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 \(\{Y_n\},\) \(n\ge 1,\) 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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2


  1. Alpuim MT (1988) Contribuições à teoria de valores extremos em sucessões dependentes. Ph.D. Thesis. DEIO. University of Lisbon

  2. Alpuim MT (1989) An extremal Markovian sequence. J Appl Probab 26:219–232

    MathSciNet  Article  Google Scholar 

  3. Crambes C, Henchiri Y (2019) Regression imputation in the functional linear model with missing values in the response. J Stat Plann Inf 201:103–119

    MathSciNet  Article  Google Scholar 

  4. Chernick MR, Hsing T, McCormick WP (1991) Calculating the extremal index for a class of stationary sequences. Adv Appl Probab 23:835–850

    MathSciNet  Article  Google Scholar 

  5. Falk M, Hüsler J, Reiss R-D (2011) Laws of small numbers: extremes and rare events, 3rd edn. Birkhäuser, Basel

    Google Scholar 

  6. Ferreira H (1994) Multivariate extreme values in T-periodic random sequences under mild oscillation restrictions. Stoch Process Appl 49:111–125

    MathSciNet  Article  Google Scholar 

  7. Ferreira M, Ferreira H (2013) Extremes of multivariate ARMAX processes. Test 22(4):606–627

    MathSciNet  Article  Google Scholar 

  8. Ferreira H, Martins AP (2003) The extremal index of sub-sampled periodic sequences with strong local dependence. REVSTAT Stat J 1:15–24

    MathSciNet  MATH  Google Scholar 

  9. Hall A, Hüsler J (2006) Extremes of stationary sequences with failures. Stoch Models 22:537–557

    MathSciNet  Article  Google Scholar 

  10. Hall A, Scotto M (2008) On the extremes of randomly sub-sampled time series. REVSTAT Stat J 6(2):151–164

    MathSciNet  MATH  Google Scholar 

  11. Hall A, Temido MG (2009) On the max-semistable limit of maxima of stationary sequences with missing values. J Stat Plan Inference 139:875–890

    MathSciNet  Article  Google Scholar 

  12. Hasllett J (1979) Problems in the stochastic storage of a solar thermal energy. In: Jacobs O (ed) Analysis and optimization of stochastic systems. Academic Press, London

    Google Scholar 

  13. Leadbetter MR (1974) On extreme values in stationary sequences. Z Wahrscheinlichkeitstheor Verw Geb 28(4):289–303

    MathSciNet  Article  Google Scholar 

  14. Martins AP, Ferreira H (2004) The extremal index of sub-sampled processes. J Stat Plan Inference 1:145–152

    MathSciNet  Article  Google Scholar 

  15. Moritz S, Sardá A, Bartz-Beielstein T, Zaefferer M, Stork J (2015) Comparison of different methods for univariate time series imputation in R. ArXiv e-prints

  16. Moritz S, Bartz-Beielstein T (2017) imputeTS: time series missing value imputation in R. R J 9(1):207–218

    Article  Google Scholar 

  17. Quintela-del-Río A, Estévez-Pérez R (2012) Nonparametric kernel distribution function estimation with kerdiest: an R package for bandwidth choice and applications. J Stat Softw 50:8

    Article  Google Scholar 

  18. Scotto M, Turkman K, Anderson C (2003) Extremes OS some sub-sampled time series. J Time Ser Anal 24:579–590

    MathSciNet  MATH  Google Scholar 

  19. Weissman I, Cohen U (1995) The extremal index and clustering of high values for derived stationary sequences. J Appl Probab 32:972–981

    MathSciNet  Article  Google Scholar 

  20. Zha R, Harel O (2019) Power calculation in multiply imputed data. Stat Pap. https://doi.org/10.1007/s00362-019-01098-8

    Article  Google Scholar 

Download references


The first two authors were partially supported by the research unit Centre of Mathematics and Applications of University of Beira Interior UIDB/00212/2020 - FCT (Fundação para a Ciência e a Tecnologia). The third author was partially supported by the Centre for Mathematics of the University of Coimbra – UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES.

Author information



Corresponding author

Correspondence to Ana Paula Martins.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ferreira, H., Martins, A.P. & da Graça Temido, M. Extremal behaviour of a periodically controlled sequence with imputed values. Stat Papers (2021). https://doi.org/10.1007/s00362-020-01217-w

Download citation


  • Missing values
  • Periodic sequence
  • Local dependence conditions
  • Extremal index