Let $X \sim N(0, \Sigma)$ be a $d$-dimensional Gaussian random vector. Suppose I have $n$ samples $X_1, \ldots, X_n$.

Consider the large sample regime where $d$ is fixed and $n$ goes to infinity, the MLE estimator of $\Sigma^{-1}$ is $\hat{\Sigma}^{-1} = (\frac{1}{n}\sum_{i=1}^nX_iX_i^\top)^{-1}$.

My question is how to establish the consistency result that $\hat{\Sigma}^{-1} \rightarrow \Sigma^{-1}$ and the asymptotic distribution of $\hat{\Sigma}^{-1} - \Sigma^{-1}$.

Thank you!


Hint: Look at each entry of $X_i X_i^T$, then, look at their average. Then, use the Law of large numbers. Finally, use the continuous mapping theorem.

  • $\begingroup$ Thank you! But how about the asymptotic normality part? $\endgroup$ – Wuchen Aug 18 '16 at 21:35
  • $\begingroup$ @Wuchen Asymptotic normality is trickier. You need to prove a couple of additional things. For instance, you can use some general results from M-estimation theory (see Theorem 5.23 from van der Vaart's Asymptotics Statistics book) $\endgroup$ – Luck Aug 18 '16 at 23:30
  • $\begingroup$ Thank you for the pointer! I will check that. So there is no exsiting results on the asymptotic normality of the inverse covariance matrix in the large sample regime? A bunch of research has been working on the high dimensional regime. $\endgroup$ – Wuchen Aug 19 '16 at 1:47

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