A few properties of sample variance

TL;DR

This paper investigates the properties of sample variance for non-i.i.d. observations, deriving formulas for its moments and exploring independence with the sample mean beyond normal distributions.

Contribution

It extends existing results by providing general formulas for the moments of sample variance without assuming independence or identical distribution.

Findings

Derived formulas for the first two moments of sample variance under general conditions.

Provided a faster proof of Lukacs' seminal result using log characteristic functions.

Explored conditions for independence of sample mean and variance beyond normal distributions.

Abstract

A basic result is that the sample variance for i.i.d. observations is an unbiased estimator of the variance of the underlying distribution (see for instance Casella and Berger (2002)). But what happens if the observations are neither independent nor identically distributed. What can we say? Can we in particular compute explicitly the first two moments of the sample mean and hence generalize formulae provided in Tukey (1957a), Tukey (1957b) for the first two moments of the sample variance? We also know that the sample mean and variance are independent if they are computed on an i.i.d. normal distribution. This is one of the underlying assumption to derive the Student distribution Student alias W. S. Gosset (1908). But does this result hold for any other underlying distribution? Can we still have independent sample mean and variance if the distribution is not normal? This paper precisely answers these questions and extends previous work of Cho, Cho, and Eltinge (2004). We are able to derive a general formula for the first two moments and variance of the sample variance under no specific assumption. We also provide a faster proof of a seminal result of Lukacs (1942) by using the log characteristic function of the unbiased sample variance estimator.