computer program for determining OSW and OSB

for a Log-normal study variable/*This program finds the optimum strata boundaries, Optimum strata widthsand optimum value of the objective function of Log-Normal distribution*/

#include <iostream>#include <math.h>#include <assert.h>#include <conio.h>#include <stdio.h>#include <stdlib.h>

using namespace std;typedef double Number;

/*Program written as per the MPP for Lognormal Study Variable*/# define z 100 //(refine to 5 dp )//# define mu 0.00009935132# define sigma 1

Number erff(Number x);

double geterf(double x) return erff(x);

/*********************************************************************Returns the error functionerf(x) = 2*(int_0^x e^-t^2 dt)/sqrt(pi) .C.A. Bertulani May/15/2000

*********************************************************************/

Number erff(Number x)

166

Number gammp(Number a, Number x);

return x < 0.0 ? -gammp(0.5,x*x) : gammp(0.5,x*x);

/*********************************************************************Returns the complementary error functionerfc(x) = 1- erf(x)= 2*(int_x^infinity e^-t^2 dt)/sqrt(pi) .C.A. Bertulani May/15/2000

*********************************************************************/

Number erffc(Number x)Number gammp(Number a, Number x);Number gammq(Number a, Number x);

return x < 0.0 ? 1.0+gammp(0.5,x*x) : gammq(0.5,x*x);

/*********************************************************************Returns the imcomplete gamma functionP(a,x) = (int_0^x e^-t t^a-1 dt)/Gamma(a) , (a > 0).C.A. Bertulani May/15/2000

*********************************************************************/

Number gammp(Number a, Number x)void gcf(Number *gammcf, Number a, Number x, Number *gln);void gser(Number *gamser, Number a, Number x, Number *gln);Number gamser,gammcf,gln;

if (x < 0.0 || a <= 0.0) cerr<< "Invalid arguments in routine gammp";if (x < (a+1.0)) gser(&gamser,a,x,&gln);return gamser; else /* Use the continued fraction representation */gcf(&gammcf,a,x,&gln); /* and take its complement. */return 1.0-gammcf;

/*********************************************************************

167

Returns the imcomplete gamma functionQ(a,x) = 1-P(a,x)

= (int_x^infinity e^-t t^a-1 dt)/Gamma(a) , (a > 0).C.A. Bertulani May/15/2000

*********************************************************************/

Number gammq(Number a, Number x)void gcf(Number *gammcf, Number a, Number x, Number *gln);void gser(Number *gamser, Number a, Number x, Number *gln);Number gamser,gammcf,gln;

if (x < 0.0 || a <= 0.0) cerr << "Invalid arguments in routine gammq";if (x < (a+1.0)) /* Use the series representation */gser(&gamser,a,x,&gln);return 1.0-gamser; /* and take its complement. */ else /* Use the continued fraction representation. */gcf(&gammcf,a,x,&gln);return gammcf;

/*********************************************************************Returns the imcomplete gamma function P(a,x) evaluated by its seriesrepresentation as gamser.Also returns ln(Gamma(a)) as gln.C.A. Bertulani May/15/2000

*********************************************************************/

#define ITMAX 100#define EPS 3.0e-7

void gser(Number *gamser, Number a, Number x, Number *gln)Number gamma_ln(Number xx);int n;Number sum,del,ap;

*gln=gamma_ln(a);if (x <= 0.0) if (x < 0.0) cerr << "x less than 0 in routine gser";*gamser=0.0;return; else

168

ap=a;del=sum=1.0/a;for (n=1;n<=ITMAX;n++) ++ap;del *= x/ap;sum += del;if (fabs(del) < fabs(sum)*EPS) *gamser=sum*exp(-x+a*log(x)-(*gln));return;cerr << "a too large, ITMAX too small in routine gser";return;#undef ITMAX#undef EPS

/*********************************************************************Returns the imcomplete gamma function Q(a,x) evaluated by itscontinued fraction representation as gammcf.Also returns ln(Gamma(a)) as gln.C.A. Bertulani May/15/2000

*********************************************************************/

#define ITMAX 100 /* Maximum allowed number of iterations. */#define EPS 3.0e-7 /* Relative accuracy */#define FPMIN 1.0e-30 /* Number near the smallest representable *//* floating point number. */

void gcf(Number *gammcf, Number a, Number x, Number *gln)Number gamma_ln(Number xx);int i;Number an,b,c,d,del,h;

