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.

翻译：我们考虑每个分量均存在单侧截断的截断多元正态分布。研究表明该分布族属于指数族。通过识别其自然参数空间 $\mathcal{D}$，我们推导出该分布族不具有正则性。我们证明该分布族的累积生成函数梯度在 $\mathcal{D}$ 的某些边界点附近保持有界，因此该分布族亦不满足陡峭性。针对非奇异截断多元正态分布的均值向量参数 $\boldsymbol{\mu}$ 与正定对称矩阵散度参数 $\boldsymbol{\Sigma}$，我们进一步讨论其极大似然估计问题。我们证明 $(\boldsymbol{\mu},\boldsymbol{\Sigma})$ 的得分方程每个解均满足矩估计方程组，并给出得分方程解存在性的必要条件。