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Suppose I want to estimate the covariance of $n$ $p$-dimensional iid random vectors $X_i$, where $n>p$. I've read in several places that if $n-p$ is small then the MLE covariance matrix estimate will be poor (especially the small eigenvalues). However the MLE estimate of the mean will be good if $n$ is large (not necessarily bigger than $p$).

Is there an intuitive reason why covariance estimation is harder than mean estimation?

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Covariance matrix estimation is some kind of 2nd moment and product moment estimation. The mean is just the first moment. That's why the the latter is more efficient to estimate, e.g. in terms of MSE (mean squared error): The empirical mean's variance is 2nd moment's size, the empirical (co-)variance's variance is 4th moment's size.

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