A Tail-Weighted Goodness-of-Fit Test for the Generalized Extreme Value Distribution Based on the Probability Integral Transform

Source 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.