Regression models for exceedance data via the full likelihood

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 datasets: maxima in the U.S.A. and minima in Brazil, and compared to other related models.