Weighted Kolmogorov-Smirnov test: Accounting for the tails

TL;DR

This paper introduces a weighted Kolmogorov-Smirnov test tailored for accurately assessing the extreme tails of empirical distributions, with precise asymptotic results relevant for fields like finance and geophysics.

Contribution

It derives exact asymptotic results for a generalized KS test focused on distribution tails, improving precision over previous approximate solutions.

Findings

Derived exact asymptotic results for tail-focused KS test

Rederived and refined previous approximate limit solutions

Enhanced understanding of tail behavior in goodness-of-fit testing

Abstract

Accurate goodness-of-fit tests for the extreme tails of empirical distributions is a very important issue, relevant in many contexts, including geophysics, insurance, and finance. We have derived exact asymptotic results for a generalization of the large-sample Kolmogorov-Smirnov test, well suited to testing these extreme tails. In passing, we have rederived and made more precise the approximate limit solutions found originally in unrelated fields, first in [L. Turban, J. Phys. A 25, 127 (1992)] and later in [P. L. Krapivsky and S. Redner, Am. J. Phys. 64, 546 (1996)].