Extremal behaviour of a periodically controlled sequence with imputed values

Helena Ferreira (), Ana Paula Martins () and Maria Graça Temido ()
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Helena Ferreira: Universidade da Beira Interior
Ana Paula Martins: Universidade da Beira Interior
Maria Graça Temido: Universidade de Coimbra, CMUC

Statistical Papers, 2021, vol. 62, issue 6, No 19, 3013 pages

Abstract: Abstract 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.

Keywords: Missing values; Periodic sequence; Local dependence conditions; Extremal index (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s00362-020-01217-w

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