Statistical Analysis for Extreme Value with Applications to Hydrological Events

Abstract

Four parameter kappa distribution (K4D) is a generalization of common three-parameter distributions and in particular of the Genelarized extreme value (GEV) distribution involved in several important real-life applications such as insurance, hydrological events, earth- quake and etc. Previous studies showed that maximum likelihood estimators (MLE) of parameters are unstable for small sample size. The method of L-moments estimation is an alternative method of estimation similar to a conventional method of moments. However, L-moment estimators are sometimes considered neither computable nor feasible. In this study, we proposed to use of maximum penalized likelihood estimation (MPLE) by adjusting the penalty function of Coles and Dixon (1999) and the penalty function of Martins and Stedinger (2000) for K4D. Monte-Carlo simulation was performed to illustrate the performance of the estimation methods, the maximum penalized likelihood estimation developed from the penalty function of Martins and Stedinger (2000) is MPLE.MS3 and MPLE.MSP3 are better than the MLE and L-moment method in terms RRMSE of all quantiles used K4D. To illustrate, its applicability including annual maximum rainfall data and annual maximum temperature was analyzed, and its fitness was compared with other estimation methods. Finally to study the r-largest order statistics for K4D with can be applied to hydrological data.

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