Research Article Bayesian Estimation of Inequality and...
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Research ArticleBayesian Estimation of Inequality and PovertyIndices in Case of Pareto Distribution Using DifferentPriors under LINEX Loss Function
Kamaljit Kaur, Sangeeta Arora, and Kalpana K. Mahajan
Department of Statistics, Panjab University, Chandigarh 160014, India
Correspondence should be addressed to Kamaljit Kaur; [email protected]
Received 29 August 2014; Accepted 7 January 2015
Academic Editor: Karthik Devarajan
Copyright Β© 2015 Kamaljit Kaur et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Bayesian estimators of Gini index and a Poverty measure are obtained in case of Pareto distribution under censored and completesetup. The said estimators are obtained using two noninformative priors, namely, uniform prior and Jeffreysβ prior, and oneconjugate prior under the assumption of Linear Exponential (LINEX) loss function. Using simulation techniques, the relativeefficiency of proposed estimators using different priors and loss functions is obtained.The performances of the proposed estimatorshave been compared on the basis of their simulated risks obtained under LINEX loss function.
1. Introduction
The Pareto distribution is a skewed, heavy-tailed distributionthat is used to model the distribution of incomes and otherfinancial variables. It was introduced by Pareto [1] which hasa probability density function of the form
π (π₯) ={{{
πΌππΌ
π₯πΌ+1, π β€ π₯ < β; π, πΌ > 0,
0, otherwise,(1)
and cumulative distribution function is
πΉ (π₯) ={{{
1 β (π
π₯)πΌ
, π₯ β₯ π,
0, otherwise.(2)
The parameter π in (2) represents the minimum income inthe population under study and assumed to be known, whilethe other parameter πΌ is assumed to be unknown.
The average income for Pareto distribution is
π =ππΌ
(πΌ β 1), πΌ > 1. (3)
In the context of income inequality and poverty, Gini indexand Poverty measure head count ratio are two most popularindices [2, 3]. Gini index is generally defined as
πΊ = 1 β twice the area under the Lorenz curve
= 1 β 2β«1
0
πΏ (π) ππ, 0 β€ π β€ 1,(4)
where πΏ(π) = (1/π) β«π
0
πΉβ1(π‘)ππ‘ is the equation of the Lorenzcurve and π = β«
1
0
πΉβ1(π‘)ππ‘ is the mean of the distribution.Equivalently, Gini index can also be defined as
πΊ =Ξ
2π, (5)
where Ξ = β«β
0
β«β
0
|π₯ β π¦|π(π₯)π(π¦)ππ₯ ππ¦ is population Ginimean difference.
The Poverty index head count ratio π0is simply the count
of the number of households whose incomes are below thepoverty line divided by the total population. In terms ofcontinuous distribution,
π0= β«π€0
0
π (π¦) ππ¦ = πΉ (π€0) , (6)
where, π€0(> π) is called Poverty Line.
Hindawi Publishing CorporationAdvances in StatisticsVolume 2015, Article ID 964824, 10 pageshttp://dx.doi.org/10.1155/2015/964824
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2 Advances in Statistics
In case of Pareto distribution,Gini index (πΊ) [4, 5] is givenby
πΊ =1
(2πΌ β 1), πΌ >
1
2, (7)
and Poverty measure (π0) is
π0= πΉ (π€
0)
= 1 β (π
π€0
)πΌ
= 1 β ππΌ0,
(8)
where, π€0(> π) and π
0= (π/π€
0).
Thus, π€0is per capita annual income representing a
minimum acceptable standard of living and π0represents the
proportion of population having income equal to or less thanπ€0.The estimation of Gini index (πΊ) and Poverty measure
(π0) and the associated inference using classical approach
(parametric and nonparametric) is available in literature [5β8]. However, in the Bayesian setup, this has not evoked theinterest of many researchers [9, 10]. In the present paper,our focus will be on the estimation of inequality and povertyindices in the Bayesian setup.
When the Bayesian method is used, the choice of appro-priate prior distribution plays an important role, whichmay be categorized as informative, noninformative, andconjugate priors [11, 12]. In the present paper, three priors(two noninformative priors and one conjugate prior) are usedto estimate shape parameter, Gini index, Average income, andPovertymeasure.The two noninformative priors areUniformprior and Jeffreysβ prior, while conjugate prior is chosen asTruncated Erlang distribution.
