# Extremal behaviour of a periodically controlled sequence with imputed values

## TL;DR

This paper studies the extremal behavior of periodically controlled sequences with imputed missing values, establishing dependence conditions, extremal index relations, and proposing an estimator, with applications to Markovian sequences.

## Contribution

It introduces a new model for sequences with periodic control and imputation, analyzes their extremal properties, and proposes a consistent estimator for the missing data parameter.

## Key findings

- Sequences are T-periodic under stationarity.
- Dependence conditions for extremal analysis are verified.
- An estimator for the missing value control parameter is proposed.

## Abstract

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.