Optimality of profile likelihood intervals for quantiles of extreme value distributions: application to environmental disasters

Abstract

Large quantiles of extreme value distributions are useful to assess the risk of environmental disasters. Profile likelihood intervals of quantiles are shown here to be optimal for samples of sizes of n ≥ 50. However, they are seldom used, notwithstanding their reasonable coverage frequencies. In contrast, asymptotic maximum likelihood confidence intervals are often used for any sample size, despite their poor coverage frequencies for moderate and small samples and their tendency to underestimate the quantile of interest. Using these intervals may have dangerous consequences in environmental applications. Calibrated interpolated bootstrap intervals have also been considered a good option for estimating quantiles of extreme value distributions but have not been compared before with profile likelihood intervals. Coverage frequencies of these three types of intervals for large quantiles are compared here through simulations for small and moderate sample sizes. The restricted likelihood function was used successfully to overcome the alleged maximum likelihood estimation problems cited in literature that arise with some Weibull and generalized extreme value distributions. Two rainfall datasets are discussed where the profile likelihood intervals were valuable tools for assessing the risk of disasters.Editor D. Koutsoyiannis