In Bayesian estimation, the criterion for good estimatorsfor the parameters of interest is the choice of appropriate lossfunction. In Bayesian estimation, two types of loss functionscommonly used are Squared error loss function (SELF) andLinear exponential (LINEX) loss function. The simplest typeof loss function is squared error, which is also referred to asquadratic loss is given as
πΏ (π) = (π β π)2
, (9)
where π is the estimator of π.The usual squared error loss function is symmetrical and
associates equal importance to the losses due to overestima-tion and underestimation of equal magnitude. However, sucha restriction may be impractical; for example, in estimationof shape parameter of Classical Pareto distribution, theoverestimation and underestimation may not be of equalimportance as over estimate of shape parameter gives anunder-estimate of inequality index which seems to be moreserious as compared to under estimate of shape parameterbecause we are often interested in reducing income inequalityindex. This leads one to think that an asymmetrical lossfunction be considered for estimation of shape parame-ter which associates greater importance to overestimation.Anumber of asymmetrical loss functions have been proposed
in statistical literature [13β16]. Varian [16] proposed a usefulasymmetrical loss function known as Linear exponential(LINEX) loss function which is given as
πΏ (π β π) = ππ(Μπβπ) β π (π β π) β 1, π ΜΈ= 0. (10)
The posterior expectation of the LINEX loss function (10) is
πΈ (πΏ (π β π)) = ππΜππΈ (πβππ) β π (π β πΈ (π)) β 1, (11)
where πΈ(β ) denotes posterior expectation with respect to theposterior density of π.
By a result of Zellner [17] the Bayes estimator of π denotedby π under the LINEX loss function is the value whichminimizes posterior expectation and is given by
π = β1
πln [πΈ (πβππ)] , (12)
provided that the expectation πΈ(πβππ) exists and is finite [18].In Figures 1(a) and 1(b), values of πΏ(π) are plotted for the
selected values of π for π = 1 and π = β1. It is seen that, forπ = 1, the function is quite asymmetricwith a value exceedingthe target being more serious than a value below the target.But, for π = β1, the function is also quite asymmetric with avalue below the target value being more serious than a valueexceeding the target.
For small value of π, the LINEX loss function can beexpanded by Taylorβs series expansion as
exp (π (π β π)) β π (π β π) β 1
=β
βπ=0
ππ (π β π)π
π!β π (π β π) β 1
=β
βπ=2
ππ (π β π)π
π!
βπ2 (π β π)
2
2.
(13)
Thus, the LINEX loss function is approximately equal tosquared error loss function for small values of b (seeFigure 1(c)).
This loss function has been considered by Zellner [17],Basu and Ebrahimi [19], and Afify [20] for different distri-butions.
In the present study, LINEX loss function is used forestimating the shape parameter, Gini index, Mean income,and a Poverty measure in the context of Pareto distributionusing noninformative priors (Uniform prior and Jeffreysβprior) and one conjugate prior (Truncated Erlang distribu-tion) along with some assumptions regarding the sampledpopulation. Bayesian approach with prior and posteriordistributions along with sampling schemes in the contextof Pareto distribution is given in Section 2. In Section 3,Bayesian estimators of shape parameter, Gini index, Meanincome, and Poverty measure using different priors under
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Advances in Statistics 3
858.5 859.0 859.5 860.0 860.5 861.0
0.0
0.5
1.0
1.5
L(π)
π
(a)
858.5 859.0 859.5 860.0 860.5 861.0
0.0
0.5
1.0
1.5
L(π)
π
(b)
858.5 859.0 859.5 860.0 860.5 861.0
0.000
0.002
0.004
0.006
L(π)
π
(c)
Figure 1: (a) LINEX Loss function when π = 859.5 and π = 1. (b) LINEX Loss function when π = 859.5 and π = β1. (c) LINEX Loss functionwhen π = 859.5 and π = 0.1.
the assumption of LINEX loss function are obtained. Finally,in Section 4, simulation is done to compare the efficiencyof three different approaches using three priors and lossfunctions. The robustness of the hyperparameters is given inSection 4.1 through simulation study. Section 5 presents theconclusion of the study.
2. Preliminary about Sampling Scheme, Priors,and Posterior Densities
The Bayesian analysis of the Pareto distribution (2) is basedon the following censored sampling scheme on personalincome data. It is assumed that annual incomes of the πpersons are under study but exact figures π₯
1, π₯2, π₯3, . . . , π₯
πare
available only for those π individuals whose annual incomedoes not exceed a prescribed annual incomeπ€
0(> π), and for
the remaining (πβπ) individuals, the exact income figures are
unknown but we do know that their annual income exceedthe prescribed figureπ€
0. Before the arrival of the sample data
on personal incomes, π is predetermined but not π, whichis a random. This censoring scheme used is referred as rightcensored sampling scheme.
