An extended Generalised Variance, with Applications

TL;DR

This paper introduces an extended measure of dispersion based on generalized variance, linking it to eigenvalues of covariance matrices, and demonstrates its applications in optimal experimental design.

Contribution

It extends Wilk's generalized variance to a new measure based on simplices, providing theoretical properties and applications in optimal experiment design.

Findings

The measure can be expressed via covariance eigenvalues.

It is concave when raised to a suitable power.

Applications include A- and D-optimal design.

Abstract

We consider a measure $\psi$ k of dispersion which extends the notion of Wilk's generalised variance, or entropy, for a d-dimensional distribution, and is based on the mean squared volume of simplices of dimension k $\le$ 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 entropy-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.