Let $X_i \sim \text{iid}(\mu, \sigma^2)$ (the question does not specify whether or not $\mu$ and $\sigma^2$ are known). I have to show the confidence interval for the variance. Since $X_i$ is not necessarily normal, I should use the asymptotic distribution. Let $\hat{\theta}^2_n = \frac1n\sum_{i=1}^n(X_i - \bar{X})^2$. Then, $$\frac{\sqrt{n}(\hat{\theta}^2_n-\sigma^2)}{\sigma_{\hat{\theta}}}\xrightarrow{d} N(0,1).$$ Then, the asymptotic confidence interval at 5% critical level should be $$P(\hat{\theta}^2_n - \frac{1.96\sigma_{\hat{\theta}}}{n} \le \sigma^2 \le \hat{\theta}^2_n + \frac{1.96\sigma_{\hat{\theta}}}{n}) = 0.95.$$ Am I on the right track? what should $\sigma_{\hat{\theta}}$ be in this case?


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