  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.
