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Suppose $X^1, \cdots, X^n \sim N_k(\mu, \Sigma)$, where $\mu$ is an unknown mean vector and $\Sigma$ is an unknown positive definite matrix.

Let $\hat \Sigma = \sum_{i=1}^{n}{(X^i-\bar X)( X^i-\bar X)^{\mathrm{T}}}/n$ be the MLE of $\Sigma$, where $\bar X = \sum_{i=1}^{n}{X^i}/n$.

How to prove it: $\mathbb{E}(\hat \Sigma) = (1-1/n)\Sigma$.

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  • $\begingroup$ Welcome to CV. If this question relates to a class exercise, please see stats.stackexchange.com/tags/self-study/info and add the tag to modify the question accordingly. $\endgroup$
    – Pitouille
    Commented Nov 17, 2021 at 14:45
  • $\begingroup$ Multiply the result in the duplicate by $1-1/n.$ $\endgroup$
    – whuber ♦
    Commented Nov 17, 2021 at 15:42

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