Some Estimators for the Pareto Distribution

We derive some shrinkage test-estimators and the Bayes estimators for the shape parameter of a Pareto distribution under the general entropy loss (GEL) function . The properties have been studied in terms of relative efficiency. The choices of shrinkage factor are also suggested. Â  Keywords: General entropy loss; Shrinkage factor; Shrinkage test-estimator; Bayes estimator; Relative efficiency. Â  Â© 2009 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. Â  DOI: 10.3329/jsr.v1i2.1642 Â  Â


Introduction
The Pareto distribution and its close relatives provide very flexible family of fat-tailed distributions which may be used as a model for income distribution of higher income group. Davis and Feldstein [1] have viewed the Pareto distribution as a potential model for the life testing problems. This distribution has established its important role in variety of other problems such as size of cities and firms [2], business mortality [3], service time in queuing system [4].
The Pareto distribution has played a major role in the investigations of previous phenomena providing a satisfactory model at the extremities. It plays an important role in socio-economic studies. It is often used as a model for analysis area including city population distribution, stock price fluctuations and oil field locations. Also, it has application in military areas and suitable for approximating the right tails of distribution with positive skewness. The Pareto law applied to study the distributions of nuclear particles [5]. Harris [6] used this distribution in determining times of maintenance service while Dyer [7] found that the two-parameter Pareto distribution transformation is equivalent to the two-parameter exponential distribution. The probability density function of the classical Pareto distribution is given as Here, be the shape parameter and θ σ is a scale parameter. Let be a random sample of size drawn from the distribution (Eq. 1). The maximum likelihood estimators (MLE) and the unbiased estimators for the parameters and Here, suffix u stands for unbiased estimator. Also, is distributed as a chi-square distribution with degrees of freedom It is recognized that a shrinkage estimator performs better if a guess value of the parameter is in the vicinity of the true value and the sample size is small. Following Thompson [8], a shrinkage estimator of is θ where 0 θ is a guess value of the parameter . The shrinkage factor lies between 0 and 1 and is specified by the experimenter according to his belief in the guess value θ k 0 θ . The shrinkage procedure has been applied in a number of problems such as mean survival time in epidemiological studies [9], forecasting of the money supply [10], estimating mortality rates [11] and improved estimation in sample surveys [12]. The performances of the shrinkage estimators utilizing a point guess value has been studied in Refs. [13][14][15][16][17][18] and others in different contexts. We know that in many real life situations, the overestimation or underestimation are not of equal consequences. For such situations a useful asymmetric loss function was introduced by Varian [19], called as the LINEX loss function. This function rises approximately exponentially on one side of zero and approximately linearly on the other side. A suitable alternative to the LINEX loss is the general entropy loss (GEL) proposed by Calabria and Pulcini [20] and is given for the parameter as θ The shape parameter allows different shapes of this loss function. For a positive error causes more serious consequences than a negative one and vice versa. Also, the minimum occurs at .
In this paper, we propose some shrinkage estimators for the shape parameter when initial estimate of is available in the form of the guess value with different choices of the shrinkage factor and study their properties in terms of relative efficiency under GEL function. The Bayes estimators of the parameter are also derived when the scale parameter θ 0 θ θ σ is known and unknown.

A Proposed Class of Estimators for the Shape Parameter
The proposed class of estimators for the shape parameter is given as may be the function of W z . The value of that minimizes is obtained as Hence, the improved estimator among the class (Eq. 4) is where and

The Proposed Shrinkage Estimator and its Properties
The risk of the estimator given in Eq. (2) under the GEL is given as T The relative efficiency of the shrinkage estimator T with respect to improved estimator is defined as The relative efficiency is the function of n, ). Also, the efficiency first increases for 1.00 < δ and then decreases in the interval as increases. It is also seen that for the moderate values of the gain in efficiency is larger in the vicinity of the true value of the parameter, i.e. ( . The value of the shrinkage factor k (say), which minimizes is obtained numerically by solving the given equality Based on the improve shrinkage estimator for is given as θ with the risk under the GEL is The relative efficiency of with respect to is given as The relative efficiency is the function of and . For the similar set of values as considered earlier, the relative efficiencies have been calculated but not presented here. The improved shrinkage estimator is more efficient than in the interval

The Proposed Shrinkage Test-Estimator and its Properties
From the above conclusion, we conclude that the shrinkage estimator performs well when guess value is approximately near to the parameter and for the moderate values of sample size as well as the shrinkage factor. This suggests that one may employ a test under the hypothesis against . A test statistic is available for testing the hypothesis against . Therefore, the proposed shrinkage test-estimator is The relative bias of the shrinkage test-estimator is obtained as The risk under the GEL for the proposed shrinkage test-estimator is obtained as The relative efficiency of the shrinkage test-estimator with respect to is The

Choices of the Shrinkage Factor
The shrinkage test-estimator based on the shrinkage factor (Eq. 9) that minimizes R(T), is given as Again, the value of the shrinkage factor (say), which minimizes the risk of the shrinkage estimator given in Eq. (13), is obtained numerically by solving the given equality Based on the proposed shrinkage test-estimator is The relative bias for the shrinkage test-estimator is obtained as The risk for the proposed shrinkage test-estimator under the GEL is given as The relative efficiency for the shrinkage test-estimator with respect to is  (Tables 1 and 2).

Case 1: When σ is known
The conjugate prior for the parameter θ can be taken as two parameters Gamma distribution, having probability density function (17) and the posterior density will be The posterior density is again a two parameter Gamma density with parameters and Here U is a sufficient statistic for the parameter θ and In particular for , 1 p − = the Bayes estimator under GEL is equal to the posterior mean and is given by The risk of these estimators under the GEL is given by  The Bayes estimator performs uniformly well with respect to for the all considered values of the parametric space and the efficiency increases as increases. Opposite trend is seen when increases (for larger values of ). The efficiency also decreases when sample size increases for