Efficient Sampling Methods for Truncated Multivariate Normal and Student-t Distributions Subject to Linear Inequality Constraints

Sampling from a truncated multivariate normal distribution subject to multiple linear inequality constraints is a recurring problem in many areas in statistics and econometrics, such as the order restricted regressions, censored data models, and shape-restricted nonparametric regressions. However, the sampling problem still appears non-trivial due to the existence of the analytically intractable normalizing constant of the truncated multivariate normal distribution. In this paper, to start with, we develop an efficient mixed rejection sampling method for the truncated univariate normal distribution, and analytically establish its superiority in terms of acceptance rates compared to some of the popular existing methods. As the full conditional distributions of a truncated multivariate normal distribution are truncated univariate normals, we employ the proposed superior univariate sampling method and implement the Gibbs sampler for sampling from a truncated multivariate normal distribution with convex polytope restriction regions. We also generalize the sampling method to truncated multivariate Student-t distributions. Empirical results 1 are presented to illustrate the superior performance of our proposed Gibbs sampler in terms of various criteria (e.g., accuracy, mixing and convergence rate).