Abstract
We obtain explicit p-Wasserstein distance error bounds between the distribution of the multi-parameter MLE and the multivariate normal distribution. Our general bounds are given for possibly high-dimensional, independent and identically distributed random vectors. Our general bounds are of the optimal order. Explicit numerical constants are given when , and in the case the bounds are explicit up to a constant factor that only depends on p. We apply our general bounds to derive Wasserstein distance error bounds for the multivariate normal approximation of the MLE in several settings; these being single-parameter exponential families, the normal distribution under canonical parametrisation, and the multivariate normal distribution under non-canonical parametrisation. In addition, we provide upper bounds with respect to the bounded Wasserstein distance when the MLE is implicitly defined.
Funding Statement
RG is supported by a Dame Kathleen Ollerenshaw Research Fellowship.
Acknowledgments
We are very grateful to Thomas Bonis for valuable discussions concerning the results from his paper [9] and for working out an explicit bound on the constant in one of the main quantitative limit theorems from his paper that we used in our paper. We would like to thank the referees for their helpful comments and suggestions that have enabled us to greatly improve our paper. In particular, we are very grateful to one of the referees for their insightful comments and explanations, which enabled us to obtain p-Wasserstein analogues of the 1-Wasserstein distance bounds given in the original submission.
Citation
Andreas Anastasiou. Robert E. Gaunt. "Wasserstein distance error bounds for the multivariate normal approximation of the maximum likelihood estimator." Electron. J. Statist. 15 (2) 5758 - 5810, 2021. https://doi.org/10.1214/21-EJS1920
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