  Extreme events are a major concern in statistical modeling. Ran\-dom 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 which cover these situations. In this paper we consider a
family of models $\{Y_n\},$ $n\geq 1,$ which 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 prove that when the underlying sequence is stationary,
$\{Y_n\}$ is $T-$periodic and if it also verifies some local dependence
conditions then $\{Y_n\}$ verifies one of the well known $D^{(s)}_T(u_n),$
$s\geq 1,$ dependence conditions for $T-$periodic sequences. We also obtain the
extremal index of $\{Y_n\}$ 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.