The likelihood function πΏ(πΌ) for complete sample in caseof Pareto distribution [4] is
πΏ (πΌ) = πΌππππΌ(π
βπ=1
π₯π)
β(πΌ+1)
. (14)
In case of censored data, the likelihood function for anydistribution [21] is
πΏ (π; πΌ) =π!
(π β π)!
π
βπ=1
π (π₯; πΌ) [1 β πΉ (π₯; πΌ)]πβπ . (15)
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4 Advances in Statistics
The likelihood function for Pareto distribution in censoredsample is
πΏ (πΌ) =πΌππππΌπ(πβπ)πΌ
π€
(βπ
π=1π₯π)(πΌ+1)
β πΌππβπΌππ€ , πΌ β (πΏ,β) , (16)
where ππ€= ln(πβππ
π€) is product income statistics [22] and
ππ€= π€πβπ(β
π
π=1π₯π).
Bayes estimators of Gini index and Average incomewill not be convergent in the interval [0, 1/2] and [0, 1],respectively, and the method will fail to work. Hence, thisdifficulty is removed by assuming πΏ > 1, to obtain differentBayes estimators.
The prior and posterior densities for noninformativepriors (Uniform prior and Jeffreysβ prior) and conjugate priorare explained below.(i) Uniform Prior. In practice, the informative priors arenot always available; for such situations, the use of nonin-formative priors is recommended. One of the most widelyused noninformative prior, due to Laplace [23], is a uniformprior. Therefore, the uniform prior has been assumed for theestimation of the shape parameter of the Pareto distribution.
Uniform prior for πΌ isππ’(πΌ) β 1. (17)
Combine likelihood function (16) with the prior density (17)by using Bayes theorem to obtain the posterior density as
πβπ’(πΌ) =
πΏ (πΌ) β π (πΌ)
β«β
πΏ
πΏ (πΌ) β π (πΌ) ππΌ
=(ππ€)π+1
Ξ (π + 1, ππ€πΏ)πΌππβπΌππ€ ,
(18)
where Ξ(π, π¦) = β«β
π¦
π’πβ1πβπ’ππ’, π¦ > 0 is the upper incompletegamma function and posterior density πβ
π’(πΌ) is left truncated
Gamma distribution.(ii) Jeffreysβ Prior. Another noninformative prior has beensuggested by Jeffreys [24] which is frequently used in situa-tions where one does not have much information about theparameters. This is defined as the distribution of the param-eters proportional to the square root of the determinants ofthe Fisher information matrix, that is, π(πΌ) β βπΌ(πΌ), whereπΌ(πΌ) = βπΈ[(π2/ππΌ2) log πΏ(πΌ | π₯)] is Fisherβs information ofthe given distribution. In case of Pareto distribution,
ππ(πΌ) β
βπ
πΌ. (19)
A motivation for Jeffreysβ prior is that Fisherβs information(πΌ(πΌ)) is an indicator of the amount of information broughtby the model (observations) about πΌ.
The posterior density is obtained as
πβπ(πΌ) =
(ππ€)π
Ξ (π, ππ€πΏ)πΌπβ1πβπΌππ€ , (πΏ β€ πΌ < β) , (20)
which is left truncated Gamma distribution.
Note: Extension of Jeffreysβ Prior. Jeffreysβ prior is a particularcase of extension of Jeffreysβ prior proposed by Al-Kutubi andIbrahim [25], defined as
π (πΌ) β [πΌ (πΌ)]π , (21)
where π is a positive constant. For π = 0.5, it reduces toJeffreysβ prior.
In case of Pareto distribution, this prior is
ππ(πΌ) β (
π
πΌ2)π
. (22)
Theposterior distribution by using extension to Jeffreysβ prioris obtained as
πβπ(πΌ) =
(ππ€)πβ2π+1
Ξ (π β 2π + 1, ππ€πΏ)πΌπβ2ππβπΌππ€ , (πΏ β€ πΌ < β) .
(23)
(iii) Conjugate Prior. The conjugate prior was introducedby Raiffa and Schlaifer [26], where the prior and posteriordistributions are from the same family, that is, the form ofthe posterior density has the same distributional form as theprior distribution. For the existence of Gini index and Meanincome for the Pareto distribution, wemust take into accounta truncated prior distribution since the random variable πΌ isdefined in (πΏ,β), where the constant πΏ > 1 is assumed to beknown.
