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arXiv · 1411.6428

An extended Generalised Variance, with Applications

Luc Pronzato·Henry Wynn·Anatoly Zhigljavsky
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Abstract

We consider a measure $\psi$ 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 $\le$ d formed by k + 1 independent copies. We show how $\psi$ 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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Luc Pronzato, Henry Wynn, Anatoly Zhigljavsky. 2014-11-24. An extended Generalised Variance, with Applications. https://arxiv.org/abs/1411.6428

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