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.

翻译：利用统计分布精确估计气候极值的T年重现水平，是未来气候预测和灾害应对工程设计中的关键步骤。本文展示了通过拟合本研究提出的{基于r-最大次序统计量的四参数kappa分布}（rK4D），如何改进此类量的估计。rK4D是{基于r-最大次序统计量的广义极值分布}（rGEVD）的扩展，类似于四参数kappa分布（K4D）是广义极值分布（GEVD）的扩展。这一新分布（rK4D）不仅适用于当GEVD中的三个参数不足以捕捉极值观测变异性时的数据拟合，还能通过利用r个最大极值观测而非仅用块最大值来降低估计的不确定性。我们推导了rK4D的联合概率密度函数（PDF）以及边缘和条件累积分布函数与PDF。为估计参数，考虑了最大似然估计和最大惩罚似然估计方法。通过蒙特卡洛模拟和对曼谷极端降雨数据的应用，说明了rK4D的实用性和有效性。作为rK4D的特例，还推导出了一些新的基于r-最大次序统计量的分布，例如r-最大逻辑分布、r-最大广义逻辑分布和r-最大广义Gumbel分布。这些基于r-最大次序统计量的分布将有助于水文学和气候学等许多研究领域的极值建模。