The Partial L-Moment of the Four Kappa Distribution

Abstract

Statistical analysis of extreme events such as flood events is often carried out to predict large return period events. The behaviour of extreme events not only involves heavy-tailed distributions but also skewed distributions, similar to the four-parameter Kappa distribution (K4D). In general, this covers many extreme distributions such as the generalized logistic distribution (GLD), the generalized extreme value distribution (GEV), the generalized Pareto distribution (GPD), and so on. To utilize these distributions, we have to estimate parameters accurately. There are many parameter estimation methods, for example, Method of Moments, Maximum Likelihood Estimator, L-Moments, or partial L-Moments. Nowadays, no researchers have applied the partial L-Moments method to estimate the parameters of K4D. Therefore, the objective of this paper is to derive the partial L-Moments (PL-Moments) for K4D, namely the PL-Moments of the K4D in order to estimate hydrological extremes from censored data. The findings of this paper are formulas of parameter estimation for K4D based on the PL-Moments approach. We have derived the Partial Probability-Weighted Moments (PPWMs) of the K4D (β'r) and derive the estimation of parameters when separated by shape parameters (k,h) conditions i.e., case k>-1 and h>0, case k>-1 and h=0 and case -1

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四Kappa分布的偏l矩

经常对洪水等极端事件进行统计分析，以预测大型重现期事件。极端事件的行为不仅涉及重尾分布，还涉及偏斜分布，类似于四参数Kappa分布（K4D）。一般来说，这涵盖了许多极值分布，如广义逻辑分布（GLD）、广义极值分布（GEV）、广义帕累托分布（GPD）等。要利用这些分布，我们必须准确估计参数。有许多参数估计方法，例如矩量法、最大似然估计、L-Moments或部分L-Moments。目前，还没有研究人员应用偏L-Moments方法来估计K4D的参数。因此，本文的目的是推导K4D的部分L矩（PL矩），即K4D的PL矩，以便从截尾数据中估计水文极值。本文的研究结果是基于PL矩方法的K4D参数估计公式。我们导出了K4D（β’r）的部分概率加权矩（PPWMs），并导出了当由形状参数（k，h）条件分离时的参数估计，即，情况k>-1和h>0，情况k>1和h=0，以及情况-1

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