Let $X_1,X_2,\cdots$ be i.i.d. from a distribution $F$ with mean $0$ and unknown variance $\sigma^2$ and having four moments. A common test for testing $H_0:\sigma^2=1$ vs $H_1:\sigma^2>1$ is to reject $H_0$ for large values of $nS_n^2=\sum_{i=1}^n(X_i-\bar{X}_n)^2$, i.e., when $nS_n^2>\chi_{\alpha,n-1}^2$ where $P(\chi_{n-1}^2>\chi_{\alpha,n-1}^2)=\alpha$. Is his test asymptotically level robust if $F$ is non-normal in such a way that kurtosis $K=\frac{\mu_4}{\mu_2^2}-3$ of $F$ is non-zero?
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Three questions asked today, probably parts of a common homework. You really need to share your thoughts on the problem and explain what you've tried. – Did Aug 27 '13 at 08:07
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I have proved that the $t$-test is asymptotically level robust for non-normal data. For that I needed nothing more than CLT. But here I am not sure what to use, and also how does kurtosis play a role in that. – QED Aug 27 '13 at 09:56