Non-Steepness and Maximum Likelihood Estimation Properties of the Truncated Multivariate Normal Distributions

This article considers exponential families of truncated multivariate normal distributions with one-sided truncation for some or all coordinates. We observe that if all components are one-sided truncated then this family is not full. The family of truncated multivariate normal distributions is extended to a full family, and the extended family is investigated in detail. We identify the canonical parameter space of the extended family and establish that the family is not regular and not even steep. We also consider maximum likelihood estimation for the location vector parameter and the positive definite (symmetric) matrix dispersion parameter of a truncated non-singular multivariate normal distribution. It is shown that if the sample size is sufficiently large then, almost surely, the maximizer of the likelihood function is unique, provided that it exists. It is also shown that each solution to the score equations for the location and dispersion parameters satisfies the method-of-moments equations. Finally, it is observed that similar results arise in the case of an arbitrary number of truncated components.