Extended generalised variances, with applications

Abstract

We consider a measure $\psi_{k}$ of dispersion which extends the notion of Wilk’s generalised variance for a $d$-dimensional distribution, and is based on the mean squared volume of simplices of dimension $k\leq d$ formed by $k+1$ independent copies. We show how $\psi_{k}$ can be expressed in terms of the eigenvalues of the covariance matrix of the distribution, also when a $n$-point sample is used for its estimation, and prove its concavity when raised at a suitable power. Some properties of dispersion-maximising distributions are derived, including a necessary and sufficient condition for optimality. Finally, we show how this measure of dispersion can be used for the design of optimal experiments, with equivalence to $A$ and $D$-optimal design for $k=1$ and $k=d$, respectively. Simple illustrative examples are presented.