Bi-additive models for extremes

Abstract

To widen the application field of mixed models we introduce bi-additive models. These models are given by the sum of a fixed term ${\mathit{X}}_0\mathit{\beta }$ and independent random effects terms ${\mathit{X}}_i{\mathit{Z}}_i,i=1,\dots ,w.$ Vectors ${\mathit{Z}}_1,\dots ,{\mathit{Z}}_w,$ will have $c_1,\dots ,c_w,i.i.d.$ components with $r-$th order cumulants ${\chi }_{r,1},\dots ,{\chi }_{r,w}.$ We now consider the case in which the distributions of these components are distributed as Gumbel, Fréchet and Weibull types, estimating their cumulants and parameters. We then obtain $1-p$ confidence ellipsoids with [approximate] probability of containing realizations of the model. These ellipsoids can be used to, trough duality, test hypothesis on the fixed effects part ${\mathit{X}}_0\mathit{\beta }$ of the models. Moreover matrices ${\mathit{X}}_0,{\mathit{X}}_1,\dots ,{\mathit{X}}_w,$ contain in their columns values of controlled variables and, for given values of the controlled variables, prediction intervals are obtained, containing future observations, with $1-p$ [approximate] probability. Three simulations studies, one for each distribution type, and an application to the Tagus river floods are included. We thus show how bi-additive models may be introduced in the important field of extreme value.

Keywords:

• Cumulants
• mixed models
• uniformly minimum variance unbiased estimator

Additional information

Funding

This work was partially supported by FCT – Fundação para a Ciência e a Tecnologia, under the scope of the projects UIDB/00212/2020 and UIDB/00297/2020.