Generalized maximum likelihood estimators for the nonstationary generalized extreme value model

Abstract

[1] The objective of the present study is to develop efficient estimation methods for the use of the GEV distribution for quantile estimation in the presence of nonstationarity. Parameter estimation in the nonstationary GEV model is generally done with the maximum likelihood estimation method (ML). In this work, we develop the generalized maximum likelihood estimation method (GML), in which covariates are incorporated into parameters. A simulation study is carried out to compare the performances of the GML and the ML methods in the case of the stationary GEV model (GEV0), the nonstationary case with a linear dependence of the location parameter on covariates (GEV1), the nonstationary case with a quadratic dependence on covariates (GEV2), and the nonstationary case with linear dependence in both location and scale parameters (GEV11). Simulation results show that the GLM method performs better than the ML method for all studied cases. The nonstationary GEV model is also applied to a case study to illustrate its potential. The case study deals with the annual maximum precipitation at the Randsburg station in California, and the covariate process is taken to be the Southern Index Oscillation.

Cento e vinte anos de totais extremos de precipitação pluvial máxima diária em Campinas, Estado de São Paulo: análises estatísticas

“…Essa indicação de Raynal (1997) é corroborada por Nadarajah e Choi, (2007) quando afirmam que a GEV possui toda a flexibilidade contida em seus casos particulares. Com base nessa última premissa, El Adlouni et al (2007), Pujol et al (2007), Méndez (2007), Furió e Meneu (2010) e Cannon (2010) utilizam esse modelo estatístico em estudos probabilísticos de dados meteorológicos extremos. Sob o aspecto matemático, é interessante ressaltar que os parâ-metros da referida função estatística são frequentemente estimados com base nos métodos da máxima verossimilhança (Nadarajah e Choi, 2007;Pujol et al, 2007;Méndez, 2007;Sansigolo, 2008) ou dos momentos-L (Hosking e Wallis, 1997;Queiroz e Chaudhry, 2006;Wilks, 2006).…”

Incorporating climate trends in the stochastic modeling of extreme minimum air temperature series of Campinas, state of São Paulo, Brazil

“…Although several authors such as Coles (2001), El Adlouni et al (2007, Méndez et al (2007), Felici et al (2007), Pujol et al (2007 and, Furió and Meneu (2011), also work under this assumption of linearity, the development of a more flexible framework that can be used to model non-linear relationships is of great interesting. In this view, the study carried out by Cannon (2010), that specifies the parameters of the GEV by using a probabilistic extension of the multilayer perceptron neural network, can be seen as an important future alternative to overcome the linear approach adopted in this study.…”

“…However, El Adlouni et al (2007) also indicate that according to the parsimony principle when the differences between two models Model i and Model j (in which; Model i ⊂ Model j if j>i) are not significant, it is better to use the simplest one (Model i ). The uncertainty in quantile estimation tends to increase as the number of parameters to be estimated increases.…”

“…As indicated by El Adlouni et al (2007), the most general model is frequently the best to represent the data sample under evaluation. However, El Adlouni et al (2007) also indicate that according to the parsimony principle when the differences between two models Model i and Model j (in which; Model i ⊂ Model j if j>i) are not significant, it is better to use the simplest one (Model i ).…”

“…It is worth emphasizing that the use of the GEV(x t ; μ, σ, ξ) model is frequently called "the stationary approach" (Coles, 2001;El Adlouni et al, 2007;Pujol et al, 2007;Méndez, 2007;Furió and Meneu, 2011) since the probability associated with any x t is independent of t. In other words, once the GEV(x t ; μ, σ, ξ) model is fitted from a (already) observed time series (for e.g. from the 1891-2010 time span) it is assumed that the values of μ, σ, ξ will remain the same during the next (t) years (for e.g.…”