ON SOME CHARACTERIZATIONS OF POPULATION DISTRIBUTIONS

Abstract

In this paper, earlier works of the present authors and a method due to Anosov for solving certain intego-functional equations are combined to show that the independence of the sample mean $\bar{X}_n$ and the $Z_n$-statistic characterizes the normal population, when the random samples are iid from a population having a continuous density function on ${\Bbb R}$, and the sample size $n\geq 3$; obviously the sample standard deviation is a $Z_n$-statistic. Further, an important subclass of $Z_n$-statistic with the form of a linear combination $\sum^n_{i=1} a_iX_{(i)}$ of order statistics is found, where $a_1 \leq \cdots \leq a_n$, not all equal and $\sum^n_{i=1}a_i=0$, which includes Gini$^\prime$s mean difference and the sample range but not the sample standard deviation. Similar approach can be applied to prove that the independence of $\bar{X}_n$ and $Z_n/\bar{X}_n$ characterizes the gamma distribution; obviously the independence of sample mean and sample coefficient of variation characterizes the gamma distribution. The study of identifying $Z_n$ to more known statistics will be the future work.