Statistical Papers

, Volume 60, Issue 2, pp 545–564 | Cite as

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

  • Luc PronzatoEmail author
  • Henry P. Wynn
  • Anatoly Zhigljavsky
Regular Article


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 \(\phi \) 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.


Simplicial distances Bregman divergence Optimal design criteria Burbea-Rao divergence Energy statistic 

Mathematics Subject Classification

62H30 62K05 



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CNRS, UCA, Laboratoire I3S, UMR 7172; 2000, route des Lucioles, Les AlgorithmesSophia AntipolisFrance
  2. 2.London School of EconomicsLondonUK
  3. 3.School of MathematicsCardiff UniversityCardiffUK

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