# Pooled Covariance Estimate if means are unequal

Assume that $X_{i,j} \sim [\mu_i, \Sigma], i=1,...n; j=1...m$ and we have realisations $x_{i,j}=X_{i,j}(\omega)$. Is the formula: $\frac{1}{(n-1)m}\sum_{i=1}^n\sum_{j=1}^m[x_{i,j}-\overline{x_{i}}][x_{i,j}-\overline{x_{i}}]^T$ then an unbiased pooled estimate for the covariance matrix $\Sigma$? I have read almost the same formula on wikipedia: https://en.wikipedia.org/wiki/Hotelling%27s_T-squared_distribution#Pooled_covariance_matrix but there they assumed that the mean-vectors are equal, but if I had to guess I would say that it should also work without the equal mean assumption.