On Estimating Scale Parameter of the Selected Pareto Population under the Generalized Stein Loss Function

Abstract

The problem of estimation after selection can be seen in numerous statistical applications. Let $X_{i1},\dots ,X_{in}$ be a random sample drawn from the population ${\Pi }_i,i=1,\dots ,k,$ where Π_i follows Pareto distribution with an unknown scale parameter θ_i and common known shape parameter β. This article is concerned with the problem of estimating θ_L (or θ_S), the scale parameter of the selected Pareto population under the generalized Stein loss function. The uniformly minimum risk unbiased (UMRU) estimators of θ_L and θ_S, scale parameters of the largest and the smallest population respectively, are determined. For k = 2, we have obtained a sufficient condition of minimaxity of θ_S and showed that the generalized Bayes estimator of θ_S is a minimax estimator for k = 2. Also, a class of linear admissible estimators of the form $d{X}_{(2)}(d{X}_{(1)})$ of θ_L and θ_S is found, and a sufficient condition for inadmissibility is provided. Further, we demonstrate that the UMRU estimator of θ_S is inadmissible. A comparison between the proposed estimators is conducted using MATLAB software and a real data set is analyzed for illustrative purposes. Finally, conclusions and discussion are reported.

KEY WORDS AND PHRASES:

• Generalized Bayes estimators
• generalized Stein loss (GSL) function
• inadmissibility
• minimaxity
• Pareto distributions
• UMRU estimator

Acknowledgment

The authors express their sincere thanks to the anonymous reviewers, the editor and the associate editor for their constructive comments and suggestions on the original version of the manuscript, which led to this improved version.