  We use the delta method and Stein's method to derive, under regularity
conditions, explicit upper bounds for the distributional distance between the
distribution of the maximum likelihood estimator (MLE) of a $d$-dimensional
parameter and its asymptotic multivariate normal distribution. Our bounds apply
in situations in which the MLE can be written as a function of a sum of i.i.d.
$t$-dimensional random vectors. We apply our general bound to establish a bound
for the multivariate normal approximation of the MLE of the normal distribution
with unknown mean and variance.
