A minimal characterization of the covariance matrix (Q1099540)

Let X be a k-dimensional random vector with mean vector \(\mu\) and non- singular covariance matrix \(\Sigma\). We show that among all pairs (a,\(\Delta)\), \(a\in {\mathbb{R}}^ k\), \(\Delta \in {\mathbb{R}}^{k\times k}\) positive definite and symmetric and \(E(X-a)'\Delta^{-1}(X-a)=k\), (\(\mu\),\(\Sigma)\) is the unique pair which minimizes det \(\Delta\). This motivates certain robust estimators of location and scale.