Research Article Estimation of Population Mean in Chain...

Research ArticleEstimation of Population Mean in Chain Ratio-TypeEstimator under Systematic Sampling

Mursala Khan1 and Rajesh Singh2

1Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan2Department of Statistics, Banaras Hindu University, Varanasi 221005, India

Correspondence should be addressed to Mursala Khan; [email protected]

Received 7 September 2015; Accepted 11 October 2015

Academic Editor: Shein-chung Chow

Copyright © 2015 M. Khan and R. Singh. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

A chain ratio-type estimator is proposed for the estimation of finite population mean under systematic sampling scheme usingtwo auxiliary variables. The mean square error of the proposed estimator is derived up to the first order of approximation and iscompared with other relevant existing estimators. To illustrate the performances of the different estimators in comparison with theusual simple estimator, we have taken a real data set from the literature of survey sampling.

1. Introduction and Literature Review

Incorporating the knowledge of the auxiliary variables is veryimportant for the construction of efficient estimators for theestimation of population parameters and increasing the effi-ciency of the estimators in different sampling design. Usingthe knowledge of the auxiliary variables, several authorshave proposed different estimation technique for the finitepopulation mean of the study variable; Cochran [1], Tripathy[2], Kadilar and Cingi [3, 4], Singh et al. [5], Khan andArunachalam [6], Lone and Tailor [7], Khan [8], Khan andHussain [9], and Khan et al. [10] have worked on the estima-tion of population parameters using auxiliary information.

In the present paper, we will work on the estimationof population mean using the knowledge of the auxiliaryvariables under systematic sampling. Various statisticianshave worked on the estimation of population mean insystematic sampling: Cochran [11], Hansen et al. [12], Robson[13], Swain [14], Singh [15], Shukla [16], Kushwaha and Singh[17], Banarasi et al. [18], R. Singh andH. P. Singh [19], Singh etal. [20], Singh and Solanki [21], Singh and Jatwa [22], Singhet al. [23, 24], Tailor et al. [25], Verma and Singh [26], andVerma et al. [27], and so forth.

Consider a finite population 𝑈 = {𝑈1, 𝑈2, . . . , 𝑈

𝑁} of size

𝑁 units, numbered from 1 to 𝑁 in some order. A sample of

size 𝑛 units is taken at random from the first 𝑘 units and every𝑘th subsequent unit; then,𝑁 = 𝑛𝑘 where 𝑛 and 𝑘 are positiveintegers; thus, there will be 𝑘 samples (clusters) each of size𝑛 and observe the study variate 𝑦 and auxiliary variate 𝑥 foreach and every unit selected in the sample. Let (𝑦

𝑖𝑗, 𝑥𝑖𝑗), for 𝑖 =

1, 2, . . . , 𝑘 and 𝑗 = 1, 2, . . . , 𝑛: denote the value of 𝑗th unit inthe 𝑖th sample.Then, the systematic samplemeans are definedas follows: 𝑦

𝑠𝑦= 𝑡0= (1/𝑛)∑

𝑛

𝑗=1𝑦𝑖𝑗and 𝑥

𝑠𝑦= (1/𝑛)∑

𝑛

𝑗=1𝑥𝑖𝑗

are the unbiased estimators of the population means 𝑌 =

(1/𝑁)∑𝑁

𝑗=1𝑦𝑖𝑗and 𝑋 = (1/𝑁)∑

𝑁

𝑗=1𝑥𝑖𝑗of 𝑦 and 𝑥, respec-

tively. Let 𝑆2𝑦= (1/(𝑁− 1))∑

𝑘

𝑖=1∑𝑛

𝑗=1(𝑦𝑖𝑗−𝑌)2, 𝑆2𝑥= (1/(𝑁−

1))∑𝑘

𝑖=1∑𝑛

𝑗=1(𝑥𝑖𝑗− 𝑋)2, and 𝑆

2

𝑧= (1/(𝑁 − 1))∑

𝑘

𝑖=1∑𝑛

𝑗=1(𝑧𝑖𝑗−

𝑍)2 be the population variances of the study variable and

the auxiliary variables, respectively, with the correspondingpopulation covariance’s 𝑆

