Pronzato, Luc, Wynn, Henry P.
ORCID: 0000-0002-6448-1080 and Zhigljavsky, Anatoly A.
(2017)
Extended generalised variances, with applications.
Bernoulli, 23 (4A).
pp. 2617-2642.
ISSN 1350-7265
Abstract
We consider a measure ψk 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≤d formed by k+1 independent copies. We show how ψk 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.
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