*gln=gamma_ln(a);b=x+1.0-a;/* etup fr evaluating continued fracion by modified Lent’z */c=1.0/FPMIN; /* method with b_0 = 0. */d=1.0/b;h=d;for (i=1;i<=ITMAX;i++) /* Iterate to convergence. */an = -i*(i-a);b += 2.0;

169

d=an*d+b;if (fabs(d) < FPMIN) d=FPMIN;c=b+an/c;if (fabs(c) < FPMIN) c=FPMIN;d=1.0/d;del=d*c;h *= del;if (fabs(del-1.0) < EPS) break;if (i > ITMAX) cerr << "a too large, ITMAX too small in gcf";*gammcf=exp(-x+a*log(x)-(*gln))*h; /* Put factors in front. */#undef ITMAX#undef EPS#undef FPMIN/********************************************************************

Returns the value of ln[Gamma(xx)] for xx > 0********************************************************************/

Number gamma_ln(Number xx)Number x,y,tmp,ser;static Number cof[6]=76.18009172947146,-86.50532032941677,24.01409824083091,-1.231739572450155,0.1208650973866179e-2,-0.5395239384953e-5;int j;

y=x=xx;tmp=x+5.5;tmp -= (x+0.5)*log(tmp);ser=1.000000000190015;for (j=0;j<=5;j++) ser += cof[j]/++y;return -tmp+log(2.5066282746310005*ser/x);/*********************************************************************/

/* Recursive function receives the parameter k and dk,yk to calculate f. */

//Please change here for the number of stages and the distance g and//initial value x0

170

int h ; // number of stagesconst double g = 13; //g is the distancedouble s = 0.00001; // s=x0, the initial valueconst double inc = 0.001; //PRECISION AMMOUNTconst double inc2 = 0.00001; //PRECISION AMMOUNTconst double prec = 1/inc;const int stages = 8;const int points = 1000 ; //Keep this to be 1/incconst int factor =4;

const int e = (int)(g*points*z+1);const int p=(int)(g*points);

//When passing parameter to function . n = your value divid by//inc to make it precise.// eg. function(3,1) will be passed as function(3,1000)

int ylimits[10];//stores the 3dp values for refining

double * minkf2[stages]; //stores minimum f to 6dp//double minkf2[stages][e]; //stores minimum f to 6dpdouble * dk2[stages]; //stores minimum d for the 6dp calculations

double RootVal(int k, double d, double y); // calculates the//value of the minimal elementsdouble fun(int,int,double ,int,int ,bool );double W(double d, double y);double MU(double d, double y, double w);double VAR(double d, double y, double w);void initArray(double ** arr, const int size, const int length)for (int i = 0; i < size; i++)arr[i] = new double[length];

for (int j = 0; j < length; j++)arr[i][j] = 0;

int main()

171

initArray(minkf2, stages, e);

initArray(dk2, stages, e);

//initialize minkfcout<<"Initializing points ...."<<endl;for (int i=0; i < stages;i++)for(int j=0;j<(p+1);j++)

minkf2[i][j]= -9999;

for (int k=0; k < stages;k++)for(int l=0;l<e;l++)

minkf2[k][l]= -9999;

cout<<"Initialiation complete"<<endl<<endl<<"Calculating...."<<endl<<endl;cout<<"enter h = Number of Stage " << endl;cin >> h;

// cout<<"enter s = Initial value " << endl;// cin >> s;

double n =100;//cout<<"enter n = Total Sample Size " << endl;//cin >> n;

double N=3369;//cout<<"enter N = Population Size " << endl;//cin >> N;

double f=fun(h,p,inc ,0,p ,true);

bool changes = false;

double d6,d5,d4,d3,d2,d1, y6,y5,y4,y3,y2,y1,x1 = 0,x2 = 0,x3 = 0,x4 = 0,x5 = 0,x6 = 0;double w6,w5,w4,w3,w2,w1 ;int temp;

//backward calculation for the 3dp resultsd6 = g;y6 = dk2[6][p];d5=d6-y6;temp = (int)(d5*points);y5=dk2[5][temp];

172

d4=d5-y5;temp = (int)(d4*points);y4=dk2[4][temp];d3=d4-y4;temp = (int)(d3*points);y3=dk2[3][temp];d2=d3-y3;temp = (int)(d2*points);y2=dk2[2][temp];d1=d2-y2;y1=d1;

