On some geometric identities involving the sample covariance matrix and its adjugate

Abstract

Several identities, involving the Lebesgue measure of data-based simplices or parallelotopes, have been obtained for functionals involving the sample covariance matrix 𝑆 or its population counterpart Σ. This is the case in particular for Wilks’ generalized variance det(𝑆), which allowed one to obtain an explicit expression for E[det(𝑆)] whenever observations are randomly sampled from a distribution with finite second-order moments. To date, however, all such results are limited to scalar functionals. In this paper, we obtain geometric identities for the adjugate adj(𝑆) of 𝑆 and for other functionals involving adj(𝑆) and the sample mean vector 𝑋¯. This allows us in particular to define uniformly minimum risk unbiased (UMRU) estimators of the corresponding population quantities. Just as the results from (Ann. Statist. 36 (2008) 2261–2283) find applications when conditional independence is of interest, our results are relevant in an elliptical framework (resp., in a general framework) where conditional independence is replaced with partial uncorrelatedness (resp., with an original concept of partial median-uncorrelatedness).