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We consider two types of averaging of complex covariance matrices, a samplemean (average) and the sample Fr\'echet mean. We analyse the performance ofthese quantities as estimators for the true covariance matrix via `intrinsic'versions of bias and mean square error, a methodology which takes account ofgeometric structure. We derive simple expressions for the intrinsic bias inboth cases, and the simple average is seen to be preferable. The same is truefor the asymptotic Riemannian risk, and for the Riemannian risk itself in thescalar case. Combined with a similar preference for the simple average usingnon-intrinsic analysis, we conclude that the simple average is preferredoverall to the sample Fr\'echet mean in this context.

Intrinsic Analysis of the Sample Fr\'echet Mean and Sample Mean of Complex Wishart Matrices