𝑦𝑥= (1/(𝑁 − 1))∑

𝑘

𝑖=1∑𝑛

𝑗=1(𝑥𝑖𝑗

−

𝑋)(𝑦𝑖𝑗

− 𝑌), 𝑆𝑦𝑧

= (1/(𝑁 − 1))∑𝑘

𝑖=1∑𝑛

𝑗=1(𝑦𝑖𝑗

− 𝑌)(𝑧𝑖𝑗

− 𝑍),and 𝑆𝑥𝑧

= (1/(𝑁 − 1))∑𝑘

𝑖=1∑𝑛

𝑗=1(𝑥𝑖𝑗− 𝑋)(𝑧

𝑖𝑗− 𝑍) among the

three variables 𝑦, 𝑥, and 𝑧, respectively. Also 𝐶2

𝑦, 𝐶2𝑥, and 𝐶

2

𝑧

are the knownpopulation coefficients of variation of the studyvariable and the auxiliary variables, respectively.

To obtain the properties of the estimators up to firstorder of approximation, we use the following errors terms:

Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2015, Article ID 248374, 5 pageshttp://dx.doi.org/10.1155/2015/248374

2 Journal of Probability and Statistics

𝑒0= (𝑦𝑠𝑦𝑠

− 𝑌)/𝑌, 𝑒1= (𝑥𝑠𝑦𝑠

− 𝑋)/𝑋, and 𝑒2= (𝑧𝑠𝑦𝑠

− 𝑍)/𝑍,such that 𝐸(𝑒

𝑖) = 0, for 𝑖 = 0, 1, and 2.

The first order of approximation of the above errors termsis given by

𝐸 (𝑒2

0) = 𝜃𝜌

∗

𝑦𝐶2

𝑦,

𝐸 (𝑒2

1) = 𝜃𝜌

∗

𝑥𝐶2

𝑥,

𝐸 (𝑒2

2) = 𝜃𝜌

∗

𝑧𝐶2

𝑧,

𝐸 (𝑒0𝑒1) = 𝜃𝑘𝐶

2

𝑥√𝜌∗𝑦𝜌∗𝑥,

𝐸 (𝑒0𝑒2) = 𝜃𝑘

∗𝐶2

𝑧√𝜌∗𝑦𝜌∗𝑧,

𝐸 (𝑒1𝑒2) = 𝜃𝑘

∗∗𝐶2

𝑧√𝜌∗𝑥𝜌∗𝑧,

(1)

where

𝜆 = (𝑁 − 1

𝑛𝑁) ,

𝜌𝑦𝑥

=

𝑆𝑦𝑥

𝑆𝑦𝑆𝑥

,

𝜌𝑦𝑧

=

𝑆𝑦𝑧

𝑆𝑦𝑆𝑧

,

𝜌𝑥𝑧

=𝑆𝑥𝑧

𝑆𝑥𝑆𝑧

,

𝑘 =

𝜌𝑦𝑥

𝐶𝑦

𝐶𝑥

,

𝑘∗=

𝜌𝑦𝑧

𝐶𝑦

𝐶𝑧

,

𝜌∗

𝑦= {1 + (𝑛 − 1) 𝜌

𝑦} ,

𝜌∗

𝑥= {1 + (𝑛 − 1) 𝜌

𝑥} ,

𝜌∗

𝑧= {1 + (𝑛 − 1) 𝜌

𝑧} ,

𝜌∗∗

=

𝜌∗

𝑦

𝜌∗𝑥

,

𝜌∗∗

2=

𝜌∗

𝑦

𝜌∗𝑧

,

𝜌∗∗

1=

𝜌∗

𝑥

𝜌∗𝑧

,

(2)

where 𝜌𝑦, 𝜌𝑥, and 𝜌

𝑧are the intraclass correlation among the

pair of units for the variables 𝑦, 𝑥, and 𝑧, respectively.The variance of the usual unbiased estimator for popula-

tion mean is

Var (𝑡0) = 𝜆𝑌

2

𝜌∗

𝑦𝐶2

𝑦. (3)

The classical ratio and product estimators for finite popula-tion mean suggested by Swain [14] and Shukla [16] are givenby

𝑡1= 𝑦𝑠𝑦

(𝑋

𝑥𝑠𝑦

) ,

𝑡2= 𝑦𝑠𝑦exp(

𝑧𝑠𝑦

𝑍

) .