//setup the limits for the 6dp calculations

temp = (int)(y6*points*z);ylimits[6] = temp;temp = (int)(y5*points*z);ylimits[5] = temp;temp = (int)(y4*points*z);ylimits[4] = temp;temp = (int)(y3*points*z);ylimits[3] = temp;temp = (int)(y2*points*z);ylimits[2] = temp;temp = (int)(y1*points*z);ylimits[1] = temp;

//printf("\n\nRefining...\n");

f=fun(h,e-1,inc2 ,ylimits[h]- factor*z,ylimits[h]+ factor*z ,false);cout <<"stage: h = " << h << " distance: g = " << g<< endl;

// printf("\n\nAccurate values derved after refining\n");

printf("\nf(h,g): %.10f \n" ,f);

//Backward calucation for the 6 dpd6=g;y6 = dk2[6][(e-1)];d5=d6-y6;temp = (int)(d5*points*z);y5=dk2[5][temp];d4=d5-y5;temp = (int)(d4*points*z);

173

y4=dk2[4][temp];d3=d4-y4;temp = (int)(d3*points*z);y3=dk2[3][temp];d2=d3-y3;temp = (int)(d2*points*z);y2=dk2[2][temp];d1=d2-y2;y1=d1;

x1=s+y1;x2=x1+y2;x3=x2+y3;x4=x3+y4;x5=x4+y5;x6=x5+y6;

w1 = W(d1, y1);w2 = W(d2, y2);w3 = W(d3, y3);w4 = W(d4, y4);w5 = W(d5, y5);w6 = W(d6, y6);

printf("\nd6: %f y6: %f x6: %f n6: %d" ,d6,y6,x6,(int)((n * RootVal(5, d6, y6)/ f)+0.5));printf("\nd5: %f y5: %f x5: %f n5: %d" ,d5,y5,x5,(int)((n * RootVal(4, d5, y5)/ f)+0.5));printf("\nd4: %f y4: %f x4: %f n4: %d" ,d4,y4,x4,(int)((n * RootVal(3, d4, y4)/ f)+0.5));printf("\nd3: %f y3: %f x3: %f n3: %d" ,d3,y3,x3,(int)((n * RootVal(2, d3, y3)/ f)+0.5));printf("\nd2: %f y2: %f x2: %f n2: %d" ,d2,y2,x2,(int)((n * RootVal(1, d2, y2)/ f)+0.5));printf("\nd1: %f y1: %f x1: %f n1: %d" ,d1,y1,x1,(int)((n * RootVal(0, d1, y1)/ f)+0.5));

cout << endl << endl << endl;

printf("\nw6: %f N6: %d mu6: %f V6: %f", w6, (int) (w6 * N + 0.5) ,MU(d6, y6, w6), VAR(d6, y6, w6) );

printf("\nw5: %f N5: %d mu5: %f V5: %f", w5, (int) (w5 * N + 0.5) ,MU(d5, y5, w5), VAR(d5, y5, w5) );

printf("\nw4: %f N4: %d mu4: %f V4: %f", w4, (int) (w4 * N + 0.5) ,

174

MU(d4, y4, w4), VAR(d4, y4, w4) );printf("\nw3: %f N3: %d mu3: %f V3: %f", w3, (int) (w3 * N + 0.5) ,MU(d3, y3, w3), VAR(d3, y3, w3) );

printf("\nw2: %f N2: %d mu2: %f V2: %f", w2, (int) (w2 * N + 0.5) ,MU(d2, y2, w2), VAR(d2, y2, w2) );

printf("\nw1: %f N1: %d mu1: %f V1: %f", w1, (int) (w1 * N + 0.5) ,MU(d1, y1, w1), VAR(d1, y1, w1) );

// while (changes);//getch();cout << endl;system("PAUSE");return 0;

//end main

double Minimum(double val1,double val2)// returns minimum of 2 numbers

if(val1<=val2)

return val1;

else

return val2;

//double fun(int k,int n,double incf,int minYk,int maxYk,bool isFirstRun)//this functions performs the same actions as "function".//it only defers in terms of the iterations of the for loop.

assert (k>=1); //Abort if k is negative

double dblRetVal;double d =n*incf; //d value for the functiondouble y;

double min;double val;double miny;int col;

