A Tail-Weighted Goodness-of-Fit Test for the Generalized Extreme Value Distribution Based on the Probability Integral Transform
Year: 2026
Type: article
Abstract: The generalized extreme value (GEV) distribution is used to model block maxima and extrapolate beyond the observed range, making adequate tail fit essential. We propose a bounded, tunable goodness-of-fit statistic for the GEV distribution with unknown parameters, based on the probability integral transform and a symmetric weight whose amplitude and concentration are controlled separately. We establish location–scale invariance, an exact relationship with the Cramér–von Mises statistic, a known-parameter limit distribution with closed-form moments, consistency against fixed alternatives, and a composite-null limit conditional on explicit regularity assumptions. Critical values are obtained by a parametric bootstrap that refits the GEV model in every resample; the same calibration is applied to the Kolmogorov–Smirnov, Anderson–Darling, Cramér–von Mises, and upper-tail modified Anderson–Darling tests. Simulations over shape parameters from −0.4 to 0.4 and sample sizes from 15 to 100 show no evidence of over-rejection, and the proposed tests are competitive with Anderson–Darling under maximum-likelihood and probability-weighted-moment estimation. All six tests have limited power against subtle upper-tail-only departures. The return-level study shows that rejection can identify samples with greater extrapolation error, whereas non-rejection does not guarantee accurate remote-quantile estimates. Applications to environmental and actuarial block maxima illustrate sensitivity to temporal trend and block length.
Source: Mathematics
DOI: 10.3390/math14213914
Authors: Preut Thanarat, Nipada Papukdee
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,137
Funder: Mahasarakham University
OpenAlex ID: W7210273324