On the Conditional Distribution of the Multivariate $t$ Distribution

TL;DR

This paper clarifies the properties of the multivariate t distribution, correcting a misconception about its conditional distribution and providing an intuitive proof to enhance understanding in statistical analysis.

Contribution

It offers a simple, intuitive proof that the conditional distribution of the multivariate t distribution is also a t distribution, correcting previous literature.

Findings

Confirmed the conditional distribution is a multivariate t distribution

Provided an intuitive proof avoiding complex density calculations

Clarified misconceptions in existing literature

Abstract

As alternatives to the normal distributions, $t$ distributions are widely applied in robust analysis for data with outliers or heavy tails. The properties of the multivariate $t$ distribution are well documented in Kotz and Nadarajah's book, which, however, states a wrong conclusion about the conditional distribution of the multivariate $t$ distribution. Previous literature has recognized that the conditional distribution of the multivariate $t$ distribution also follows the multivariate $t$ distribution. We provide an intuitive proof without directly manipulating the complicated density function of the multivariate $t$ distribution.