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

We consider the truncated multivariate normal distributions for which every component is one-sided truncated. We show that this family of distributions is an exponential family. We identify $\mathcal{D}$, the corresponding natural parameter space, and deduce that the family of distributions is not regular. We prove that the gradient of the cumulant-generating function of the family of distributions remains bounded near certain boundary points in $\mathcal{D}$, and therefore the family also is not steep. We also consider maximum likelihood estimation for $\boldsymbol{\mu}$, the location vector parameter, and $\boldsymbol{\Sigma}$, the positive definite (symmetric) matrix dispersion parameter, of a truncated non-singular multivariate normal distribution. We prove that each solution to the score equations for $(\boldsymbol{\mu},\boldsymbol{\Sigma})$ satisfies the method-of-moments equations, and we obtain a necessary condition for the existence of solutions to the score equations.