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

TL;DR

The proposed CCDF test turned out to perform well for both small and large samples, comparable to the spacings and entropy difference tests, and was extended to a nonstationary rGEV model.

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{é}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.

Figures (5)

• Figure 1: Histograms for the selected $r$ for five parameter sets of rGEV with $r=5$ (true), from 1,000 random samples with $n=30,\, 50,\, 80$. The shape parameters from top to bottom are par1: $k=-0.35$, par2: $k=-0.2$, par3: $k=0.0$, par4: $k=0.2$), and par5: $k=0.35$.
• Figure 2: The same as Figure \ref{['fig:rsel_sim_gev']} but for the Wakeby distribution. The parameters from top to bottom are from WA-1 to WA-6 in which the details are provided in the Supplementary Information.
• Figure 3: Scatterplot matrix of histograms, time series plots, and scatterplots of $r$ largest order statistics for $r=1,\dots,5$, drawn from daily rainfall data (unit: $mm$ ) in Sancheong, Korea.
• Figure 4: Probability plots for $r=1,\dots,8$ for extreme rainfall data of Sancheong in Korea, drawn from the conditional cumulative distribution function (CCDF) method.
• Figure 5: Same as Figure \ref{['fig:pp_ccdf_101']} but using the spacings method.