Let πΌ have Truncated Erlang distribution [22]
ππ(πΌ) =
(π½)π
Ξ (π, πΏπ½)πΌπβ1πβπ½πΌ
βΌ TED (π½, π; πΏ) ,
(πΏ < πΌ < β, πΏ > 1, π½ > 0, π = 1, 2, . . .) ,
(24)
where π½ and π are the hyperparameters.The posterior density for πΌ is
πβπ(πΌ) =
(π½ + ππ€)π+1
Ξ (π + π, (π½ + ππ€) πΏ)
πΌπ+πβ1πβ(π½+ππ€)πΌ
βΌ TED ((π½ + ππ€) , (π + π) ; πΏ) .
(25)
The posterior density (πβπ(πΌ)) follows Truncated Erlang dis-
tribution with parameters (π½ + ππ€) and (π + π).
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Advances in Statistics 5
3. Bayesian Estimation under LinearExponential (LINEX) Loss Function UsingDifferent Priors
3.1. Bayesian Estimators Using Uniform Prior. Bayesian esti-mator οΏ½ΜοΏ½ of πΌ using uniform prior (17) and posterior density(18), under the assumption of the LINEX loss function (ref.(12)) is obtained as
οΏ½ΜοΏ½π’= β
1
πlogπΈ [πβππΌ] ,
πΈ [πβππΌ] = β«β
πΏ
πβππΌπβπ’(πΌ) ππΌ
=(ππ€)π+1
Ξ (π + 1, ππ€πΏ)
β«β
πΏ
πΌππβ(π+ππ€)πΌππΌ
=(ππ€)π
Ξ (π + 1, ππ€πΏ)
Ξ (π + 1, (π + ππ€) πΏ)
(π + ππ€)π
=Ξ (π + 1, (π + π
π€) πΏ)
Ξ (π + 1, ππ€πΏ)
(ππ€
π + ππ€
)π+1
.
(26)
Therefore,
οΏ½ΜοΏ½π’= β
1
πlog(
Ξ (π + 1, (π + ππ€) πΏ)
Ξ (π + 1, ππ€πΏ)
(ππ€
π + ππ€
)π+1
) . (27)
The Bayes estimator πΊ of πΊ, using uniform prior is
πΊπ’= β
1
πlogπΈ [πβππΊ] ,
πΈ [πβππΊ] = πΈ [πβπ/(2πΌβ1)]
=(ππ€)π+1
Ξ (π + 1, ππ€πΏ)
β«β
πΏ
πΌππβ(π/(2πΌβ1)+πΌππ€)ππΌ
(28)
putting π‘ = 2πΌ β 1
= (ππ€
2)π+1
πβππ€/2
Ξ (π + 1, ππ€πΏ)
β π
βπ=0
(ππ)β«β
2πΏβ1
π‘ππβ(π/π‘+π‘ππ€/2)ππ‘
(By Binomial expansion)
=(ππ€)πβ1
πβππ€/2
2πΞ (π + 1, ππ€πΏ)
β π
βπ=0
(ππ)(
2π
ππ€
)(π+1)/2
πΎπ+1
(β2πππ€)
(29)
(using formula (9) of 3.471, page 368 of Gradshteyn andRyzhik [27] β«
β
0
π₯]β1πβπ½/π₯βπΎπ₯ππ₯ = 2(π½/πΎ)]/2πΎ](2βπ½πΎ),[Reπ½ > 0, Re πΎ > 0] whereπΎ](β ) is modified Bessel functionof third kind).
Thereby,
πΊπ’= β
1
πlog(
(ππ€)πβ1
πβππ€/2
2πΞ (π + 1, ππ€πΏ)
β π
βπ=0
(ππ)(
2π
ππ€
)(π+1)/2
πΎπ+1
(β2πππ€)) .
