An extended Generalised Variance, with Applications

Abstract : We consider a measure ψ k of dispersion which extends the notion of Wilk's generalised variance, or entropy, 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 entropy-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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https://hal.archives-ouvertes.fr/hal-01086442
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Submitted on : Monday, November 24, 2014 - 11:19:24 AM
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Luc Pronzato, Henry Wynn, Anatoly Zhigljavsky. An extended Generalised Variance, with Applications. 2014. ⟨hal-01086442⟩

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