Characterization of Power Function Distribution through Expectation of Function of Order Statistics

Bhatt Milind B

doi:10.13189/ms.2015.030602

Abstract

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.. are various approaches available in the literature for the characterization of the power function distribution. In this research note different path breaking approach for the characterization of power function distribution through the expectation of function of order statistics is given and provides a method to characterize the power function distribution which needs any arbitrary non constant function only.

Highlights

Notable attempt to characterized Power function distribution through independence of suitable function of order statistics and distributional properties of transformation of exponential are Basu [1], Govindarajulu [2], Desu [3] and Dallas [4] where as of exponential and related distributions assuming linear relation of conditional expectation by Beg [5], characterization based on record values 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 of continuous distributions through lower record statistics by Faizan [9] included the characterization of power function distribution as special case
Direct characterization for power function distribution has been given in Fisz [10] who use independence properties of order statistics where as Arslan [11] used product of order statistic. [contraction is a particular where −∞ < a < b < ∞ are known constants, xc−1 is positive absolutely continuous function and
The aim of the present research note is to give the new characterization through the expectation of function of order statistics, using identity and equality of expectation

Summary

Introduction

Notable attempt to characterized Power function distribution through independence of suitable function of order statistics and distributional properties of transformation of exponential are Basu [1], Govindarajulu [2], Desu [3] and Dallas [4] where as of exponential and related distributions assuming linear relation of conditional expectation by Beg [5], characterization based on record values 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 of continuous distributions through lower record statistics by Faizan [9] included the characterization of power function distribution as special case. Direct characterization for power function distribution has been given in Fisz [10] who use independence properties of order statistics where as Arslan [11] used product of order statistic. [contraction is a particular where −∞ < a < b < ∞ are known constants, xc−1 is positive absolutely continuous function and Since derivative of xc−1 being positive and since range is truncated by θ from right for f (x; θ) defined in (1.1), ac c. The aim of the present research note is to give the new characterization through the expectation of function of order statistics, using identity and equality of expectation.

Characterization theorem

Illustrative Examples