A Derivation of the Wishart and Singular Wishart Distributions

Abstract

Multivariate statistical analysis is the area of statistics that is concerned with observations made on many variables. Determining how variables are related is a main objective in multivariate analysis. The covariance matrix is an essential part of understanding the dependence between variables. The distribution of the sample covariance matrix for a sample from a multivariate normal distribution, known as the Wishart distribution, is fundamental to multivariate statistical analysis. An important assumption of the well-known Wishart distribution is that the number of variables is smaller than the number of observations. In high-dimensions when the number of variables exceeds the number of observations, the Wishart matrix is singular and has a singular Wishart distribution. The purpose of this research is to rederive the Wishart and singular Wishart distributions and understand the mathematics behind each derivation.