Random missing data can constitute a problem when modelling rare events. Imputation is crucial in these situations and therefore models that describe different imputation functions enhance possible applications and enlarge the few known families of models which cover these situations. In this talk, we consider a family of models $\{Y_n\}, n\geq 1$, that can be associated to automatic systems which have a periodic control, in the sense that it is guaranteed that at instants multiple of $T$, $T\geq 2$, no value is lost. Random missing values are here replaced by the biggest of the previous observations up to the one surely registered.

We characterize the extremal behaviour of $\{Y_n\}, n\geq 1$, and obtain its extremal index expression. A consistent estimator for the model parameter is also proposed and its finite sample behaviour analysed.

Joint work with Helena Ferreira (UBI) and Maria da Graça Temido (UC).