Distances between distributions via Stein's method

Abstract

We build on the formalism developed in [21] to propose new representations of solutions to Stein equations. We provide new uniform and non uniform bounds on these solutions (a.k.a. Stein factors). We use these representations to obtain representations for differences between expectations in terms of solutions to the Stein equations. We apply these to compute abstract Stein-type bounds on Kolmogorov, Total Variation and Wasserstein distances between arbitrary distributions. We apply our results to several illustrative examples, and compare our results with current literature on the same topic, whenever possible. In all occurrences our results are competitive.

Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I

“…Remark 4.9. The following bounds will appear in the supplementary material of the arXiv version of the preprint [10]. For n ≥ 2,…”

“…(We define 0 0 := 1, but this is irrelevant because the bound (4.51) is greater than 1 in this case.) These bounds were obtained using a recent technique of [10] for bounding distances between distributions that builds upon the formalism of [9] for new representations of solutions to Stein equations. For another recent approach to bounding distances between distributions, see [8].…”

“…The author is supported by a Dame Kathleen Ollerenshaw Research Fellowship. The author would like to thank Yvik Swan for generously sharing some results that will be added to the supplementary material of the arXiv version of his preprint [10], which are stated in Remark 4.9.…”