(30)
The Bayes estimator οΏ½ΜοΏ½ of π, using uniform prior is
οΏ½ΜοΏ½π’= β
1
πlogπΈ [πβππ] ,
πΈ [πβππ] = πΈ [πβπππΌ/(πΌβ1)]
=(ππ€)π+1
Ξ (π + 1, ππ€πΏ)
β«β
πΏ
πΌππβ(πππΌ/(πΌβ1)+πΌππ€)ππΌ
putting π‘ = πΌ β 1
=(ππ€)π+1
πβ(ππ+ππ€)
Ξ (π + 1, ππ€πΏ)
β π
βπ=0
(ππ)β«β
πΏβ1
π‘ππβ(ππ/π‘+π‘ππ€)ππ‘
=(ππ€)π+1
πβ(ππ+ππ€)
Ξ (π + 1, ππ€πΏ)
β π
βπ=0
(ππ) 2(
ππ
ππ€
)(π+1)/2
πΎπ+1
(2βππππ€)
(31)
(using formula (9) of 3.471, page 368 of Gradshteyn andRyzhik [27] β«
β
0
π₯]β1πβπ½/π₯βπΎπ₯ππ₯ = 2(π½/πΎ)]/2πΎ](2βπ½πΎ),[Reπ½ > 0, Re πΎ > 0] whereπΎ](β ) is modified Bessel functionof third kind)
οΏ½ΜοΏ½π’= β
1
πlog(
(ππ€)π+1
πβ(ππ+ππ€)
Ξ (π + 1, ππ€πΏ)
β π
βπ=0
(ππ) 2(
ππ
ππ€
)(π+1)/2
πΎπ+1
(2βππππ€)) .
(32)
The Bayes estimator οΏ½ΜοΏ½0of π0, using uniform prior, is
οΏ½ΜοΏ½0π’= β
1
πlogπΈ [πβππ0] ,
πΈ [πβππ0] = πΈ [πβπ(1βππΌ
0)]
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6 Advances in Statistics
=(ππ€)π+1
Ξ (π + 1, ππ€πΏ)
β«β
πΏ
πβπ(1βππΌ
0)πΌππβπΌππ€ππΌ,
οΏ½ΜοΏ½0π’= β
1
πlog(
(ππ€)π+1
Ξ (π + 1, ππ€πΏ)
β«β
πΏ
πβπ(1βππΌ
0)πΌπβ1πβπΌππ€ππΌ) .
(33)
3.2. Bayesian Estimators Using Jeffreysβ Prior. In case ofJeffreysβ prior (19) and using posterior density (20), theBayesian estimators of πΌ,πΊ,π, and π
0under the assumption
of the LINEX loss function are obtained as follows:
οΏ½ΜοΏ½π= β
1
πlogπΈ [πβππΌ]
= β1
πlog(β«
β
πΏ
πβππΌπβπ(πΌ) ππΌ)
= β1
πlog(
Ξ (π, (π + ππ€) πΏ)
Ξ (π, ππ€πΏ)
(ππ€
π + ππ€
)π
) ,
πΊπ= β
1
πlogπΈ [πβππΊ]
= β1
πlog(β«
β
πΏ
πβπ/(2πΌβ1)πβπ(πΌ) ππΌ)
= β1
πlog(
(ππ€)π
πβππ€/2
2πβ1Ξ (π, ππ€πΏ)
β πβ1
βπ=0
(π β 1π
)(2π
ππ€
)(π+1)/2
πΎπ+1
(β2πππ€)) ,
οΏ½ΜοΏ½π= β
1
πlogπΈ [πβππ]
= β1
πlog(β«
β
πΏ
πβπππΌ/(πΌβ1)πβπ(πΌ) ππΌ)
= β1
πlog(
(ππ€)π
πβ(ππ+ππ€)
Ξ (π, ππ€πΏ)
β πβ1
βπ=0
(π β 1π
) 2(ππ
ππ€
)(π+1)/2
β πΎπ+1
(2βππππ€)) ,
οΏ½ΜοΏ½0π= β
1
πlogπΈ [πβππ0]
= β1
πlog(β«
β
πΏ
πβπ(1βππΌ
0)πβπ(πΌ) ππΌ)
= β1
πlog(
(ππ€)π
Ξ (π, ππ€πΏ)
β«β
πΏ
πβπ(1βππΌ
0)πΌπβ1πβπΌππ€ππΌ) .
(34)
Note. The expression for extension of Jeffreysβ prior can beobtained with some modifications in Jeffreysβ prior and arelisted below:
οΏ½ΜοΏ½π= β
1
πlog(
Ξ (π β 2π + 1, (π + ππ€) πΏ)
Ξ (π β 2π + 1, ππ€πΏ)
(ππ€
π + ππ€
)πβ2π+1
) ,
πΊπ= β
1
πlog(
(ππ€)πβ2π+1
πβππ€/2
2πβ2πΞ (π β 2π + 1, ππ€πΏ)
β πβ2π
βπ=0
(π β 2ππ
)(2π
ππ€
)(π+1)/2
πΎπ+1
(β2πππ€)),
οΏ½ΜοΏ½π= β
1
πlog(
(ππ€)πβ2π+1
πβ(ππ+ππ€)
Ξ (π β 2π + 1, ππ€πΏ)
β πβ2π
βπ=0
(π β 2ππ
) 2(ππ
ππ€
)(π+1)/2
β πΎπ+1
(2βππππ€)) ,
οΏ½ΜοΏ½0π= β
1
πlog(
(ππ€)πβ2π+1
Ξ (π β 2π + 1, ππ€πΏ)
β β«β
πΏ
πβπ(1βππΌ
0)πΌπβ2ππβπΌππ€ππΌ) .
