### Convergence of Sample Variance

Sample mean behaves like the normal distribution when appropriately viewed, and the classical Berry-Esseen theorem bounds the deviation for each sample size n (not just as n tends to infty); the convergence rate is like \sqrt{n}. We know that as n tends to infty, sample variance tends to the true variance. Is there a reference for bounding their deviation for each sample size n? If the samples are drawn from a normal distribution, then scaled chi-squared distribution can be used to bound the deviation.

## 8 Comments:

Can't you do it in two steps? First look at E[X^2] vs SampAv(X^2). The sample value of the second moment is the average of sample from the distribution X^2, and behaves like a Gaussian with mean E[X^2]. Berry Esseen will tell you how close (assuming you have good bounds on higher moments). Similarly, sample average is concentrated around E[X] so you will get some concentration for the square of the sample average. Put together, you should get some reasonable bounds on sample variance vs. variance.

Dear Anom,

That is the kind of analysis I have been playing around with, so email me if you have a clear bound. I was hoping someone would just point me to an authoritative source and save me the work.

-- Metoo

Hi Muthu. I always smile when I see the words Berry-Esseen theorem (having used it more than once myself). I'm not sure where a good pointer to answer your question is offhand; if you hear from someone, be sure to post it though!

Sample variance also "behaves like the normal distribution when appropriately viewed" i.e. subtract the true value and rescale. The Berry-Esseen can then be applied, although now there is a remainder term but this is smaller than the dominant 1/sqrt(n).

Sample average is concentrated around E[X] so you will get some concentration for the square of the sample average. Put together, you should get some reasonable bounds on sample variance vs. variance.

Well, I believe in the 75% world that you share, I have been reading about samples since I was a little boy !

shijun 6.5

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