Extremal behaviour of a periodically controlled sequence with imputed values

Extreme events are a major concern in statistical modeling. Random missing data can constitute a problem when modeling such 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 that cover these situations. In this paper we consider a family of models $$\{Y_n\},$$ { Y n } , $$n\ge 1,$$ n ≥ 1 , that can be associated to automatic systems which have a periodic control, in the sense that at instants multiple of T, $$T\ge 2,$$ T ≥ 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 prove that when the underlying sequence is stationary, $$\{Y_n\}$$ { Y n } is T-periodic and, if it also verifies some local dependence conditions, then $$\{Y_n\}$$ { Y n } verifies one of the well known $$D^{(s)}_T(u_n),$$ D T ( s ) ( u n ) , $$s\ge 1,$$ s ≥ 1 , dependence conditions for T-periodic sequences. We also obtain the extremal index of $$\{Y_n\}$$ { Y n } and relate it to the extremal index of the underlying sequence. A consistent estimator for the parameter that “controls” the missing values is here proposed and its finite sample properties are analysed. The obtained results are illustrated with Markovian sequences of recognized interest in applications.