Extended generalised variances, with applications

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

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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.

Item Type:	Article
Official URL:	http://www.bernoulli-society.org/index.php/publica...
Additional Information:	© 2017 International Statistical Institute/Bernoulli Society for Mathematical Statistics and Probability
Divisions:	Centre for Analysis of Time Series
Subjects:	Q Science > QA Mathematics
Date Deposited:	02 Jun 2017 16:18
Last Modified:	11 Sep 2025 09:37
URI:	http://eprints.lse.ac.uk/id/eprint/79804

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