(35)
3.3. Bayesian Estimators Using Conjugate Prior. Using theBayesian posterior density (25), the Bayes estimators of πΌ, πΊ,π, and π
0, under the assumption of the LINEX loss function
are
οΏ½ΜοΏ½π= β
1
πlogπΈ [πβππΌ]
= β1
πlog(β«
β
πΏ
πβππΌπβπ(πΌ) ππΌ)
= β1
πlog(
Ξ (π + π, (π + π½ + ππ€) πΏ)
Ξ (π + π, (π½ + ππ€) πΏ)
(π½ + π
π€
π + π½ + ππ€
)π+π
) ,
πΊπ= β
1
πlogπΈ [πβππΊ]
= β1
πlog(β«
β
πΏ
πβπ/(2πΌβ1)πβπ(πΌ) ππΌ)
= β1
πlog(
(π½ + ππ€)π+π
πβ(π½+ππ€)/2
2π+πβ1Ξ (π + π, (π½ + ππ€) πΏ)
β π+πβ1
βπ=0
(π + π β 1
π)(
2π
π½ + ππ€
)
(π+1)/2
β πΎπ+1
(β2πππ€)) ,
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Advances in Statistics 7
οΏ½ΜοΏ½π= β
1
πlogπΈ [πβππ]
= β1
πlog(β«
β
πΏ
πβπππΌ/(πΌβ1)πβπ(πΌ) ππΌ)
= β1
πlog(
(π½ + ππ€)π+π
πβ(ππ+π½+ππ€)
Ξ (π + π, (π½ + ππ€) πΏ)
β π+πβ1
βπ=0
(π + π β 1
π) 2(
ππ
π½ + ππ€
)
(π+1)/2
β πΎπ+1
(2βππππ€)) ,
οΏ½ΜοΏ½0π= β
1
πlogπΈ [πβππ0]
= β1
πlog(β«
β
πΏ
πβπ(1βππΌ
0)πβπ(πΌ) ππΌ)
= β1
πlog(
(π½ + ππ€)π+π
Ξ (π + π, (π½ + ππ€) πΏ)
β β«β
πΏ
πβπ(1βππΌ
0)πΌπ+πβ1πβ(π½+ππ€)πΌππΌ) .
(36)
Note: Case of Complete Sample. The Bayesian estimatorsfor complete sample can be obtained using noninformativepriors and conjugate prior by simply substituting π = π in theabove estimators.
4. Simulation Study
In order to assess the statistical performance of these esti-mators of shape parameter, Gini index, Mean income, andPoverty measure using LINEX loss function, a simulationstudy is conduced. The estimated losses are computed usinggenerated random samples from Pareto distribution of dif-ferent sizes. These estimated losses are computed for samplesizes π = 20 (20) 100, πΌ = 2.5 (1) 4.5, π = 1, πΏ = 1.5,and π = 450. The value of π€
0= 859.6 should be taken
from Poverty line given by the Government of India in 2009-10 for urban people. For the conjugate prior, the values ofhyperparameter are taken as π½ = 0.5, π = 2; π½ = 2, andπ = 2. The estimated losses of πΌ, πΊ, π, and π
0with LINEX
loss function by using noninformative (Uniform prior andJeffreysβ prior) and conjugate priors are tabulated in Tables1, 2, 3, and 4, respectively.
It is observed from the above simulation study (ref. Tables1, 2, 3, and 4) that
(i) Bayesian estimators with conjugate prior (hyperpa-rameter π½ = 0.5, π = 2) perform better as compared tononinformative priors as it has smaller estimated lossfor πΌ, πΊ,π, and π
0;
(ii) in case of noninformative priors, Jeffreysβ prior hasless estimated loss than uniform prior, which impliesthat Bayesian methods with Jeffreysβ prior are better;
Table 1: Estimated loss functions for πΌ using LINEX loss function.
