A measure of dispersion based on average exponentiated volumes, with application in experimental design


Séminaire Probabilités & Statistique

7/01/2016 - 14:00 Mr Luc PRONZATO (Université Nice Sophia Antipolis) Salle 1 - Tour IRMA

(joint work with Henry P. Wynn, LSE, and Anatoly A. Zhigljavsky, Cardiff University)

We consider a measure of dispersion in dimension d which is based on the mean exponentiated volume of k-dimensional simplices formed by k+1 independent copies, with k less than or equal to d. The mean squared volume is related to the (d-k)-th coefficient of the characteristic polynomial of the covariance matrix and forms an extension of the notion of Wilk's generalised variance. We prove its concavity when raised at power 1/k, and some properties of dispersion-maximising distributions are derived, including a necessary and sufficient condition for optimality. The application of this measure of dispersion to the design of optimal experiments for parameter estimation is considered, with A and D-optimal design coinciding with the special cases obtained for k=1 and k=d respectively. Means of volumes raised to some power different from two, including negative values, will be considered too, with application to space-filling design for computer experiments.