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Estimation of variance and standard deviation
Posted:
Jun 7, 2013 12:07 PM


Hi,
Hi, I know that variance estimation for normal variables X is best done with bootstrap, and that the confidence intervals based on the $$\chi^2$$ distribution aren't that good. Still, for a variety of reasons I would like to be able to have a formula to get at least an idea of the confidence interval during a preliminary analysis. Of course, I would afterwards refine my estimates by loading samples in R and using bootstrap. Basic idea: variance of sample variance is
$$Var[S^2]=\frac{1}{N}\left(\mu_4\frac{N3}{N1}\sigma^4\right)$$
This doesn't assume normality of X, only that samples are i.i.d. and that $$\mu_4$$ (the fourth central moment) as well as $$\sigma$$ are finite. It is possible to prove that S^2 is asymptotically normally distributed, so I can derive the 95% confidence interval
$$S^2 \pm 1.96\sqrt{\frac{1}{N}\left(\mu_4\frac{N3}{N1}\sigma^4\right)}$$
The problem of course is that $$\mu_4$$ and $$\sigma^4$$ are unknown. $$S^2$$ can be substituted for $$\sigma^2$$ because of Slutsky theorem, so I hope it's correct to substitute $$\frac{1}{N1}\sum\limits_i^N(x_i\bar{x})^4$$ for $$\mu_4$$. Is that true? If so, I think that this formula for the confidence interval should work at least for large samples. Thanks,
Best Regards
deltaquattro