175

if(k==1) //base casey = d;

dblRetVal = RootVal(k,d,y);elsefor(int i=minYk;i<=maxYk;i++)

y = i*incf;//this sets to precission of y to 6dp

double root;

root = RootVal(k,d,y); //calculate the root.

if(root != -1) //if root is valid

col =n-i;//get the current d value

if(minkf2[k-1][col]==-9999) //check if the result has been//previously calculated

if(isFirstRun)val = root+ fun((k-1),col,incf,0,col,true);//if not,//calculate the resultelseval = root+ fun((k-1),col,incf,ylimits[k-1]-factor*z,ylimits[k-1]+ factor*z,false);

elseval = root+ minkf2[k-1][col];

if (i==minYk)min =val;//base case

elsemin = Minimum(min,val);

176

if(min == val)miny=y;

//end for

dblRetVal = min;

//end else

//store the f and the d value of the minimum calculated.col = n;minkf2[k][col] = dblRetVal;dk2[k][col]=miny;

return dblRetVal;

//end function

double RootVal(int k, double d, double y)double rtval;double calc;

calc=((-0.5*exp(2*sigma*sigma)*(geterf((2*sigma*sigma-log(d+s))/(sigma*sqrt(2)))-geterf((2*sigma*sigma-log(d-y+s))/(sigma*sqrt(2)))))*(0.5*(geterf((log(d+s))/(sigma*sqrt(2)))-geterf((log(d-y+s))/(sigma*sqrt(2)))))-pow((0.5*exp(((sigma*sigma)*0.5))*(geterf((sigma*sigma-log(d+s))/(sigma*sqrt(2)))-geterf((sigma*sigma-log(d-y+s))/(sigma*sqrt(2))))),2));

if(calc<0)// cout<<"\nError: Negative Root\n";// rtval = -1;

else

177

calc = sqrt(calc);rtval = calc;

return rtval;

double VAR(double d, double y, double w)/*calculate the root//value of the current distribution*/

double rtval;double calc;

calc=((-0.5*exp(2*sigma*sigma)*(geterf((2*sigma*sigma-log(d+s))/(sigma*sqrt(2)))-geterf((2*sigma*sigma-log(d-y+s))/(sigma*sqrt(2)))))*(0.5*(geterf((log(d+s))/(sigma*sqrt(2)))-geterf((log(d-y+s))/(sigma*sqrt(2)))))-pow((0.5*exp((sigma*sigma*0.5))*(geterf((sigma*sigma-log(d+s))/(sigma*sqrt(2)))-geterf((sigma*sigma-log(d-y+s))/(sigma*sqrt(2))))),2));

calc = calc / pow(W(d,y), 2);

rtval = calc;return rtval;

double MU(double d, double y, double w)/*calculate the rootvalue of the current distribution*/double rtval;double calc;

calc = -1 * exp((sigma*sigma*0.5)) *(geterf((sigma*sigma-log(d+s))/(sigma*sqrt(2)))

- geterf((sigma*sigma-log(d-y+s))/(sigma*sqrt(2))))/( geterf((log(d+s))/(sigma*sqrt(2)))

- geterf((log(d-y+s))/(sigma*sqrt(2))) );rtval = calc;

return rtval;

178

double W(double d, double y)/*calculate the root value of thecurrent distribution*/double rtval;double calc;

//calc= pow(xm,a) * ( 1/pow(d-y+s,a) - 1/pow(d+s, a) );calc = 0.5 * (geterf((log(d+s))/(sigma*sqrt(2)))

-geterf((log(d-y+s))/(sigma*sqrt(2))));rtval = calc;

return rtval;

179

B.2 C++ Computer program for determining OSW and OSB

for a Pareto auxiliary variable

/*This program finds the optimum strata boundaries, Optimum strata widths,optimum value of objective function, for Pareto distribution*/

#include <iostream>#include <math.h>#include <assert.h>#include <conio.h>#include <stdio.h>using namespace std;typedef double Number;

