Advances in Statistics

Volume 2015, Article ID 964824, 10 pages

http://dx.doi.org/10.1155/2015/964824

## Bayesian Estimation of Inequality and Poverty Indices in Case of Pareto Distribution Using Different Priors under LINEX Loss Function

Department of Statistics, Panjab University, Chandigarh 160014, India

Received 29 August 2014; Accepted 7 January 2015

Academic Editor: Karthik Devarajan

Copyright © 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.

#### Abstract

Bayesian estimators of Gini index and a Poverty measure are obtained in case of Pareto distribution under censored and complete setup. The said estimators are obtained using two noninformative priors, namely, uniform prior and Jeffreys’ prior, and one conjugate prior under the assumption of Linear Exponential (LINEX) loss function. Using simulation techniques, the relative efficiency of proposed estimators using different priors and loss functions is obtained. The performances of the proposed estimators have been compared on the basis of their simulated risks obtained under LINEX loss function.

#### 1. Introduction

The Pareto distribution is a skewed, heavy-tailed distribution that is used to model the distribution of incomes and other financial variables. It was introduced by Pareto [1] which has a probability density function of the form
and cumulative distribution function is
The parameter in (2) represents the minimum income in the population under study and assumed to be known, while the other parameter *α* is assumed to be unknown.

The average income for Pareto distribution is In the context of income inequality and poverty, Gini index and Poverty measure head count ratio are two most popular indices [2, 3]. Gini index is generally defined as where is the equation of the Lorenz curve and is the mean of the distribution.

Equivalently, Gini index can also be defined as where is population Gini mean difference.

The Poverty index head count ratio is simply the count of the number of households whose incomes are below the poverty line divided by the total population. In terms of continuous distribution, where, is called Poverty Line.

In case of Pareto distribution, Gini index [4, 5] is given by and Poverty measure is where, and .

Thus, is per capita annual income representing a minimum acceptable standard of living and represents the proportion of population having income equal to or less than .

The estimation of Gini index and Poverty measure () and the associated inference using classical approach (parametric and nonparametric) is available in literature [5–8]. However, in the Bayesian setup, this has not evoked the interest of many researchers [9, 10]. In the present paper, our focus will be on the estimation of inequality and poverty indices in the Bayesian setup.

When the Bayesian method is used, the choice of appropriate prior distribution plays an important role, which may be categorized as informative, noninformative, and conjugate priors [11, 12]. In the present paper, three priors (two noninformative priors and one conjugate prior) are used to estimate shape parameter, Gini index, Average income, and Poverty measure. The two noninformative priors are Uniform prior and Jeffreys’ prior, while conjugate prior is chosen as Truncated Erlang distribution.

In Bayesian estimation, the criterion for good estimators for the parameters of interest is the choice of appropriate loss function. In Bayesian estimation, two types of loss functions commonly used are Squared error loss function (SELF) and Linear exponential (LINEX) loss function. The simplest type of loss function is squared error, which is also referred to as quadratic loss is given as
where is the estimator of *θ*.

The usual squared error loss function is symmetrical and associates equal importance to the losses due to overestimation and underestimation of equal magnitude. However, such a restriction may be impractical; for example, in estimation of shape parameter of Classical Pareto distribution, the overestimation and underestimation may not be of equal importance as over estimate of shape parameter gives an under-estimate of inequality index which seems to be more serious as compared to under estimate of shape parameter because we are often interested in reducing income inequality index. This leads one to think that an asymmetrical loss function be considered for estimation of shape parameter which associates greater importance to overestimation. A number of asymmetrical loss functions have been proposed in statistical literature [13–16]. Varian [16] proposed a useful asymmetrical loss function known as Linear exponential (LINEX) loss function which is given as The posterior expectation of the LINEX loss function (10) is where denotes posterior expectation with respect to the posterior density of .

By a result of Zellner [17] the Bayes estimator of denoted by under the LINEX loss function is the value which minimizes posterior expectation and is given by provided that the expectation exists and is finite [18].

In Figures 1(a) and 1(b), values of are plotted for the selected values of for and . It is seen that, for , the function is quite asymmetric with a value exceeding the target being more serious than a value below the target. But, for , the function is also quite asymmetric with a value below the target value being more serious than a value exceeding the target.