Multivariate Normal Distributions Parametrized as a Riemannian Symmetric Space

Miroslav Lovric, Maung Min-Oo and Ernst A. Ruh

Journal of Multivariate Analysis, 2000, vol. 74, issue 1, 36-48

Abstract: The construction of a distance function between probability distributions is of importance in mathematical statistics and its applications. The distance function based on the Fisher information metric has been studied by a number of statisticians, especially in the case of the multivariate normal distribution (Gaussian) on n. It turns out that, except in the case n=1, where the Fisher metric describes the hyperbolic plane, it is difficult to obtain an exact formula for the distance function (although this can be achieved for special families with fixed mean or fixed covariance). We propose to study a slightly different metric on the space of multivariate normal distributions on n. Our metric is based on the fundamental idea of parametrizing this space as the Riemannian symmetric space SL(n+1)/SO(n+1). Symmetric spaces are well understood in Riemannian geometry, allowing us to compute distance functions and other relevant geometric data.

Keywords: multivariate normal distributions; geodesic distance; Riemannian symmetric space; curvature; center of mass (search for similar items in EconPapers)
Date: 2000
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Citations: View citations in EconPapers (8)

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