/*Program written as per the MPP for Pareto auxiliary Variable*/# define z 100 //(refine to 5 dp )# define a 1.472 //alpha value# define b 1.000527 //x_m valueint h ; // number of stagesconst double c = 1; //Expected value of error in h-th stratumconst double b0 = 2; //b0 and b1 are the regression coefficientsconst double b1 = 0.5;const double g = 27.146593; // g is the distance of the distributionconst double s = b; // s=x0, the initial valueconst double inc = 0.001; //PRECISION AMMOUNTconst double inc2 = 0.00001; //PRECISION AMMOUNTconst double prec = 1/inc;const int stages = 8;const int points = 1000; //Keep this to be 1/incconst int factor = 4;const int e=(int)(g*points*z+1);const int p=(int)(g*points);int ylimits[10]; //stores the 3dp values for refiningdouble * minkf2[stages]; //stores minimum f to 6dpdouble * dk2[stages]; //stores minimum d for the 6dp calculationsdouble RootVal(int k, double d, double y);double fun(int,int,double ,int,int ,bool );double W(double d, double y);double MU(double d, double y, double w);double VAR(double d, double y, double w);

void initArray(double ** arr, const int size, const int length)for (int i = 0; i < size; i++)

180

arr[i] = new double[length];

for (int j = 0; j < length; j++)arr[i][j] = 0;

int main()initArray(minkf2, stages, e);initArray(dk2, stages, e);

cout<<"Initializing points ...."<<endl;for (int i=0; i < stages;i++)for(int j=0;j<(p+1);j++)

minkf2[i][j]= -9999;

for (int k=0; k < stages;k++)for(int l=0;l<e;l++)

minkf2[k][l]= -9999;

cout<<"Initialiation complete"<<endl<<endl<<"Calculating...."<<endl<<endl;cout<<"enter h = Number of Stage " << endl;cin >> h;double n;cout<<"enter n = Total Sample Size " << endl;cin >> n;double N;cout<<"enter N = Population Size " << endl;cin >> N;

double f=fun(h,p,inc ,0,p ,true);bool changes = false;

do double d6,d5,d4,d3,d2,d1, y6,y5,y4,y3,y2,y1,x1 = 0,x2 = 0,x3 = 0,x4 = 0,x5 = 0,x6 = 0;double w6,w5,w4,w3,w2,w1,z1,z2,z3,z4,z5,z6;int temp;//backward calculation for the 3dp resultsd6 = g;y6 = dk2[6][p];

181

d5=d6-y6;temp = (int)(d5*points);y5=dk2[5][temp];d4=d5-y5;temp = (int)(d4*points);y4=dk2[4][temp];d3=d4-y4;temp = (int)(d3*points);y3=dk2[3][temp];d2=d3-y3;temp = (int)(d2*points);y2=dk2[2][temp];d1=d2-y2;y1=d1;//setup the limits for the 6dp calculationstemp = (int)(y6*points*z);ylimits[6] = temp;temp = (int)(y5*points*z);ylimits[5] = temp;temp = (int)(y4*points*z);ylimits[4] = temp;temp = (int)(y3*points*z);ylimits[3] = temp;temp = (int)(y2*points*z);ylimits[2] = temp;temp = (int)(y1*points*z);ylimits[1] = temp;f=fun(h,e-1,inc2 ,ylimits[h]- factor*z,ylimits[h]+ factor*z ,false);cout <<"stage: h = " << h << " distance: g = " << g<< endl;printf("\nf(h,g): %.10f \n" ,f);//Backward calucation for the 6 dpd6=g;y6 = dk2[6][(e-1)];d5=d6-y6;temp = (int)(d5*points*z);y5=dk2[5][temp];d4=d5-y5;temp = (int)(d4*points*z);y4=dk2[4][temp];d3=d4-y4;temp = (int)(d3*points*z);y3=dk2[3][temp];if( h > 2)

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double c3;w3 = W(d3, y3);c3 = (n* pow(VAR(d3, y3, w3),0.5))/f;if (c3>500)

cout << "y3 value does not satisfy rule. Enter new value:";cin >> y3;changes = true;

d2=d3-y3;temp = (int)(d2*points*z);y2=dk2[2][temp];

if( h > 1)double c2;w2 = W(d2, y2);c2 = (n* pow(VAR(d2, y2, w2),0.5))/f;

if (c2>500)

cout << "y2 value does not satisfy rule. Enter new value:";cin >> y2;changes = true;

w1 = W(d1, y1),w2 = W(d2, y2),w3 = W(d3, y3),w4 = W(d4, y4),w5 = W(d5, y5),w6 = W(d6, y6);d1=d2-y2;y1=d1;x1=b+y1;z1=b0+b1*x1;if( h > 2)

x2=x1+y2;z2=b0+b1*x2;if (h > 3)

