Take the 2-minute tour ×
Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It's 100% free, no registration required.

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: $$ \mathrm{Var}(S^2)={\mu_4\over n}-{\sigma^4\,(n-3)\over n\,(n-1)} $$ 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?

share|improve this question
    
I don't think that formula is correct. –  cardinal Jun 6 '12 at 15:45
    
Related: math.stackexchange.com/a/73080/7003 –  cardinal Jun 6 '12 at 15:46
    
that related question was asked by @byron-schmuland –  Abe Jun 6 '12 at 16:57
1  
I think you mean answered, not asked. The formula given in this question is incorrect; as Byron's answer nicely demonstrates. :) –  cardinal Jun 6 '12 at 16:58
    
@cardinal yes, I meant answered. I just wanted to ping him –  Abe Jun 6 '12 at 17:08

2 Answers 2

up vote 9 down vote accepted

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:

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

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

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

share|improve this answer
    
(+1) Nearly 40 years since the last edition, MGB is still the best beginning/intermediate introduction to math stat. It's a shame it has been out of print in the Western world for so long. –  cardinal Jun 6 '12 at 21:02
    
Many thanks for figuring this out. –  Abe Jun 6 '12 at 21:47
    
I found a pdf of MGD, but there is no citation to the original proof. Which is okay, but it would be nice to know where to find it. –  Abe Jun 6 '12 at 22:50
    
The actual derivation of the result is not in MGB, but rather us relegated to problem 5(b) on page 266. –  cardinal Jun 6 '12 at 22:54
1  
@Abe: You will almost certainly not find an "original" reference for this. It is not the sort of standalone "publishable" result found in academic journals. It is simply a (rather tedious) calculation following from the basic properties of mathematical expectation. Quoting a textbook like MGB is perfectly reasonable and acceptable. –  cardinal Jun 6 '12 at 22:57

It is not clear if this will suit your needs for a definitive reference, but this question comes up in the exercises of Casella and Berger:

(page 364, exercise 7.45 b):

enter image description here

With reference to exercise 5b that provides another variant, in which $\Theta_2$ and $\Theta_4$ are the second and fourth moments ($\sigma^2$ and $\kappa$), respectively:

enter image description here

These are equivalent to the equation given in an answer on math.SE:

$\mbox{Var}(S^2)={\mu_4\over n}-{\sigma^4\,(n-3)\over n\,(n-1)}$

share|improve this answer
    
It is interesting that your link and my link (in the comments to the OP) are different, but point to the same place. –  cardinal Jun 6 '12 at 16:30
2  
@cardinal - I just copy-pasted from the OP - but the last digits are the user id of the person who copies the link, e.g. my link would be math.stackexchange.com/a/73080/3733 –  David Jun 6 '12 at 16:33
    
Aha! (+1) I did not notice that the last part of the link was one's own id! Thanks for pointing that out. We're being followed... –  cardinal Jun 6 '12 at 16:39
    
it is good to have a trustworthy reference, but would still be nice to track down the original. +1 for looking through the exercises. –  Abe Jun 6 '12 at 16:39
    
@cardinal one justification for / use of tracking is the badges for sharing links (announcer, booster, publicist) –  David Jun 6 '12 at 16:44

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.