# Covariance estimation

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?