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x3=x2+y3;z3=b0+b1*x3;if ( h > 4)

x4=x3+y4;z4=b0+b1*x4;if (h > 5)x5=x4+y5;z5=b0+b1*x5;if ( h > 6)

x6=x5+y6;z6=b0+b1*x6;

double sum_WhSh = RootVal(0, d3, y3) + RootVal(0, d2, y2) +RootVal(0, d1, y1);printf("\nd6: %f l6: %f x6: %f y6: %f n6: %d" ,d6,y6,x6,z6,(int)(n * RootVal(5, d6, y6)/ sum_WhSh +0.5));printf("\nd5: %f l5: %f x5: %f y5: %f n5: %d" ,d5,y5,x5,z5,(int)(n * RootVal(4, d5, y5)/ sum_WhSh +0.5));printf("\nd4: %f l4: %f x4: %f y4: %f n4: %d" ,d4,y4,x4,z4,(int)(n * RootVal(3, d4, y4)/ sum_WhSh +0.5));printf("\nd3: %f l3: %f x3: %f y3: %f n3: %d" ,d3,y3,x3,z3,(int)(n * RootVal(2, d3, y3)/ sum_WhSh +0.5));printf("\nd2: %f l2: %f x2: %f y2: %f n2: %d" ,d2,y2,x2,z2,(int)(n * RootVal(1, d2, y2)/ sum_WhSh +0.5));printf("\nd1: %f l1: %f x1: %f y1: %f n1: %d" ,d1,y1,x1,z1,(int)(n * RootVal(0, d1, y1)/ sum_WhSh +0.5));cout << endl << endl << endl;printf("\nw6: %f N6: %d mu6: %f V6: %f", w6, (int) (w6 * N + 0.5) ,MU(d6, y6, w6), VAR(d6, y6, w6) );printf("\nw5: %f N5: %d mu5: %f V5: %f", w5, (int) (w5 * N + 0.5) ,MU(d5, y5, w5), VAR(d5, y5, w5) );printf("\nw4: %f N4: %d mu4: %f V4: %f", w4, (int) (w4 * N + 0.5) ,MU(d4, y4, w4), VAR(d4, y4, w4) );printf("\nw3: %f N3: %d mu3: %f V3: %f", w3, (int) (w3 * N + 0.5) ,MU(d3, y3, w3), VAR(d3, y3, w3) );printf("\nw2: %f N2: %d mu2: %f V2: %f", w2, (int) (w2 * N + 0.5) ,MU(d2, y2, w2), VAR(d2, y2, w2) );printf("\nw1: %f N1: %d mu1: %f V1: %f", w1, (int) (w1 * N + 0.5) ,MU(d1, y1, w1), VAR(d1, y1, w1) );

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while (changes);getch();

//end main

double Minimum(double val1,double val2) // returns minimum of 2 numbersif(val1<=val2)return val1;elsereturn val2;

double VAR(double d, double y, double w)/*calculate the root value of thecurrent distribution*/double rtval;double calc;calc=((-1*pow(b1,2)*pow(a,1)*pow(b,0))/(pow(2-a,1)))*(pow(d-y+s,2-a)-pow(d+s,2-a))/(pow(d-y+s,-1*a)-pow(d+s,-1*a))-((pow(b1,2)*pow(a,2)*pow(b,0))/(pow(1-a,2)))*pow((pow(d-y+s,1-a)-pow(d+s,1-a)),2)/pow((pow(d-y+s,-1*a)-pow(d+s,-1*a)),2)+c;

rtval = calc;return rtval;

double MU(double d, double y, double w)/*calculate the root value of thecurrent distribution*/double rtval;double calc;

calc= b0-((b1*a*( pow(d-y+s,1-a) - pow(d+s, 1-a) ))/((1-a)*( pow(d-y+s,-1*a)- pow(d+s, -1*a) )));

rtval = calc;return rtval;

double W(double d, double y)/*calculate the root value of thecurrent distribution*/

