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Suppose $X_1, ..., X_n$ are independent random variables with expected value $\mu$ and variance $\sigma^2$:

  • I would like to show the $$ \hat\sigma^2 = \frac{1}{n} \sum_{i=1}^{n}\left(X_i - \bar X_n\right)^2\quad \mbox{is consistent} $$
  • So far I have been able to show that $$ \mathbb{E}\left[\hat\sigma^2\right] = \frac{n-1}{n} \sigma^2 $$
  • All the proofs I have seen for consistency rely on the continuous mapping theorem.
  • However, I think it should also be straightforward to show directly, that $$ \hat\sigma^2\ \mbox{converges to}\ \sigma^2\ \mbox{in probability} $$

I would appreciated any help.

Felix Marin
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user007
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1 Answers1

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From the expectation you computed, you can see that $\widehat{\sigma}^{2}$ is asymptotically unbiased for $\sigma^{2}$. If you compute its variance, and show that it goes to zero as $n \rightarrow \infty$, this is enough to say that it is a consistent estimator for $\sigma^{2}$.

aprobabilityspace
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