Statistical Inference for Stochastic Processes

, Volume 8, Issue 2, pp 151–184 | Cite as

On the Non-parametric Prediction of Conditionally Stationary Sequences

  • S. CairesEmail author
  • J. A. Ferreira


We prove the strong consistency of estimators of the conditional distribution function and conditional expectation of a future observation of a discrete time stochastic process given a fixed number of past observations. The results apply to conditionally stationary processes (a class of processes including Markov and stationary processes) satisfying a strong mixing condition, and they extend and bring together the work of several authors in the area of non-parametric estimation. One of our goals is to provide further justification for the growing practical application of non-parametric estimators in non-stationary time series and in other `non-i.i.d.' settings. Some arguments as to why such estimators should work very generally in practice, often in a nearly `optimal' way, are given. Two numerical illustrations are included, one with simulated data and the other with oceanographic data.


non-parametric prediction conditional distribution function conditional expectation time series data analysis 


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  1. Ango Nze, P., Doukhan, P 2002Weak dependence: models and applicationsDehling, H.Mikösh, T.Sorensen, M. eds. Empirical Process Techniques for Dependent Data.BirkhäuserBasel117136Google Scholar
  2. Ash, R., Dol´eans-Dade, C 2000Probability and Measure Theory, 2nd editionAcademic PressNew YorkGoogle Scholar
  3. Bosq, D 1997Nonparametric Statistics for Stochastic Processes, 2nd edition, Lecture Notes in Statistics 110Springer-VerlagNew YorkGoogle Scholar
  4. Brockwell, P., Davis, R 1987Time Series: Theory and MethodsSpringer-VerlagNew YorkGoogle Scholar
  5. Caires, S. and Sterl, A.: Validation of ocean wind and wave data using triple collocation, J. Geophys. Res. 108(C3) (2003), 43.1-43.16, doi:10.1029/2002JC001491.Google Scholar
  6. Carbon, M 1983Inégalité de Bernstein pour les processus fortement m´elangeants, non n´ecessairement stationairesApplications, C. R. Acad. Sci. Paris, I, t.297303306zbMATHMathSciNetGoogle Scholar
  7. Collomb, G 1984de convergence presque compl‘ete du pr´edicteur ‘anoyau, ZWahrscheinlichkeitstheorie verw. Gebiete66441460CrossRefzbMATHMathSciNetGoogle Scholar
  8. Collomb, G 1985Nonparametric regression: an up-to-date bibliographyStatistics16309324zbMATHMathSciNetGoogle Scholar
  9. Devroye, L 1981On the almost everywhere convergence of nonparametric regression function estimatesAnn. Stat.913101319zbMATHMathSciNetGoogle Scholar
  10. Grenander, U, Szegö, G 1958Toeplitz Forms and Their ApplicationsUniversity of California PressCaliforniaGoogle Scholar
  11. Gy örfi, L, Härdle, W, Sarda, P, Vieu, P 1989Nonparametric Curve Estimation from Time Series, Lecture Notes in Statistics 60Springer-VerlagNew YorkGoogle Scholar
  12. Härdle, W 1989Applied Nonparametric RegressionCambridge University PressCambridgeGoogle Scholar
  13. Ibragimov, I., Rozanov, Y 1978Gaussian Random ProcessesSpringer-VerlagNew YorkGoogle Scholar
  14. Knopp, K 1928Theory and Application of Infinite SeriesBlackie and SonLondon GlasgowGoogle Scholar
  15. Nadaraya, E 1964On estimating regressionTheor. Probab. Appl.9141142CrossRefGoogle Scholar
  16. Roussas, G 1969Nonparametric estimation of the transition distribution function of a Markov process, AnnMath. Stat.4013861400zbMATHMathSciNetGoogle Scholar
  17. Roussas, G 1990Nonparametric regression estimation under mixing conditions StochastProcess. Appl.36107116CrossRefzbMATHMathSciNetGoogle Scholar
  18. Roussas, G. 1991Estimation of transition distribution function and its quantiles in Markov processes: strong consistency and asymptotic normalityRoussas, G. eds. Nonparametric Funcional Estimation and Related Topics Vol 335.KluwerDordrecht443462Google Scholar
  19. Roussas, G. (ed): Nonparametric Funcional Estimation and Related Topics, Vol. 335, Kluwer, Dordrecht, 1991b.Google Scholar
  20. Stute, W 1986On almost sure convergence of conditional empirical distribution functionsAnn. Probab.14891901zbMATHMathSciNetGoogle Scholar
  21. Tucker, H, Graduate, A 1967Course in ProbabilityAcademic PressNew York LondonGoogle Scholar
  22. Watson, G. 1964Smooth regression analysisSankhya26359372zbMATHGoogle Scholar
  23. Yakowitz, S 1979Nonparametric estimation of Markov transition functions AnnStat.7671679zbMATHMathSciNetGoogle Scholar
  24. Yakowitz, S 1985Nonparametric density estimation, prediction and regression for Markov sequencesAm. Statist. Soc.80215221zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.KNMIRoyal Netherlands Meteorological InstituteDe BiltThe Netherlands
  2. 2.CWICentrum voor Wiskunde en InformaticaAmsterdamThe Netherlands

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