Automated Threshold Selection for Extreme Value Analysis via Goodness-of-Fit Tests with Application to Batched Return Level Mapping

Abstract

Threshold selection is a critical issue for extreme value analysis with threshold-based approaches. Under suitable conditions, exceedances over a high threshold have been shown to follow the generalized Pareto distribution (GPD) asymptotically. In practice, however, the threshold must be chosen. If the chosen threshold is too low, the GPD approximation may not hold and bias can occur. If the threshold is chosen too high, reduced sample size increases the variance of parameter estimates. To process batch analyses, commonly used selection methods such as graphical diagnosis are subjective and cannot be automated, while computational methods may not be feasible. We propose to test a set of thresholds through the goodness-of-fit of the GPD for the exceedances, and select the lowest one, above which the data provides adequate fit to the GPD. Previous attempts in this setting are not valid due to the special feature that the multiple tests are done in an ordered fashion. We apply two recently available stopping rules that control the false discovery rate or familywise error rate to ordered goodness-of-fit tests to automate threshold selection. Various model specification tests such as the Cramer-von Mises, Anderson-Darling, Moran's, and a score test are investigated. The performance of the method is assessed in a large scale simulation study that mimics practical return level estimation. This procedure was repeated at hundreds of sites in the western US to generate return level maps of extreme precipitation.