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Lognormal Distributions and Geometric Averages of Symmetric Positive Definite Matrices
Author(s) -
Schwartzman Armin
Publication year - 2016
Publication title -
international statistical review
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.051
H-Index - 54
eISSN - 1751-5823
pISSN - 0306-7734
DOI - 10.1111/insr.12113
Subject(s) - mathematics , log normal distribution , univariate , statistics , matrix (chemical analysis) , bivariate analysis , limit (mathematics) , combinatorics , mathematical analysis , multivariate statistics , materials science , composite material
Summary This article gives a formal definition of a lognormal family of probability distributions on the set of symmetric positive definite (SPD) matrices, seen as a matrix‐variate extension of the univariate lognormal family of distributions. Two forms of this distribution are obtained as the large sample limiting distribution via the central limit theorem of two types of geometric averages of i.i.d. SPD matrices: the log‐Euclidean average and the canonical geometric average. These averages correspond to two different geometries imposed on the set of SPD matrices. The limiting distributions of these averages are used to provide large‐sample confidence regions and two‐sample tests for the corresponding population means. The methods are illustrated on a voxelwise analysis of diffusion tensor imaging data, permitting a comparison between the various average types from the point of view of their sampling variability.