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Given an arbitrary sample, sample variance would be calculated. But how the variance of sample variance should be estimated? I tried to do some simulations using influence functions estimation methods. The influence function is $IF(x, T, F) = (x-\mu)^{2} - \sigma^2$, where $\mu$ and $\sigma^2$ were substituted by sample mean and sample variance. My simulation showed that this method worked very well for some common distributions, like exponentional, normal, uniform, etc. But not for t(1) distribution, because the forth moment doesn't exist.

In my current project, the sample numbers are a bunch of simulated survival probabilities (by Fine-Gray model, or Gerds Model with censoring). So the forth moment does exist (Considering the probability is within 1 and 0. Even though the probabilities will exceed this range by both models, they are within acceptable bounds), but the influence function method failed to converge to the empirical variance of sample variance. I know it's vague to describe the question this way, but is there any more limitations for using influence function to estimate variance of sample variance? And what else method should I use to adapt to a wider choice of samples?

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Given $n$ sample values, the true variance of the sample variance (see e.g., O'Neill 2014, p. 284) is:

$$\mathbb{V}(S_n^2) = \Big( \kappa - \frac{n-3}{n-1} \Big) \frac{\sigma^4}{n},$$

where $\sigma^2$ is the true variance and $\kappa$ is the true kurtosis of the underlying distribution for the sample values. From this result, we can obtain a simple result for the variance of the ratio of the sample variance to the true variance:

$$\mathbb{V} \Big( \frac{S_n^2}{\sigma^2} \Big) = \Big( \kappa - \frac{n-3}{n-1} \Big) \frac{1}{n}.$$

This expression depends on the sample size $n$ (which is known) and is an affine function of the kurtosis $\kappa$ (which is unknown). Hence, estimation of the variance of this ratio is equivalent to estimation of an affine function of the kurtosis parameter. Any estimator of the kurtosis can be used to yield a corresponding estimator of the variance of this ratio.


O'Neill, B. (2014) Some useful moment results for sampling problems. The American Statistician 68(4), pp. 282-296 (edited by corrigendum).

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