A Few Properties of Sample Variance

Abstract

A basic result is that the sample mean for i.i.d. observations is an unbiased estima- tor of the variance of the underlying distribution (see for instance Casella and Berger (2002)). But what happens if the observations are neither independent nor identi- cally distributed. What can we say? Can we in particular compute explicitly the rst two moments of the sample mean and hence generalize standard 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. But what about 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, Moon, and Eltinge (2004).

Keywords: Sample Variance, Variance of Sample Variance, Independence Between Sample Mean and Variance