An extended Generalised Variance, with Applications

Luc Pronzato (1, 2) , Henry P. Wynn (3) , Anatoly A. Zhigljavsky (4)

1 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques
2 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe SYSTEMES
3 LSE - Department of Statistics - London School of Economics
4 School of Mathematics [Cardiff]

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.

Mots clés: optimal design; design of experiments; maximum-entropy measure; quadratic entropy; generalised variance; dispersion

Domaines: Statistiques [math.ST]

Luc Pronzato, Henry P. Wynn, Anatoly A. Zhigljavsky. An extended Generalised Variance, with Applications. 2014. ⟨hal-01086442⟩