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1990 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

The Multivariate Normal Distribution

verfasst von: Y. L. Tong

Verlag: Springer New York

Buchreihe : Springer Series in Statistics

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
The multivariate normal distribution is undoubtedly one of the most well-known and useful distributions in statistics, playing a predominant role in many areas of applications. In multivariate analysis, for example, most of the existing inference procedures for analyzing vector-valued data have been developed under the assumption of normality. In linear model problems, such as the analysis of variance and regression analysis, the error vector is often assumed to be normally distributed so that statistical analysis can be performed using distributions derived from the normal distribution. In addition to appearing in these areas, the multivariate normal distribution also appears in multiple comparisons, in the studies of dependence of random variables, and in many other related areas.
Y. L. Tong
Chapter 2. The Bivariate Normal Distribution
Abstract
In the univariate case, a random variable X is said to have a normal distribution with mean μ and variance σ2 > 0 (in symbols, N(μ, σ2)) if its density function is of the form
$$ f\left( {x;\mu ,{\sigma ^2}} \right) = \frac{1}{{\sqrt {2\pi \sigma } }}{e^{ - {Q_1}\left( {x;\mu ,{\sigma ^2}} \right)/2}},x \in \Re , $$
where
$$ {Q_1}\left( {x;\mu ,{\sigma ^2}} \right) = \frac{1}{{{\sigma ^2}}}{\left( {x - \mu } \right)^2} = \left( {x - \mu } \right){\left( {{\sigma ^2}} \right)^{ - 1}}\left( {x - \mu } \right), $$
μ ∈ ℜ, and σ2 ∈ (0, ∞). The bivariate normal density function given below is a natural extension of this univeriate normal density.
Y. L. Tong
Chapter 3. Fundamental Properties and Sampling Distributions of the Multivariate Normal Distribution
Abstract
In this chapter we study some fundamental properties of the multivariate normal distribution, including distribution properties and related sampling distributions.
Y. L. Tong
Chapter 4. Other Related Properties
Abstract
In Chapter 3 we have studied the fundamental properties and related distribution theory of the multivariate normal distribution using the specific functional form of its density function. Most of the results given there have been obtained by direct algebraic calculations. In this chapter we study some related properties with a more general approach. We show that the multivariate normal density function belongs to certain large classes of density functions. Consequently, it has all the common properties possessed by density functions in those classes. This approach allows us to apply more general and powerful mathematical tools for deriving useful results.
Y. L. Tong
Chapter 5. Positively Dependent and Exchangeable Normal Variables
Abstract
The study of concepts of positive dependence of random variables, started in the late 1960s, has yielded numerous useful results in both statistical theory and applications. The results are generally given for large classes of distributions and are not just for the normal family. However, when the underlying distribution is multivariate normal, then many special results follow and most of them involve only the covariance matrix of the distribution.
Y. L. Tong
Chapter 6. Order Statistics of Normal Variables
Abstract
The theory and applications of order statistics have been studied extensively in the literature, a convenient reference is David (1981). In this chapter we present some results concerning the distributions and moments of order statistics when the parent distribution is multivariate normal.
Y. L. Tong
Chapter 7. Related Inequalities
Abstract
As noted by Pólya (1967), “Inequalities play a role in most branches of mathematics and have widely different applications.” This is certainly true in statistics and probability. From the viewpoint of applications, inequalities have become a useful tool in estimation and hypothesis-testing problems (such as for yielding bounds on the variances of estimators and on the probability contents of confidence regions, and for establishing monotonicity properties of the power functions of certain tests), in multivariate analysis, in reliability theory, and so forth. Perhaps the usefulness of inequalities in multivariate analysis can be best illustrated by the following situation: Suppose that in an applied problem the confidence probability of a given confidence region for the mean vector is difficult to evaluate. If an inequality in the form of a lower bound on the confidence probability can easily be obtained, and if the lower bound already meets the required level of specification, then we know for sure that the true confidence probability meets or exceeds the required level.
Y. L. Tong
Chapter 8. Statistical Computing Related to the Multivariate Normal Distribution
Abstract
In this chapter we discuss some useful methods concerning statistical computing related to the multivariate normal distribution. Section 8.1 deals with methods for generating random variates from a multivariate normal distribution in simulation studies. The methods involve linear transformations of i.i.d. univariate normal variables, and the linear transformation used in a given application depends on the covariance matrix of the distribution. In Sections 8.2 and 8.3 we discuss numerical methods for evaluating probability integrals under a multivariate normal density function. Special attention will be focused on the computation of the distribution functions of normal variables and of their absolute values (called one-sided and two-sided probability integrals). Equi-coordinate percentage points and probability integrals for exchangeable normal variables and for their absolute values have been tabulated numerically, and the tables are given in the Appendix of this volume. The accuracy and uses of the tables are discussed in Section 8.4.
Y. L. Tong
Chapter 9. The Multivariate t Distribution
Abstract
If Z is an N(0, 1) variable and independent of S, where vS2 has a chi-square distribution with v degrees of freedom, then the random variable t = Z/S is called a Student’s t variable with v degrees of freedom. The distribution of t can be found in elementary textbooks, and it plays a central role in statistical inference problems concerning the mean of a univariate normal distribution with unknown variance. The multivariate t distribution, defined below and studied in this chapter, is a multivariate generalization of Student’s t distribution.
Y. L. Tong
Backmatter
Metadaten
Titel
The Multivariate Normal Distribution
verfasst von
Y. L. Tong
Copyright-Jahr
1990
Verlag
Springer New York
Electronic ISBN
978-1-4613-9655-0
Print ISBN
978-1-4613-9657-4
DOI
https://doi.org/10.1007/978-1-4613-9655-0