Bayesian inference in extremes using the four-parameter kappa distribution

Abstract

Maximum likelihood estimation (MLE) of the four-parameter kappa distribution (K4D) is known to be occasionally unstable for small sample sizes and to be very sensitive to outliers. To overcome this problem, this study proposes Bayesian analysis of the K4D. Bayesian estimators are obtained by virtue of a posterior distribution using the random walk Metropolis–Hastings algorithm. Five different priors are considered. The properties of the Bayesian estimators are verified in a simulation study. The empirical Bayesian method turns out to work well. Our approach is then compared to the MLE and the method of the L-moments estimator by calculating the 20-year return level, the confidence interval, and various goodness-of-fit measures. It is also compared to modeling using the generalized extreme value distribution. We illustrate the usefulness of our approach in an application to the annual maximum wind speeds in Udon Thani, Thailand, and to the annual maximum sea-levels in Fremantle, Australia. In the latter example, non-stationarity is modeled through a trend in time on the location parameter. We conclude that Bayesian inference for K4D may be substantially useful for modeling extreme events.