The 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.
1 Introduction.- 1.1. Some Fundamental Properties.- 1.2. Historical Remarks.- 1.3. Characterization.- 1.4. Scope and Organization.- 2 The Bivariate Normal Distribution.- 2.1. Some Distribution Properties.- 2.2. The Distribution Function and Sampling Distributions.- 2.3. Dependence and the Correlation Coefficient.- Problems.- 3 Fundamental Properties and Sampling Distributions of the Multivariate Normal Distribution.- 3.1. Preliminaries.- 3.2. Definitions of the Multivariate Normal Distribution.- 3.3. Basic Distribution Properties.- 3.4. Regression and Correlation.- 3.5. Sampling Distributions.- Problems.- 4 Other Related Properties.- 4.1. The Elliptically Contoured Family of Distributions and the Multivariate Normal.- 4.2. Log-Concavity and Unimodality Properties.- 4.3. MTP2 and MRR2 Properties.- 4.4. Schur-Concavity Property.- 4.5. Arrangement-Increasing Property.- Problems.- 5 Positively Dependent and Exchangeable Normal Variables.- 5.1. Positively Dependent Normal Variables.- 5.2. Permutation-Symmetric Normal Variables.- 5.3. Exchangeable Normal Variables.- Problems.- 6 Order Statistics of Normal Variables.- 6.1. Order Statistics of Exchangeable Normal Variables.- 6.2. Positive Dependence of Order Statistics of Normal Variables.- 6.3. Distributions of Certain Partial Sums and Linear Combinations of Order Statistics.- 6.4. Miscellaneous Results.- Problems.- 7 Related Inequalities.- 7.1. Introduction.- 7.2. Dependence-Related Inequalities.- 7.3. Dimension-Related Inequalities.- 7.4. Probability Inequalities for Asymmetric Geometric Regions.- 7.5. Other Related Inequalities.- Problems.- 8 Statistical Computing Related to the Multivariate Normal Distribution.- 8.1. Generation of Multivariate Normal Variates.- 8.2. Evaluation and Approximations of Multivariate Normal Probability Integrals.- 8.3. Computation of One-Sided and Two-Sided Multivariate Normal Probability Integrals.- 8.4. The Tables.- Problems.- 9 The Multivariate t Distribution.- 9.1. Distribution Properties.- 9.2. Probability Inequalities.- 9.3. Convergence to the Multivariate Normal Distribution.- 9.4. Tables for Exchangeable t Variables.- Problems.- References.- Appendix-Tables.- Author Index.