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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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Identification Number: 10.3150/16-BEJ821


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
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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 Jul 2024 03:09

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