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

Abstract

Sampling from a truncated multivariate 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 remains nontrivial due to the analytically intractable normalizing constant of the truncated multivariate distribution. We first develop an efficient 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. We then extend our methodology to obtain samples from a truncated multivariate normal distribution subject to convex polytope restriction regions. Finally, we generalize the sampling method to truncated scale mixtures of multivariate normal distributions. Empirical results are presented to illustrate the superior performance of our proposed Gibbs sampler in terms of various criteria (e.g., mixing and integrated auto-correlation time).

AMS Subject Classification:

• 62E99

Keywords:

• Gibbs sampler
• Rejection sampling
• Truncated multivariate normal distribution
• Truncated multivariate Student-t distribution
• Truncated scale mixture of multivariate normal distribution