We discuss three estimation methods: the method of moments, probability weighted moments, and L-moments for the scale parameter and the extreme value index in the generalized Pareto distribution under linear normalization. Moreover, we adapt these methods to use for the generalized Pareto distribution under power and exponential normalizations. A simulation study is conducted to compare the three methods on the three models and determine which is the best, which turned out to be the probability weighted moments. A new computational technique for improving fitting quality is proposed and tested on two real-world data sets using the probability weighted moments. We looked back at various maximal data sets that had previously been addressed in the literature and for which the generalized extreme value distribution under linear normalization had failed to adequately explain them. We use the suggested procedure to find good fits.
The main focus of this paper is to find a suitable distribution for the hydrology series of six catchments in Pakistan. Among others Gumbel and Generalized Extreme Value (GEV) distributions were implemented for frequency analysis using Probability weighted moments (PWM) and maximum likelihood (ML) methods for estimating the parameters. Based on goodness of fit tests it was found that GEV distribution fits closely and PWM method is best suited for estimating the parameters. Peaks over threshold (POT) series model was also tried which resulted in favor of GEV distribution. The quantile estimates based on aforementioned distributions also revealed that GEV distribution has been found close to the observed values of annual maximum peak (AMP) flows. Power comparison studies using various goodness of fit tests for Gumbel and GEV distributions using Log-normal and Weibull as alternative distributions at different levels of significance and for different sample sizes n = 10, 30, 50 showed that Anderson Darling (AD) test is more powerful test followed by Modified Anderson Darling (MAD), Cramer Von Mises (CVM) and Kolmogorov Smirnov (KS) test. and Príspevok je zameraný na odvodenie vhodného rozdelenia pravdepodobností hydrologických radov šiestich povodí v Pakistane. Spomedzi iných sa použili rozdelenia Gumbela a jeho štandardnej extrémnej hodnoty (GEV), s odvodením parametrov metódou momentov vážených pravdepodobnosťou (PWM) a metódou maximálnej pravdepodobnosti (ML) pre analýzu početností. Na základe testov zhody sa zistilo, že na odhad parametrov je najvhodnejšia metóda PWM, a že najlepšia zhoda je pre GEV rozdelenie. Model s výberom maxím nad určitou hranicou (POT) sa tiež použil, s najlepšou zhodou tiež pre rozdelenie GEV. Odhady kvantilov uskutočnené podľa vyššie uvedených rozdelení tiež potvrdili, že tieto pre rozdelenie GEV sú blízke pozorovaným hodnotám ročných maxím (AMP). Porovnanie mocnín (PC) sa tiež uskutočnili pri použití rôznych testov zhody. Pre rozdelenia podľa Gumbela a GEV, alternatívne tiež rozdelenia podľa Weibulla a Logaritmicko-normálne, sa pre súbory s veľkosťou n = 10, 30, 50 členov ako najúčinnejšie ukázali nasledujúce testy na rôznych hladinách významnosti: podľa Andersona Darlinga (AD) ako najúčinnejšieho, nasledoval modifikovaný Anderson Darling (MAD), podľa Cramera von Missesa (CVM) a Kolmogorova-Smirnova (KS).
We suggest a nonparametric version of the probability weighted empirical characteristic function (PWECF) introduced by Meintanis {et al.} \cite{meiswaall2014} and use this PWECF in order to estimate the parameters of arbitrary transformations to symmetry. The almost sure consistency of the resulting estimators is shown. Finite-sample results for i.i.d. data are presented and are subsequently extended to the regression setting. A real data illustration is also included.