Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stein's method for Fréchet approximation: a regularly varying functions approach

Paul Mansanarez, Guillaume Poly, Yvik Swan

TL;DR

The paper develops a Stein’s method framework for Fréchet approximation that bounds distributional distances between normalized maxima and Fréchet laws using a reverse hazard rate–based discrepancy. It connects convergence to Fréchet limits with tail regular variation of the underlying distribution and provides a density-derivative–free approach with explicit distance bounds for Kolmogorov, total variation, and Wasserstein metrics. The main contributions include a concrete discrepancy that governs these distances, a precise link between Fréchet convergence and tail regular variation, and explicit computations for maxima from Pareto, Cauchy, and Burr XII distributions. This methodology yields practical, fixed-n approximations in extreme value theory and broadens the applicability of Stein’s method in the analysis of maxima.

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 Fréchet 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.

Stein's method for Fréchet approximation: a regularly varying functions approach

TL;DR

The paper develops a Stein’s method framework for Fréchet approximation that bounds distributional distances between normalized maxima and Fréchet laws using a reverse hazard rate–based discrepancy. It connects convergence to Fréchet limits with tail regular variation of the underlying distribution and provides a density-derivative–free approach with explicit distance bounds for Kolmogorov, total variation, and Wasserstein metrics. The main contributions include a concrete discrepancy that governs these distances, a precise link between Fréchet convergence and tail regular variation, and explicit computations for maxima from Pareto, Cauchy, and Burr XII distributions. This methodology yields practical, fixed-n approximations in extreme value theory and broadens the applicability of Stein’s method in the analysis of maxima.

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 Fréchet 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.
Paper Structure (4 sections, 13 theorems, 88 equations, 1 figure, 1 table)

This paper contains 4 sections, 13 theorems, 88 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Stein's method and a proof of Theorems \ref{['mainresult']} and \ref{['mainresultwass']}
  3. Proof of Theorem \ref{['borneXalpha']}
  4. Regularly varying functions

Key Result

Theorem 1.1

Let $P, Q$ be two cdfs satisfying Assumption 0, with pdfs $p, q$ and supports $[c_P, +\infty)$ and $[c_Q, +\infty)$, respectively. Suppose that $P$ has no atom (i.e. $p_0 = 0$) and that $c_Q\ge c_P\ge -\infty$. Introduce the Stein discrepancy Then with $q_0 = Q(\left\{ c_Q \right\})$ the starting mass of $Q$.

Figures (1)

  • Figure 1: Numerical evaluation (with Mathematica) of $n^{1/\tau} \Delta(\Phi_{\alpha}\mid F_n)$ (left plot) and $n^{1/\tau} \Delta_w(\Phi_{\alpha}\mid F_n)$ (right plot) for $n= 10^{5}$, as a function of $\alpha \in [1, 10]$ for $\tau = 3$ (blue curve), $\tau=3.5$ (orange curve) and $\tau = 4$ (green curve) when $F$ is the Burr XII distribution.

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 1.4: Maxima of independent Pareto
  • Example 1.5: Comparison of Fréchet
  • Example 1.6: Maxima of independent Cauchy
  • Example 1.7: Maxima of independent Burr XII
  • Proposition 2.1
  • proof
  • Definition A.1: B.1.1, haan2006extreme
  • ...and 13 more