π πΌUniformprior
Jeffreyβsprior
Conjugate priorπ½ = 0.5π = 2
π½ = 2π = 2
202.5 0.200543 0.173893 0.112013 0.1132993.5 0.423936 0.357678 0.281125 0.3718234.5 0.719781 0.456351 0.311154 0.710794
402.5 0.110843 0.077269 0.050112 0.0728723.5 0.207535 0.204212 0.145707 0.1743984.5 0.324085 0.228739 0.207738 0.344289
602.5 0.065696 0.061891 0.058858 0.0593363.5 0.135812 0.104322 0.102511 0.1235644.5 0.283127 0.211419 0.149228 0.224148
802.5 0.048582 0.052477 0.044407 0.0452433.5 0.094729 0.094126 0.081215 0.0898614.5 0.146575 0.140906 0.126948 0.163061
1002.5 0.047068 0.040324 0.034990 0.0383363.5 0.072414 0.071366 0.065080 0.0705024.5 0.112283 0.104459 0.099383 0.131260
Table 2: Estimated loss functions for G using LINEX loss function.
π πΌUniformprior
Jeffreyβsprior
Conjugate priorπ½ = 0.5π = 2
π½ = 2π = 2
202.5 0.003944 0.0031157 0.002672 0.0573223.5 0.000849 0.0007378 0.000700 0.0167334.5 0.000671 0.0005303 0.000463 0.009637
402.5 0.001503 0.0011873 0.000963 0.0083623.5 0.000642 0.0005590 0.000516 0.0037824.5 0.000314 0.0002975 0.000197 0.002782
602.5 0.000811 0.0007397 0.000692 0.0063733.5 0.000415 0.0003852 0.000319 0.0017834.5 0.000200 0.0001726 0.000159 0.000873
802.5 0.000687 0.0006286 0.000586 0.0026373.5 0.000298 0.0002746 0.000189 0.0009784.5 0.000141 0.0001403 0.000116 0.000512
1002.5 0.000611 0.0005395 0.000483 0.0010323.5 0.000231 0.0002250 0.000102 0.0008224.5 0.000115 0.0001073 0.000083 0.000421
(iii) a change in the value of π½ on higher side does resultin an increase in the loss; the loss remains unaffectedby the change in the value of π.
In Table 5 simulation study is taken to find estimated lossfor πΌ, πΊ, π, and π
0under the assumptions of SELF using
different priors by considering small as well as large samplesfor comparisons purpose with the LINEX loss function.
From Table 5 and its comparison with LINEX loss func-tion (ref. Tables 1, 2, 3, and 4), it is observed that LINEXloss function gives smaller loss in comparison with SELF for
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8 Advances in Statistics
Table 3: Estimated loss functions forM using LINEX loss function.
π πΌUniformprior
Jeffreyβsprior
Conjugate priorπ½ = 0.5π = 2
π½ = 2π = 2
202.5 0.073957 0.0657402 0.026145 0.0564653.5 0.061835 0.0558135 0.015994 0.0467434.5 0.056649 0.0418561 0.012289 0.035673
402.5 0.073204 0.0555914 0.025888 0.0436743.5 0.060616 0.0466542 0.016802 0.0403014.5 0.055089 0.0435518 0.013802 0.031533
602.5 0.072393 0.0458580 0.026035 0.0393733.5 0.059386 0.0376830 0.017845 0.0363734.5 0.053528 0.0352241 0.015377 0.025377
802.5 0.071778 0.0360502 0.026361 0.0300123.5 0.058222 0.0286558 0.018818 0.0297334.5 0.051894 0.0267040 0.016845 0.020345
1002.5 0.071070 0.0263228 0.020575 0.0279733.5 0.057185 0.0196096 0.019812 0.0287324.5 0.030343 0.0183061 0.018161 0.019637
Table 4: Estimated loss functions for π0using LINEX loss function.
π πΌUniformprior
Jeffreyβsprior
Conjugate priorπ½ = 0.5π = 2
π½ = 2π = 2
202.5 0.003918 0.0016639 0.0015042 0.00373603.5 0.007714 0.0012619 0.0011718 0.00335104.5 0.006892 0.0006375 0.0006100 0.0033324
402.5 0.003030 0.0012092 0.0011452 0.00135733.5 0.001033 0.0007198 0.0007159 0.00162994.5 0.001099 0.0003325 0.0003235 0.0011149
602.5 0.002237 0.0009517 0.0008915 0.00112493.5 0.001652 0.0004774 0.0004483 0.00089254.5 0.001019 0.0002165 0.0002040 0.0005813
802.5 0.001769 0.0007372 0.0007191 0.00090473.5 0.001163 0.0003889 0.0003795 0.00063154.5 0.000659 0.0001677 0.0001622 0.0003924
1002.5 0.001009 0.0006512 0.0005704 0.00071863.5 0.000465 0.0002854 0.0002770 0.00042694.5 0.000287 0.0001354 0.0001240 0.0002843
noninformative priors and conjugate prior for small as wellas large sample sizes. When sample size increases estimatedloss decreases in all cases.