(4)

The mean square errors of the estimators, to the first order ofapproximation, are given as follows:

MSE (𝑡1) = 𝜆𝑌

2

[𝜌∗

𝑦𝐶2

𝑦+ 𝜌∗

𝑥𝐶2

𝑥(1 − 2𝑘√𝜌∗∗)] , (5)


2

[𝜌∗

𝑦𝐶2

𝑦+ 𝜌∗

𝑧𝐶2

𝑧(1 + 2𝑘

∗√𝜌∗∗

2)] . (6)

The usual regression estimator, using single auxiliary variableand its variance, is given as follows:

𝑡3= 𝑦𝑠𝑦

+ 𝑏 (𝑋 − 𝑥𝑠𝑦) , (7)

MSE (𝑡3) = 𝜆𝜌

∗

𝑦𝑆2

𝑦[1 − 𝜌

2

𝑦𝑥] . (8)

Utilizing the known knowledge of the auxiliary variable,Singh et al. [20] suggested the following ratio and producttype exponential estimators:


𝑋 − 𝑥𝑠𝑦

𝑋 + 𝑥𝑠𝑦

) ,


𝑥𝑠𝑦

− 𝑋

𝑥𝑠𝑦

+ 𝑋) .

(9)

The mean square errors of the estimators up to first order ofapproximation are given by


2

[𝜌∗

𝑦𝐶2

𝑦+

𝜌∗

𝑥𝐶2

𝑥

4(1 − 4𝑘√𝜌∗∗)] , (10)


2

[𝜌∗

𝑦𝐶2

𝑦+

𝜌∗

𝑥𝐶2

𝑥

4(1 + 4𝑘√𝜌∗∗)] . (11)

After that, Tailor et al. [25] define the following ratio-cum-product estimator for the population mean 𝑌:

𝑡6= 𝑦𝑠𝑦

(𝑋

𝑥𝑠𝑦

)(

𝑧𝑠𝑦

𝑍

) . (12)

The mean square error of the estimator 𝑡6, up to first order of

approximation, is given by


2

[𝜌∗

𝑦𝐶2

𝑦+ 𝜌∗

𝑥𝐶2

𝑥(1 − 2𝑘√𝜌∗∗)

+ 𝜌∗

𝑧𝐶2

𝑧(1 − 2𝑘

∗∗√𝜌∗∗

1) + 2𝑘

∗𝐶2

𝑧√𝜌∗𝑦𝜌∗𝑧] .

(13)

Journal of Probability and Statistics 3

2. Proposed Estimator

In this section, we have proposed the following regression inratio-cum-product type estimator for the unknown popula-tion mean under systematic sampling:

𝑡𝑚

= 𝑦𝑠𝑦

(𝑋

𝑋 + 𝑏𝑦𝑥

(𝑥𝑠𝑦

− 𝑋)

)

𝛿1

⋅ (

𝑍 + 𝑏𝑦𝑧

(𝑧𝑠𝑦

− 𝑍)

𝑍

)

𝛿2

,

(14)

where 𝛿1and 𝛿2are the unknown constants, whose values are

to be found for the minimummean square error.The mean square error (MSE) of the estimator up to first

order of approximation is

MSE (𝑡𝑚) = 𝜆𝑌

2

[𝜌∗

𝑦𝐶2

𝑦+ 𝛿2

1𝛽2

𝑦𝑥𝜌∗

𝑥𝐶2

𝑥+ 𝛿2

2𝛽2

𝑦𝑧𝜌∗

𝑧𝐶2

𝑧

− 2𝛿1𝛽𝑦𝑥

𝑘𝐶2

𝑥√𝜌∗𝑦𝜌∗𝑥+ 2𝛿2𝛽𝑦𝑧

𝑘∗𝐶2

𝑧√𝜌∗𝑦𝜌∗𝑧

− 2𝛿1𝛿2𝛽𝑦𝑥

𝛽𝑦𝑧

𝑘∗∗

𝐶2

𝑧√𝜌∗𝑥𝜌∗𝑧] .

