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 \(\{Y_n\},\) \(n\ge 1,\) that can be associated to automatic systems which have a periodic control, in the sense that at instants multiple of T, \(T\ge 2,\) 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 \(D^{(s)}_T(u_n),\) \(s\ge 1,\) 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.

极端事件是统计建模中的主要问题。对此类罕见事件进行建模时，随机丢失的数据可能会构成问题。插补在这些情况下至关重要，因此，描述不同插补功能的模型可以增强可能的应用，并扩大涵盖这些情况的少数已知模型系列。在本文中，我们考虑一类模型\（\ {Y_n \}，\） \（n \ ge 1，\），该模型可以与具有周期控制的自动系统相关联，即在T的瞬间倍数下，\（T \ ge 2，\），不会丢失任何值。在这里，随机遗漏值将替换为以前最大的观察值，直至已确定的记录为止。我们证明当基础序列是固定的时，\（\ {Y_n \} \）是T周期的，并且，如果它还验证了某些局部依赖条件，则\（\ {Y_n \} \）会验证一个众所周知的\（D ^ {（s）} _ T T周期序列的（u_n），\）\（s \ ge 1，\）依赖条件。我们还获得\（\ {Y_n \} \）的极值索引，并将其与基础序列的极值索引相关联。本文提出了一个用于“控制”缺失值的参数的一致估计量，并分析了其有限样本属性。所获得的结果通过在应用程序中被认可的马尔可夫序列进行说明。