Automated selection of $r$ in $r$-largest order statistics models: finite-sample performance and return-level inference

Abstract

Selecting the number $r$ of upper order statistics is a key practical issue in fitting $r$-largest order statistics models. The entropy difference test for selecting $r$ relies on a large-sample approximation. The score test can be used for small samples but is computationally demanding. To obtain a fast method that is also applicable to small samples, we propose a method based on the conditional cumulative distribution function (CCDF), which applies a sequence of Cramér--von Mises goodness-of-fit tests. The tests are calibrated by a parametric bootstrap in which each refit is replaced by a single Newton step, so the computation remains fast. We also revisit the spacings test and show that the two test statistics become nearly identical as $r$ increases. Monte Carlo simulations show that the CCDF test has accurate size and good power in small and moderate samples. Further simulations show that a few additional order statistics can reduce the error of return-level estimates compared with those obtained from the block maxima only. The method is illustrated using rainfall data from Sancheong, Korea.

arXiv commentRevised version