Stein's method for Fr\'echet approximation: a regularly varying functions approach

Abstract

We develop a variant of Stein's method of comparison of generators to bound the Kolmogorov, total variation, and Wasserstein-1 distances between distributions on the real line. Our discrepancy is expressed in terms of the ratio of reverse hazard rates; it therefore remains tractable even when density derivatives are intractable. Our main application concerns the approximation of normalized extremes by Fr\'echet laws. In this setting, the new discrepancy provides a quantitative measure of distributional proximity in terms of the average regular variation at infinity of the underlying cumulative distribution function. We illustrate the approach through explicit computations for maxima of Pareto, Cauchy, and Burr~XII distributions. Our new discrepancy also opens the way to statistical applications which we outline.