Characterizations of the Normal Distribution via the Independence of the Sample Mean and the Feasible Definite Statistics with Ordered Arguments

TL;DR

This paper explores how the independence of the sample mean and certain new feasible definite statistics characterizes the normal distribution, extending classical results using Anosov's theorem.

Contribution

It introduces a new class of feasible definite statistics based on ordered arguments and demonstrates their role in characterizing the normal distribution.

Findings

Characterization of the normal distribution via independence of sample mean and new statistics

Extension of classical results to a broader class of statistics

Simplified derivation of known characterizations like sample variance and range

Abstract

It is well known that the independence of the sample mean and the sample variance characterizes the normal distribution. By using Anosov's theorem, we further investigate the analogous characteristic properties in terms of the sample mean and some feasible definite statistics. The latter statistics introduced in this paper for the first time are based on nonnegative, definite and continuous functions of ordered arguments with positive degree of homogeneity. The proposed approach seems to be natural and can be used to derive easily characterization results for many feasible definite statistics, such as known characterizations involving the sample variance, sample range as well as Gini's mean difference.