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Bregman divergences based on optimal design criteria and simplicial measures of dispersion

Pronzato, Luc, Wynn, Henry P. and Zhigljavsky, Anatoly (2019) Bregman divergences based on optimal design criteria and simplicial measures of dispersion. Statistical Papers, 60 (2). pp. 195-214. ISSN 0932-5026

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Identification Number: 10.1007/s00362-018-01082-8

Abstract

In previous work the authors defined the k-th order simplicial distance between probability distributions which arises naturally from a measure of dispersion based on the squared volume of random simplices of dimension k. This theory is embedded in the wider theory of divergences and distances between distributions which includes Kullback–Leibler, Jensen–Shannon, Jeffreys–Bregman divergence and Bhattacharyya distance. A general construction is given based on defining a directional derivative of a function ϕ from one distribution to the other whose concavity or strict concavity influences the properties of the resulting divergence. For the normal distribution these divergences can be expressed as matrix formula for the (multivariate) means and covariances. Optimal experimental design criteria contribute a range of functionals applied to non-negative, or positive definite, information matrices. Not all can distinguish normal distributions but sufficient conditions are given. The k-th order simplicial distance is revisited from this aspect and the results are used to test empirically the identity of means and covariances.

Item Type: Article
Additional Information: © 2019 Springer-Verlag GmbH Germany
Divisions: Centre for Analysis of Time Series
Subjects: Q Science > QA Mathematics
Date Deposited: 20 Mar 2019 15:57
Last Modified: 18 Jul 2019 23:08
URI: http://eprints.lse.ac.uk/id/eprint/100297

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