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为单侧截断的截断多元正态分布。我们证明了该分布族是一种指数族。我们确定了 $\mathcal{D}$，相应的自然参数空间，并推断该分布族也不是正则的。我们证明了该分布族的累积生成函数的梯度在$\mathcal{D}$中某些边界点附近保持有界，因此该族也不是陡峭的。我们还考虑了截断非奇异多元正态分布的位置向量参数 $\boldsymbol{\mu}$ 和正定（对称）矩阵分散参数 $\boldsymbol{\Sigma}$ 的最大似然估计。我们证明了 $(\boldsymbol{\mu},\boldsymbol{\Sigma})$ 的得分方程的每个解都满足矩估计方程，并得到了得分方程存在解的必要条件。