# 4. The Multivariate Normal Distribution

1. Alvin C. Rencher

Published Online: 27 MAR 2003

DOI: 10.1002/0471271357.ch4

## Methods of Multivariate Analysis, Second Edition

#### How to Cite

Rencher, A. C. (2002) The Multivariate Normal Distribution, in Methods of Multivariate Analysis, Second Edition, John Wiley & Sons, Inc., New York, NY, USA. doi: 10.1002/0471271357.ch4

#### Author Information

1. Brigham Young University, USA

#### Publication History

1. Published Online: 27 MAR 2003
2. Published Print: 22 FEB 2002

#### Book Series:

1. Wiley Series in Probability and Statistics

#### ISBN Information

Print ISBN: 9780471418894

Online ISBN: 9780471271352

## SEARCH

### Keywords:

• generalized population variance;
• multicollinearity;
• maximum likelihood;
• Wishart distribution;
• Q-Q plot;
• skewness;
• kurtosis;
• mean slippage;
• variance slippage;
• elliptically symmetric distribution

### Summary

Most inferential procedures in the book are based on the multivariate normal distribution. The properties of multivariate normal random variables include the following: linear functions are normal, certain quadratic functions have a chi-square distribution, any subset of variables has a normal distribution, any two subvectors are independent if their covariances are all zero, the conditional distribution of a subvector adjusted for another is normal.

Estimates of the mean vector and covariance matrix are given, along with the distribution of these estimators. There is a multivariate central limit theorem corresponding to the univariate central limit theorem. Methods are given for assessing both univariate and multivariate normality of a sample. Some of these methods involve test statistics, and tables of critical values are given in Appendix A. Methods for detecting outliers in either univariate or multivariate data are discussed.

Numerical and graphical illustrations are given, and the problems at the end of the chapter call for derivation of some of the results in the chapter and provide additional numerical illustrations.