Empirical covariance operator ($\hat{C}$)

For Hilbert-space-valued random variables $X_1,\dots ,X_n$ and a fixed deterministic centre $m$, the empirical covariance operator is ${\hat{C}}_m=n^{-1}{\sum }_i(X_i-m)\otimes (X_i-m)$. If the observations share mean $\mu$ and covariance $C$, then $\mathbb{E}{\hat{C}}_m=C+(\mu -m)\otimes (\mu -m)$. This follows by expanding each centred vector and cancelling the mean-zero cross terms. A fitted random centre changes that expectation: for an IID sample centred at its sample mean, $\mathbb{E}{\hat{C}}_{\overline{X}}=(n-1)C/n$. Multiplying by $n/(n-1)$ gives the usual unbiased estimate when $n>1$.

A pooled covariance operator averages the within-group empirical covariance operators from several samples that are assumed to share one population covariance. Centering within each group prevents differences between group means from contaminating the covariance estimate.