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double rtval;double calc;calc=pow(b,a) *(pow(d-y+s,-1*a)-pow(d+s,-1*a));

rtval = calc;return rtval;

double RootVal(int k, double d, double y)/*calculate the root value of thecurrent distribution*/double rtval;double calc;calc=((-1*pow(b1,2)*pow(a,1)*pow(b,2*a))/(pow(2-a,1)))*(pow(d-y+s,2-a)-pow(d+s,2-a))*(pow(d-y+s,-1*a)-pow(d+s,-1*a))-((pow(b1,2)*pow(a,2)*pow(b,2*a))/(pow(1-a,2)))*(pow(d-y+s,1-a)-pow(d+s,1-a))*(pow(d-y+s,1-a)-pow(d+s,1-a))+(c*pow(b,2*a)*pow((pow(d-y+s,-1*a)-pow(d+s,-1*a)),2));

if(calc<0)elsecalc = sqrt(calc);

rtval = calc;

return rtval;

double fun(int k,int n,double incf,int minYk,int maxYk,bool isFirstRun)/*this functions performs the same actions as "function". It only defers interms of the iterations of the for loop.*/assert (k>=1); //Abort if k is negativedouble dblRetVal;

double d =n*incf; //d value for the functiondouble y;double min;double val;double miny;int col;

if(k==1) //base case

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y = d;dblRetVal = RootVal(k,d,y);

elsefor(int i=minYk;i<=maxYk;i++)/*iterate over the interval allowed to

calculate the 6dp results*/

y = i*incf;//this sets to precission of y to 6dpdouble root;root = RootVal(k,d,y); //calculate the root.if(root != -1) //if root is valid

col =n-i;//get the current d value

if(minkf2[k-1][col]==-9999) /*check if the result has beenpreviously calculated*/

if(isFirstRun)val = root+ fun((k-1),col,incf,0,col,true);//if not, calculate the resultelseval = root+ fun((k-1),col,incf,ylimits[k-1]- factor*z,ylimits[k-1]+ factor*z,false);

elseval = root+ minkf2[k-1][col];//if result exists, use it for calculations

if (i==minYk)

min =val;//base case

else

min = Minimum(min,val);//get the minimum if the result and the current mininmum

if(min == val)miny=y;//get the position of the current minimum//end for

dblRetVal = min;

//end else//store the f and the d value of the minimum calculated.

col = n;minkf2[k][col] = dblRetVal;dk2[k][col]=miny;

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return dblRetVal;

//end function

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B.3 MATLAB Computer program for simulation study on cali-

bration estimators

t = cputime;% TimeNUM_SIMULATIONS = 100000; %No of samples/iterationsNUMEL = 106;%Population sizeSTRATA_BOUNDARIES = [6/NUMEL 6/NUMEL 8/NUMEL 10/NUMEL 12/NUMEL 4/NUMEL30/NUMEL 17/NUMEL 10/NUMEL 3/NUMEL ];%Statum weight WhSAMPLES = 40; %Total Sample sizeSAMPLES_PER_STRATA = [3/SAMPLES 3/SAMPLES 3/SAMPLES 3/SAMPLES 4/SAMPLES2/SAMPLES 11/SAMPLES 6/SAMPLES 3/SAMPLES 2/SAMPLES];V= (STRATA_BOUNDARIES./(SAMPLES_PER_STRATA*SAMPLES));%Values of Vh=Wh/nhybar = zeros(NUM_SIMULATIONS,13);ystar = zeros(NUM_SIMULATIONS,13);

%Open csv files to write datafid_data = fopen(’data.csv’,’w’);fid_rao = fopen(’data_rao.csv’,’w’);fid_singh_03 = fopen(’data_singh_03.csv’,’w’);fid_tracy = fopen(’data_tracy.csv’,’w’);fid_standard = fopen(’data_standard.csv’,’w’);fid_singh_98 = fopen(’data_singh_98.csv’,’w’);fid_new_01 = fopen(’data_new_01.csv’,’w’);fid_new_02 = fopen(’data_new_02.csv’,’w’);fid_jacek1 = fopen(’data_jacek1.csv’,’w’);fid_jacek2 = fopen(’data_jacek2.csv’,’w’);fid_Mtracy = fopen(’data_Mtracy.csv’,’w’);fid_rao1 = fopen(’data_rao1.csv’,’w’);fid_rao3 = fopen(’data_rao3.csv’,’w’);fid_rao4 = fopen(’data_rao4.csv’,’w’);

for i=1:NUM_SIMULATIONSfileData = csvread(’Tabacco.csv’,0,1);X1 = fileData(:,1);X2 = fileData(:,2);Y = fileData(:,3);% Selecting sample unitsstrata = zeros(numel(STRATA_BOUNDARIES), 1);for j=1:numel(STRATA_BOUNDARIES)

if (j==1)start = 1;finish = STRATA_BOUNDARIES(j)*NUMEL;

elsestart = sum(STRATA_BOUNDARIES(1:j-1))*NUMEL+1;