(15)

On differentiating (15), with respect to 𝛿1and 𝛿

2, we obtain

the minimummean squared error of the estimator 𝑡𝑚, which

is given by

MSE (𝑡𝑚)

= 𝜆𝑌2

𝜌∗

𝑦[

[

𝐶2

𝑦− 𝑘2∗

𝑥𝐶2

𝑧−

(𝑘∗𝑘∗∗

𝐶2

𝑧− 𝑘𝐶2

𝑥)2

(𝐶2𝑥− 𝑘2∗∗𝐶2

𝑧)

]

]

,

(16)

where the optimum values are 𝛿1

= √𝜌∗𝑦(𝑘𝐶2

𝑥− 𝑘∗𝑘∗∗

𝐶2

𝑧)/

𝛽𝑦𝑥√𝜌∗𝑥(𝐶2

𝑥− 𝑘2∗∗

𝐶2

𝑧) and 𝛿

2= (√𝜌∗

𝑦/𝛽𝑦𝑧√𝜌∗𝑧){𝑘∗∗

(𝑘𝐶2

𝑥−

𝑘∗𝑘∗∗

𝐶2

𝑧)/(𝐶2

𝑥− 𝑘2∗∗

𝐶2

𝑧) − 𝑘∗}.

3. Comparison

In this section, we have compared the MSE of the proposedestimator with the MSEs of simple estimator, Swain [14]estimator, Shukla [16] estimator, Singh et al. [20] estimators,and Tailor et al. [25] estimator and found some theoreticalconditions under which the proposed estimator will alwaysperform better:

(i) By (16) and (3), MSE(𝑡𝑚) ≤ MSE(𝑡

0) if

[

[

𝑘2∗

𝑥𝐶2

𝑧+

(𝑘∗𝑘∗∗

𝐶2

𝑧− 𝑘𝐶2

𝑥)2

(𝐶2𝑥− 𝑘2∗∗𝐶2

𝑧)

]

]

≥ 0. (17)

(ii) By (16) and (5), MSE(𝑡𝑚) ≤ MSE(𝑡

1) if

[

[

𝜌∗

𝑦

{

{

{

𝑘2∗

𝑥𝐶2

𝑧+

(𝑘∗𝑘∗∗

𝐶2

𝑧− 𝑘𝐶2

𝑥)2

(𝐶2𝑥− 𝑘2∗∗𝐶2

𝑧)

}

}

}

+ 𝜌∗

𝑥𝐶2

𝑥(1 − 2𝑘√𝜌∗∗)]

]

≥ 0.

(18)

(iii) By (16) and (6), MSE(𝑡𝑚) ≤ MSE(𝑡

2) if

[

[

𝜌∗

𝑦

{

{

{

𝑘2∗

𝑥𝐶2

𝑧+

(𝑘∗𝑘∗∗

𝐶2

𝑧− 𝑘𝐶2

𝑥)2

(𝐶2𝑥− 𝑘2∗∗𝐶2

𝑧)

}

}

}

+ 𝜌∗

𝑧𝐶2

𝑧(1 + 2𝑘

∗√𝜌∗∗

2)]

]

≥ 0.

(19)

(iv) By (16) and (8), MSE(𝑡𝑚) ≤ MSE(𝑡

3) if

[

[

𝑘2∗

𝑥𝐶2

𝑧− 𝐶2

𝑦𝜌2

𝑦𝑥+

(𝑘∗𝑘∗∗

𝐶2

𝑧− 𝑘𝐶2

𝑥)2

(𝐶2𝑥− 𝑘2∗∗𝐶2

𝑧)

]

]

≥ 0. (20)

(v) By (16) and (10), MSE(𝑡𝑚) ≤ MSE(𝑡

4) if

[

[

𝜌∗

𝑦

{

{

{

𝑘2∗

𝑥𝐶2

𝑧+

(𝑘∗𝑘∗∗

𝐶2

𝑧− 𝑘𝐶2

𝑥)2

(𝐶2𝑥− 𝑘2∗∗𝐶2

𝑧)

}

}

}

+𝜌∗

𝑥𝐶2

𝑥

4(1 − 4𝑘√𝜌∗∗)]

]

≥ 0.

