The Multivariate Normal DistributionThe multivariate normal distribution has played a predominant role in the historical development of statistical theory, and has made its appearance in various areas of applications. Although many of the results concerning the multivariate normal distribution are classical, there are important new results which have been reported recently in the literature but cannot be found in most books on multivariate analysis. These results are often obtained by showing that the multivariate normal density function belongs to certain large families of density functions. Thus, useful properties of such families immedi ately hold for the multivariate normal distribution. This book attempts to provide a comprehensive and coherent treatment of the classical and new results related to the multivariate normal distribution. The material is organized in a unified modern approach, and the main themes are dependence, probability inequalities, and their roles in theory and applica tions. Some general properties of a multivariate normal density function are discussed, and results that follow from these properties are reviewed exten sively. The coverage is, to some extent, a matter of taste and is not intended to be exhaustive, thus more attention is focused on a systematic presentation of results rather than on a complete listing of them. |
Contents
| 4 | |
CHAPTER | 9 |
CHAPTER | 13 |
CHAPTER 3 | 23 |
7 | 60 |
དཎྜཎྜསྤྱ | 93 |
CHAPTER 8 | 181 |
19 | 200 |
The Multivariate t Distribution | 202 |
| 218 | |
| 225 | |
Appendix Tables | 229 |
| 261 | |
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Common terms and phrases
a₁ a₂ applications arbitrary but fixed b₁ bivariate normal distribution c₁ C₂ canonical correlation common correlation coefficient common mean common variance o² conditional distribution convex convex set Corollary defined degrees of freedom denote density function f(x distribution function exchangeable normal variables exchangeable random variables Fact function of X1 given X₂ holds identically distributed independent inequalities joint density function k₁ Let X1 log-concave M-matrix marginal distribution Marshall and Olkin mean vector mean µ MTP2 multiple correlation multiple correlation coefficient multivariate normal distribution n-dimensional random variable nondecreasing function nonnegative nonsingular normal density function obtained order statistics P₁ P₂ permutation permutation-symmetric positive definite positive dependence probability content probability integrals problem Proposition Proschan real numbers result right-hand side sample covariance matrix Schur-concave function Show symmetric t₁ tion Tong U₁ univariate V₁ values Verify X₁ Y₁ Y₂ Z₁ µ₁ σ₁ σ²