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finish = sum(STRATA_BOUNDARIES(1:j))*NUMEL;endtemp = randperm(uint32(finish-(start-1)));temp = temp(1:SAMPLES*SAMPLES_PER_STRATA(j));dlmwrite (’output.rao2.csv’, temp(1:SAMPLES*SAMPLES_PER_STRATA(j)),’-append’);index_listj = uint32(sort( temp + (start - 1)));strata(j) = finish - start + 1;

end% Find basic calculations for each variableNUM_STRATA = numel(STRATA_BOUNDARIES);% Calcualations for X1[ x1 ] = single_var( X1, index_list, strata );% Calcualations for X2[ x2 ] = single_var( X2, index_list, strata );% Calcualations for Y[ y ] = single_var( Y, index_list, strata );print_data(fid_data, x1, x2, y);[ybar(i,1), ystar(i,1)] = standard(STRATA_BOUNDARIES, fid_standard, x1, y);[ybar(i,2), ystar(i,2)] = singh_98(STRATA_BOUNDARIES, fid_singh_98, x1, y);[ybar(i,3), ystar(i,3)] = singh_03(STRATA_BOUNDARIES, fid_singh_03, x1, y);[ybar(i,4), ystar(i,4)] = tracy(STRATA_BOUNDARIES, fid_tracy, x1, y);[ybar(i,5), ystar(i,5)] = Mtracy(STRATA_BOUNDARIES, fid_Mtracy, x1, y);[ybar(i,6), ystar(i,6)] = new_01(STRATA_BOUNDARIES, V, fid_new_01, x1, y);[ybar(i,7), ystar(i,7)] = new_02(STRATA_BOUNDARIES, V, fid_new_02, x1, y);[ybar(i,8), ystar(i,8)] = jacek1(STRATA_BOUNDARIES, fid_jacek1, x1, y);[ybar(i,9), ystar(i,9)] = jacek2(STRATA_BOUNDARIES, fid_jacek2, x1, y);[ybar(i,10), ystar(i,10)] = rao1(STRATA_BOUNDARIES, fid_rao1, x1, x2, y);[ybar(i,11), ystar(i,11)] = mvc_rao(STRATA_BOUNDARIES, fid_rao, x1, x2, y);[ybar(i,12), ystar(i,12)] = rao3(STRATA_BOUNDARIES, fid_rao3, x1, x2, y);[ybar(i,13), ystar(i,13)] = rao4(STRATA_BOUNDARIES, fid_rao4, x1, x2, y);

end

%Close csv files after writing datafclose(fid_rao);fclose(fid_singh_03);fclose(fid_tracy);fclose(fid_standard);fclose(fid_singh_98);fclose(fid_data);fclose(fid_new_01);fclose(fid_new_02);fclose(fid_jacek1);fclose(fid_jacek2);

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fclose(fid_Mtracy);fclose(fid_rao1);fclose(fid_rao3);fclose(fid_rao4);

%Calculations of MSE, Bias, VarianceMSE = mean((ystar-y.BAR).^2);bias = mean(ystar)-y.BAR;rel.bias = (mean(ystar)-y.BAR)/y.BAR;

%Print MSE, Bias, Variancestr = ’stratified’;’singh 98’; ’singh 03’; ’tracy’; ’Mtracy’; ’new 01’;’new 02’;’jacek1’; ’jacek2’; ’rao1’; ’ra02’;’rao3’;’rao4’;;disp(’MSE =>>’);for i=1:numel(str)

fprintf(’%s = \t%f\n’, stri, MSE(i));enddisp(’Bias =>>’);for i=1:numel(str)

fprintf(’%s = \t%f\n’, stri, bias(i));enddisp(’RB =>>’);for i=1:numel(str)

fprintf(’%s = \t%f\n’, stri, rel.bias(i));enddisp(’VAR =>>’);for i=1:numel(str)

fprintf(’%s = \t%f\n’, stri, MSE(i)-bias(i)^2);endfprintf(’CPU TIME: %f\n’, cputime-t);%clear *;

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