(21)

(vi) By (16) and (11), MSE(𝑡𝑚) ≤ MSE(𝑡

5) if

[

[

𝜌∗

𝑥𝐶2

𝑥

4(1 + 4𝑘√𝜌∗∗)

+ 𝜌∗

𝑦

{

{

{

𝑘2∗

𝑥𝐶2

𝑧+

(𝑘∗𝑘∗∗

𝐶2

𝑧− 𝑘𝐶2

𝑥)2

(𝐶2𝑥− 𝑘2∗∗𝐶2

𝑧)

}

}

}

]

]

≥ 0.

(22)

(vii) By (16) and (13), MSE(𝑡𝑚) ≤ MSE(𝑡

6) if

[

[

𝜌∗

𝑥𝐶2

𝑥(1 − 2𝑘√𝜌∗∗) + 𝜌

∗

𝑧𝐶2

𝑧(1 − 2𝑘

∗∗√𝜌∗∗

1)

+ 2𝑘∗𝐶2

𝑧√𝜌∗𝑦𝜌∗𝑧

+ 𝜌∗

𝑦

{

{

{

𝑘2∗

𝑥𝐶2

𝑧+

(𝑘∗𝑘∗∗

𝐶2

𝑧− 𝑘𝐶2

𝑥)2

(𝐶2𝑥− 𝑘2∗∗𝐶2

𝑧)

}

}

}

]

]

≥ 0.

(23)

4 Journal of Probability and Statistics

Table 1: The percent relative efficiency of different estimators withrespect to 𝑡

0.

PopulationEstimator MSE(𝑡

𝛼) PRE(𝑡

𝛼, 𝑡0)

𝑡0

1455.08 100.00𝑡1

373.32 389.62𝑡2

768.06 189.45𝑡3

45.52 3196.57𝑡4

820.09 177.43𝑡5

1044.42 139.32𝑡6

187.08 777.79𝑡𝑚

22.74 6400.00

4. Numerical Comparison

For comparing the theoretical efficiency conditions of thedifferent estimators numerically, we have used the followingreal data set.

Population 1 (source: Tailor et al. [25]). Consider

𝑁 = 15,

𝑛 = 3,

𝑋 = 44.47,

𝑌 = 80,

𝑍 = 48.40,

𝐶𝑦= 0.56,

𝐶𝑥= 0.28,

𝐶𝑧= 0.43,

𝑆2

𝑦= 2000,

𝑆2

𝑥= 149.55,

𝑆2

𝑧= 427.83,

𝑆𝑦𝑥

= 538.57,

𝑆𝑦𝑧

= −902.86,

𝑆𝑥𝑧

= −241.06,

𝜌𝑦𝑥

= 0.9848,

𝜌𝑦𝑧

= −0.9760,

𝜌𝑥𝑧

= −0.9530,

𝜌𝑦= 0.6652,

𝜌𝑥= 0.707,

𝜌𝑧= 0.5487.

(24)

For the percent relative efficiencies (PREs) of the estimator,we use the following formula and the results are shown inTable 1:

PRE(𝑡𝛼, 𝑡0) = MSE(𝑡

0)/MSE(𝑡

𝛼) × 100, for 𝛼 = 0, 1, 2,

3, 4, 5, 6, and 𝑚.

5. Conclusion

A chain ratio-type estimator is proposed under double sam-pling scheme using two auxiliary variables, and the propertiesof the proposed estimator are derived up to first order ofapproximations. Both theoretically and empirically, it hasbeen shown that the recommended estimator performedbetter than the other competing estimators in terms of higherpercent relative efficiency. Hence, looking on the dominancenature of the proposed estimator may be suggested for itspractical applications.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

The authors are thankful to the anonymous learned refereesfor their valuable suggestions regarding the improvement ofthe paper.

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