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 (Formula presented.) and independent random effects terms (Formula presented.) Vectors (Formula presented.) will have (Formula presented.) components with (Formula presented.) th order cumulants (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) of the models. Moreover matrices (Formula presented.) contain in their columns values of controlled variables and, for given values of the controlled variables, prediction intervals are obtained, containing future observations, with (Formula presented.) [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.