## Saturday, August 22, 2009

### 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.

Anonymous said...

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.

11:16 AM
Anonymous said...

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

11:41 AM
Michael Mitzenmacher said...

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!

12:19 PM
Anonymous said...

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).

3:58 AM
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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.

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