Solved – Reference for $mathrm{Var}[s^2]=sigma^4 left(frac{2}{n-1} + frac{kappa}{n}right)$

In his answer to my previous question, @Erik P. gives the expression
mathrm{Var}[s^2]=sigma^4 left(frac{2}{n-1} + frac{kappa}{n}right) >,
where $kappa$ is the excess kurtosis of the distribution. A reference to the Wikipedia entry on the distribution of the sample variance is given, but the wikipedia page says "citation needed".

My primary question is, is there a reference for this formula? Is it 'trivial' to derive, and if so, can it be found in a textbook? (@Erik P. couldn't find it in Mathematical statistics and data analysis nor I in Statistical Inference by Casella and Berger. Even though the topic is covered.

It would be nice to have a textbook reference, but even more useful to have a (the) primary reference.

(A related question is: What is the distribution of the variance of a sample from an unknown distribution?)

Update: @cardinal pointed out another equation on math.SE:
mathrm{Var}(S^2)={mu_4over n}-{sigma^4,(n-3)over n,(n-1)}
where $mu_4$ is the fourth central moment.

Is there some way that to rearranged the equations and resolve the two, or is the equation in the title wrong?

Source: Introduction to the Theory of Statistics, Mood, Graybill, Boes, 3rd Edition, 1974, p. 229.

Derivation: Note that in the OP's Wikipedia link, $kappa$ is not the kurtosis but the excess kurtosis, which is the "regular" kurtosis – 3. To get back to the "regular" kurtosis we have to add 3 in the appropriate place in the Wikipedia formula.

We have, from MGB:

$text{Var}[S^2] = {1over{n}}(mu_4 – {{n-3}over{n-1}}sigma^4)$

which, using the identity $mu_4 = (kappa + 3)sigma^4$, can be arranged to (derivation mine, so any errors are too):

$ = {1over{n}}(kappa sigma^4 + {{n-1}over{n-1}}3sigma^4 -{{n-3}over{n-1}}sigma^4) = sigma^4left({kappa over{n}}+{3(n-1)-(n-3)over{n(n-1)}}right) = sigma^4left({kappaover{n}} + {{2}over{n-1}}right) $

Similar Posts:

Rate this post

Leave a Comment