Characterization of Power-Function Distribution through Expectation

Milind Bhanuprasad Bhatt

doi:10.4236/ojs.2013.36052

Abstract

For the characterization of the power function distribution, one needs any arbitrary non constant function only in place of independence of suitable function of order statistics, linear relation of conditional expectation, recurrence relations between expectations of function of order statistics, distributional properties of exponential distribution, record valves, lower record statistics, product of order statistics and Lorenz curve, etc. available in the literature. The goal of this research is not to give a different path-breaking approach for the characterization of power function distribution through the expectation of non constant function of random variable and provide a method to characterize the power function distribution as remark. Examples are given for the illustrative purpose.

Highlights

Other attempts were made for the characterization of exponential and related distributions assuming linear relation of conditional expectation by Beg [5], characterization based on record valves by Nagraja [6], characterization of some types of distributions using recurrence relations between expectations of function of order statistics by Alli [7], characterization results on exponential and related distributions by Tavangar [8], and characterization continuous distributions through lower record statistics by Faizan [9] included the characterization of power function distribution
Direct characterization for power function distribution has been given in Arslan [10] who used the product of order statistics [contraction is a particular case of product of order statistics which has interesting applications such as in economic modeling and reliability see Alamatsaz
This research note provides the characterization based on identity of distribution and equality of expectation of function of random variable for power-function distribution with the probability density function (p.d.f.)

Summary

Introduction

Several characterizations of power function distribution have been made notably by Fisz [1], Basu [2], Govindarajulu [3] and Dallas [4] using independence of suitable function of order statistics and distributional properties of transformation of exponential variable. [Graph of fraction of total income owned by lowest pth fraction of the population is Lorenz curve of distribution of income [15]. This research note provides the characterization based on identity of distribution and equality of expectation of function of random variable for power-function distribution with the probability density function (p.d.f.). The aim of the present research note is to give the new characterization through the expectation of function x for the power function distribution.

Characterization

Illustrative Examples