Constructing confidence intervals of extreme rainfall quantiles using Bayesian, bootstrap, and profile likelihood approaches

Abstract

Hydrological risk is highly dependent on the occurrence of extreme rainfalls. This fact has led to a wide range of studies on the estimation and uncertainty analysis of the extremes. In most cases, confidence intervals (CIs) are constructed to represent the uncertainty of the estimates. Since the accuracy of CIs depends on the asymptotic normality of the data and is questionable with limited observations in practice, a Bayesian highest posterior density (HPD) interval, bootstrap percentile interval, and profile likelihood (PL) interval have been introduced to analyze the uncertainty that does not depend on the normality assumption. However, comparison studies to investigate their performances in terms of the accuracy and uncertainty of the estimates are scarce. In addition, the strengths, weakness, and conditions necessary for performing each method also must be investigated. Accordingly, in this study, test experiments with simulations from varying parent distributions and different sample sizes were conducted. Then, applications to the annual maximum rainfall (AMR) time series data in South Korea were performed. Five districts with 38-year (1973-2010) AMR observations were fitted by the three aforementioned methods in the application. From both the experimental and application results, the Bayesian method is found to provide the lowest uncertainty of the design level while the PL estimates generally have the highest accuracy but also the largest uncertainty. The bootstrap estimates are usually inferior to the other two methods, but can perform adequately when the distribution model is not heavy-tailed and the sample size is large. The distribution tail behavior and the sample size are clearly found to affect the estimation accuracy and uncertainty. This study presents a comparative result, which can help researchers make decisions in the context of assessing extreme rainfall uncertainties.