Modeling climate extremes using the four-parameter kappa distribution for $r$-largest order statistics

Accurate estimation of the T-year return levels of climate extremes using statistical distribution is a critical step in the projection of future climate and in engineering design for disaster response. We show how the estimation of such quantities can be improved by fitting {the four-parameter kappa distribution for $r$-largest order statistics} (rK4D), which was developed in this study. The rK4D is an extension of {the generalized extreme value distribution for $r$-largest order statistics} (rGEVD), similar to the four-parameter kappa distribution (K4D), which is an extension of the generalized extreme value distribution (GEVD). This new distribution (rK4D) can be useful not only for fitting data when three parameters in the GEVD are not sufficient to capture the variability of the extreme observations, but also in reducing the estimation uncertainty by making use of the r-largest extreme observations instead of only the block maxima. We derive a joint probability density function (PDF) of rK4D and the marginal and conditional cumulative distribution functions and PDFs. To estimate the parameters, the maximum likelihood estimation and the maximum penalized likelihood estimation methods were considered. The usefulness and practical effectiveness of the rK4D are illustrated by the Monte Carlo simulation and by an application to the Bangkok extreme rainfall data. A few new distributions for $r$-largest order statistics are also derived as special cases of the rK4D, such as the $r$-largest logistic, the $r$-largest generalized logistic, and the $r$-largest generalized Gumbel distributions. These distributions for $r$-largest order statistics would be useful in modeling extreme values for many research areas, including hydrology and climatology.