The multivariate normal distribution
This book represents a comprehensive and coherent treatment of the results related to the multivariate normal distribution. In addition to the classical topics on distribution theory, correlation analysis and sampling distributions, it also contains important results reported recently in the literature, but which cannot be found in most books on multivariate analysis. The material is organized in a unified modern approach, and the main themes are dependence, probability inequalities, and their roles in theory and applications. Some of the properties (such as log-concavity, unimodality, Schurconcavity and total positivity) of a multivariate normal density function are discussed, and results that follow from these properties and reviewed extensively. The volume also includes tables of the equi-coordinate percentage points and probability inequalities for exchangeable normal variables. The volume is accessible to graduate students and advanced undergraduates in statistics, mathematics, and related applied areas, and can be used as a reference in a course on multivariate analysis.
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applications arbitrary but fixed bivariate normal distribution canonical correlation common correlation coefficient common mean common variance a2 conditional distribution convex convex set Corollary correlation coeff1cient correlation coefficient p e defined deflne degrees of freedom denote distribution function example exchangeable normal variables exchangeable random variables Fact following theorem function of x2 given X2 holds identically distributed joint density function joint distribution Karlin and Rinott Let Xl log-concave M-matrix marginal distribution Marshall and Olkin mean vector multiple correlation multiple correlation coefficient multivariate normal density multivariate normal distribution multivariate normal variable n x n n-dimensional random variable nondecreasing function nonsingular normal density function obtained order statistics orthogonal matrix partial ordering permutation-symmetric positive definite positive dependence probability content Probability Integral problem Proposition Proschan PUOD real numbers right-hand side sample covariance matrix Schur-concave function Section Show Student's t distribution tion univariate values variables with means Verify Xu X2