Characterizations of Populations Using Regression Properties

Abstract

In the present work, we study characterizations of populations obtained by using regression properties of one statistic on another. We extend some of the results of Lukacs and Laha by considering the cubic regression (polynomial regression of order 3) of a cubic statistic $S$ on a linear one $L$. This assumption of cubic regression is used to derive a third order non-linear differential equation in the characteristic function $h(t)$ of a set of $n$ independently and identically distributed random variables. The coefficients in this differential equation represent certain fixed relationships between the coefficients of the statistic $S$ and the regression coefficients. For appropriate choices of the coefficients in the fundamental differential equation, the resulting equation can be solved to yield the characteristic function of a particular distribution. In this way, we are able to obtain a series of characterization theorems for each of a variety of populations including Normal, Gamma, Binomial, Poisson, Geometric and several others. Moreover, all of the results obtained by Lukacs and Laha for characterizing populations using quadratic and constant regression are shown to be special cases of the theorems obtained in the present work. Finally, in the last section, we present an outline of a technique which can be used to study any general $r \leqq m$th order polynomial regression of any $m$th order statistic on a linear one. The approach used to generate the $m$th order differential equation is indicated and a method for determining the appropriate conditions on the coefficients is discussed.