Moments Calculation For the Doubly Truncated Multivariate Normal Density

TL;DR

This paper derives explicit formulas for the mean, variance, and bivariate marginals of doubly truncated multivariate normal distributions, extending existing methods and providing algorithms for practical computation.

Contribution

It introduces a comprehensive approach to calculate moments and marginals for doubly truncated multivariate normal distributions, including new formulas and invariance properties.

Findings

Explicit formulas for truncated mean and variance

Algorithms implemented in R for practical computation

Demonstrations with three example cases

Abstract

In the present article we derive an explicit expression for the trun- cated mean and variance for the multivariate normal distribution with ar- bitrary rectangular double truncation. We use the moment generating ap- proach of Tallis (1961) and extend it to general {\mu}, {\Sigma} and all combinations of truncation. As part of the solution we also give a formula for the bivari- ate marginal density of truncated multinormal variates. We also prove an invariance property of some elements of the inverse covariance after trunca- tion. Computer algorithms for computing the truncated mean, variance and the bivariate marginal probabilities for doubly truncated multivariate normal variates have been written in R and are presented along with three examples.