The four-parameter kappa distribution (K4D) is a generalized form of some commonly used distributions such as generalized logistic, generalized Pareto, generalized Gumbel, and generalized extreme value (GEV) distributions. Owing to its flexibility, the K4D is widely applied in modeling in several fields such as hydrology and climatic change. For the estimation of the four parameters, the maximum likelihood approach and the method of L-moments are usually employed. The L-moment estimator (LME) method works well for some parameter spaces, with up to a moderate sample size, but it is sometimes not feasible in terms of computing the appropriate estimates. Meanwhile, the maximum likelihood estimator (MLE) is optimal for large samples and applicable to a very wide range of situations, including non-stationary data. However, using the MLE of K4D with small sample sizes shows substantially poor performance in terms of a large variance of the estimator. We therefore propose a maximum penalized likelihood estimation (MPLE) of K4D by adjusting the existing penalty functions that restrict the parameter space. Eighteen combinations of penalties for two shape parameters are considered and compared. The MPLE retains modeling flexibility and large sample optimality while also improving on small sample properties. The properties of the proposed estimator are verified through a Monte Carlo simulation, and an application case is demonstrated taking Thailand's annual maximum temperature data. Based on this study, we suggest using combinations of penalty functions in general.

翻译：四参数Kappa分布（K4D）是广义逻辑分布、广义帕累托分布、广义冈贝尔分布和广义极值分布（GEV）等常用分布的广义形式。由于其灵活性，K4D在水文学和气候变化等多个领域的建模中得到广泛应用。对于四个参数的估计，通常采用最大似然法和L矩法。L矩估计量（LME）方法在某些参数空间下表现良好，适用于中等样本量，但有时在计算适当估计量方面不可行。同时，最大似然估计量（MLE）在大样本下具有最优性，适用于非常广泛的情况，包括非平稳数据。然而，在小样本情况下使用K4D的MLE会表现出估计量方差较大的显著性能缺陷。因此，我们通过调整现有的限制参数空间的惩罚函数，提出了K4D的最大惩罚似然估计（MPLE）。研究考虑并比较了针对两个形状参数的十八种惩罚函数组合。MPLE在保持建模灵活性和大样本最优性的同时，也改善了小样本特性。通过蒙特卡洛模拟验证了所提估计量的性质，并以泰国年最高气温数据为例进行了应用演示。基于本研究，我们建议在一般情况下使用惩罚函数的组合。