Extended generalised variances, with applications

Pronzato, Luc, Wynn, Henry P. and Zhigljavsky, Anatoly ORCID: https://orcid.org/0000-0003-0630-8279 2017. Extended generalised variances, with applications. Bernoulli 23 (4A) , pp. 2617-2642. 10.3150/16-BEJ821

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Abstract

We consider a measure ψ k ψk of dispersion which extends the notion of Wilk’s generalised variance for a d d -dimensional distribution, and is based on the mean squared volume of simplices of dimension k≤d k≤d formed by k+1 k+1 independent copies. We show how ψ k ψk can be expressed in terms of the eigenvalues of the covariance matrix of the distribution, also when a n 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 A and D D -optimal design for k=1 k=1 and k=d k=d , respectively. Simple illustrative examples are presented.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Schools > Mathematics
Subjects:	Q Science > QA Mathematics
Uncontrolled Keywords:	design of experiments dispersion generalised variance maximum-dispersion measure optimal design quadratic entropy
Publisher:	Bernoulli Society for Mathematical Statistics and Probability
ISSN:	1350-7265
Date of First Compliant Deposit:	26 May 2017
Date of Acceptance:	10 March 2017
Last Modified:	18 Apr 2025 16:00
URI:	https://orca.cardiff.ac.uk/id/eprint/100905

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