Multivariate Normal Distribution
Hemant Ishwaran · 2025
A multivariate normal (or Gaussian) distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. It describes a random vector whose components are jointly normally distributed. The multivariate normal distribution is a cornerstone of multivariate statistical analysis. Its mathematical properties and ability to describe the joint behavior of multiple random variables make it a critical tool for developing and applying statistical methods across a wide range of disciplines. Understanding the multivariate normal distribution is essential for effectively modeling, analyzing, and interpreting multivariate data. This chapter reviews the properties of the multivariate normal distribution and provides various applications of its use, including the famous regression to the mean phenomenon, normal mixture models, Bayesian multivariate normal hierarchical models, and unbiased estimating equations with Godambe information. The latter illustrates how the multivariate normal distribution often arises in asymptotic theory. Readers will also find an appendix at the end that conveniently lists several classical statistics related to the univariate and multivariate normal distributions.