  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.
