Bi-additive models for extremes

To widen the application field of mixed models we introduce bi-additive models. These models are given by the sum of a fixed term X0β and independent random effects terms XiZi, i=1,…,w. Vectors Z1,…,Zw, will have c1,…,cw,i.i.d. components with r−th order cumulants χr,1,…,χr,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 X0β of the models. Moreover matrices X0,X1,…,Xw, 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.