4.1. Choice of Hyperparameters. Sinha and Howlader [28]suggested that a Bayes estimate is robust with respect to itshyperparameter if it leads to a high (min /max) index of theestimate for the varying values of those hyperparameter. Tocheck results, simulations are done by taking different values
Table 5: Estimated loss functions for πΌ, G, M, and π0using different
priors under the assumptions of SELF.
π πΌUniformprior
Jeffreyβsprior
Conjugate priorπ½ = 0.5π = 2
π½ = 2π = 2
ForπΌ
402.5 0.198417 0.188773 0.105229 0.1490433.5 0.545553 0.315654 0.301779 0.3030944.5 0.636095 0.546807 0.511984 0.662855
1002.5 0.081056 0.080694 0.065290 0.0722333.5 0.178339 0.192881 0.138684 0.1395104.5 0.261753 0.299142 0.231038 0.215135
ForπΊ
402.5 0.002541 0.002135 0.001879 0.0534373.5 0.001215 0.001071 0.001055 0.0336834.5 0.000989 0.000629 0.000222 0.026677
1002.5 0.001347 0.001311 0.001054 0.0113183.5 0.000604 0.000408 0.000407 0.0069674.5 0.000228 0.000236 0.000165 0.005405
Forπ
402.5 0.085215 0.075152 0.061571 0.0972153.5 0.092519 0.085051 0.070570 0.1023104.5 0.157210 0.115720 0.095721 0.105721
1002.5 0.105721 0.097121 0.050712 0.0987213.5 0.097215 0.070125 0.033710 0.0597134.5 0.080712 0.052325 0.092530 0.082173
Forπ0
402.5 0.003513 0.004420 0.003916 0.0051923.5 0.001382 0.003596 0.002156 0.0049214.5 0.001224 0.001993 0.001057 0.003051
1002.5 0.001152 0.001907 0.001805 0.0019823.5 0.000538 0.000914 0.000705 0.0015724.5 0.000260 0.000896 0.000679 0.000971
of hyperparameter and keeping πΌ and π fixed (ref. Tables 6and 7).
The ratio (min /max) in case of both Gini index andPoverty measure is close to 1 for different combinations ofπ and π½ indicating thereby the Bayes estimates are robustwith respect to hyperparameters, which justifies the use ofhyperparameters in simulation study.
5. Conclusion
The simulation study as carried out in Section 4 suggests thatBayesian estimators using conjugate prior (hyperparameterπ½ = 0.5, π = 2) perform better than two noninformativepriors (Uniform prior and Jeffreysβ prior) in general. It is alsoobserved that LINEX loss function results in smaller loss thanthe SELF for both small and large samples irrespective of thechoice of the priors taken for the Bayesian estimators. Hence,the combinations of conjugate prior and LINEX loss results insmaller loss than the choice of other two priors and squarederror loss function. One can further infer that as sample sizeincreases the expected loss function decreases for all cases.
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Advances in Statistics 9
Table 6: Bayes estimate of Gini index using conjugate prior (π = 100, πΌ = 3.5).
π½π
(Min /Max) | π½1 2 3 4 5
0.5 0.21247 0.22623 0.24944 0.23980 0.24143 0.8531 0.23816 0.25030 0.22015 0.25817 0.23342 0.8521.5 0.20394 0.22034 0.21269 0.23569 0.22392 0.8652 0.22687 0.24029 0.22722 0.26901 0.25348 0.8432.5 0.21976 0.23529 0.25022 0.24789 0.26048 0.845(Min /Max) | π 0.856 0.880 0.850 0.879 0.859
Table 7: Bayes estimate of Poverty measure using conjugate prior (π = 100, πΌ = 3.5).
π½π
(Min /Max) | π½1 2 3 4 5
0.5 0.89102 0.89271 0.89536 0.89844 0.89987 0.9981 0.88525 0.88800 0.89170 0.89269 0.89549 0.9891.5 0.88163 0.88560 0.88582 0.88885 0.89141 0.9892 0.87639 0.87720 0.88162 0.88555 0.88619 0.9882.5 0.87005 0.87451 0.87786 0.87947 0.88246 0.985(Min /Max) | π 0.976 0.979 0.980 0.978 0.980
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
The authors are thankful to the anonymous referees and theeditor for their valuable suggestions and comments.
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10 Advances in Statistics
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