A minimal characterization of the covariance matrix

SummaryLetX be ak-dimensional random vector with mean vectorμ and non-singular covariance matrix Σ. We show that among all pairs (a, Δ),a ∈ IRk, Δ ∈ IRk×k positive definite and symmetric andE(X−a)′ Δ−1(X−a)=k, (μ, Σ) is the unique pair which minimizes det Δ. This motivates certain robust estimators of location and scale.