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We obtain explicit $p$-Wasserstein distance error bounds between thedistribution of the multi-parameter MLE and the multivariate normaldistribution. Our general bounds are given for possibly high-dimensional,independent and identically distributed random vectors. Our general bounds areof the optimal $\mathcal{O}(n^{-1/2})$ order. Explicit numerical constants aregiven when $p\in(1,2]$, and in the case $p>2$ the bounds are explicit up to aconstant factor that only depends on $p$. We apply our general bounds to deriveWasserstein distance error bounds for the multivariate normal approximation ofthe MLE in several settings; these being single-parameter exponential families,the normal distribution under canonical parametrisation, and the multivariatenormal distribution under non-canonical parametrisation. In addition, weprovide upper bounds with respect to the bounded Wasserstein distance when theMLE is implicitly defined.

Wasserstein distance error bounds for the multivariate normal approximation of the maximum likelihood estimator