Automated selection of r for stationary and nonstationary models for r largest order statistics

2026-02-27

AI Summary

This study addresses the lack of efficient and reliable methods for automatically selecting the optimal threshold order $ r $ in the $ r $-th order generalized extreme value (rGEV) model under small-sample conditions. The authors propose a sequential goodness-of-fit test based on the conditional cumulative distribution function (CCDF), which—by introducing the Cramér–von Mises statistic into the selection of $ r $—offers both computational efficiency and robustness in small samples. The approach is further extended to nonstationary settings. Monte Carlo simulations demonstrate that the method performs competitively with established techniques such as the spacing metric and entropy difference methods across both small and large sample sizes, while requiring significantly less computation time. The proposed procedure has been successfully applied to the analysis of daily rainfall extremes in South Korea.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

In generalized extreme value model for the r largest order statistics, denoted by rGEV, the selection of r is critical. The existing entropy difference test for selecting r is applicable to large sample. Another existing method (the score test with parametric bootstrap) is applicable to small sample, but computationally demanding. To address this problem for small sample, we propose a new method using a sequence of the goodness-of-fit tests based on the conditional cumulative distribution function (CCDF). The proposed CCDF test is easy to implement and computationally fast. The Cram{\'e}r-von Mises test was employed for the goodness-of-fit purpose. The proposed method is compared via Monte Carlo simulations with existing methods including the spacings, the score, and the entropy difference tests. The proposed CCDF test turned out to perform well for both small and large samples, comparable to the spacings and entropy difference tests. The utility of the proposed method is illustrated by an application to the r largest daily rainfall data in Korea. Additionally, we extended the existing methods and the CCDF test to a nonstationary rGEV model. Wide applicability of the proposed method are discussed.

Problem

• r largest order statistics
• generalized extreme value model
• selection of r
• small sample
• nonstationary model

Innovation

• r largest order statistics
• goodness-of-fit test
• conditional CDF
• nonstationary rGEV
• automatic r selection