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Extended generalised variances, with applications

Abstract : We consider a measure ψ k 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 ψ 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 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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Luc Pronzato, Henry Wynn, Anatoly Zhigljavsky. Extended generalised variances, with applications. Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2017, 23 (4A), pp.2617-2642. ⟨10.3150/16-BEJ821⟩. ⟨hal-01308092⟩

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