Regression models for exceedance data via the full likelihood

Abstract

Many situations in practice require appropriate specification of operating characteristics under extreme conditions. Typical examples include environmental sciences where studies include extreme temperature, rainfall and river flow to name a few. In these cases, the effect of geographic and climatological inputs are likely to play a relevant role. This paper is concerned with the study of extreme data in the presence of relevant auxiliary information. The underlying model involves a mixture distribution: a generalized Pareto distribution is assumed for the exceedances beyond a high threshold and a non-parametric approach is assumed for the data below the threshold. Thus, the full likelihood including data below and above the threshold is considered in the estimation. The main novelty is the introduction of a regression structure to explain the variation of the exceedances through all tail parameters. Estimation is performed under the Bayesian paradigm and includes model choice. This allows for determination of higher quantiles under each covariate configuration and upper bounds for the data, where appropriate. Simulation results show that the models are appropriate and identifiable. The models are applied to the study of two temperature F. F. do Nascimento (B) 123 496 Environ Ecol Stat (2011) 18:495-512 datasets: maxima in the U.S.A. and minima in Brazil, and compared to other related models.

Bayesian time‐varying quantile regression to extremes

“…The main motivation of using the quantile regression model is to write the quantile as a function of the covariates to allow us to interpret which factors influence the high quantiles. In this subsection, the proposed model is composed by adding the regression models into the three parameters ( q ( p ), σ , ξ ) of the distribution, using a similar approach to Nascimento et al () for the GPD.…”

“…They use asymptotic properties to find the distribution of the parameters, whereas the posterior distribution is precise. The priors of the regression coefficients are chosen to be normal with a large variance for β p , ν , and α , and are given by Nascimento et al () show that, for the GPD, these prior densities work well in both simulations and environmental data applications. By joining the likelihood with the prior, we obtain the posterior distribution.…”

“…The main motivation of using the quantile regression model is to write the quantile as a function of the covariates to allow us to interpret which factors influence the high quantiles. In this subsection, the proposed model is composed by adding the regression models into the three parameters (q(p), , ) of the distribution, using a similar approach to Nascimento et al (2011) for the GPD. Let Y 1 , … , Y n be the n independent random variables, where each Y i , i = 1, … , n, follows the GEV distribution with quantile parameter q(p), scale parameter , and shape parameter .…”

“…Cabras, Castellanos, and Gamerman (2011) consider adding seasonal covariates to the GPD parameter, applied to rain data. Nascimento, Gamerman, and Lopes (2011) model the exceedance of temperature considering altitude and latitude as covariates, describing a regression to its parameters.…”

Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach

“…In the context of extreme values, we can obtain any quantile p not requiring modeling for each p by using equation (2). Nascimento, Gamerman & Lopes (2011) considered regression structures in relation to generalized Pareto distribution (GPD) parameters. Yet another approach is to propose the variances of high quantiles over time through a structure of dynamic models.…”