Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-25T09:05:49.040Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform consistency of the partitioning estimate under ergodic conditions

Part of: Limit theorems Inference from stochastic processes Nonparametric inference

Published online by Cambridge University Press:  09 April 2009

Naâmane Laib
Naâmane Laib
Affiliation:
L.S.T.A. Université Paris 6 Aile 45-55, 3ème étage 4, Place Jussieu 75252 Paris Cedex 05 France e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish the uniform almost sure convergence of the partitioning estimate, which is a histogram-like mean regression function estimate, under ergodic conditions for a stationary and unbounded process. The main application of our results concerns time series analysis and prediction in the Markov processes case.

Keywords

Autoregressionergodic processesmartingale differencepartitioning estimatepredictionregressionstationarity.

MSC classification

Secondary: 62G05: Estimation 60F15: Strong theorems 62M10: Time series, auto-correlation, regression, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 67 , Issue 1 , August 1999 , pp. 1 - 14
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Andrews, D. W. K., ‘Non-strong mixing autoregressive processes’, J. Appl. Probab. 21 (1984), 930–934.CrossRefGoogle Scholar
[2]Ash, R. and Gardner, M., Topics in stochastic processes (Academic Press, New York, 1975).Google Scholar
[3]Bosq, D., ‘Nonparametric prediction for unbounded almost stationary processes’, in: Nonparametric functional estimation and related topics (Kluwer Acad. Publ., Amsterdam, 1991) pp. 389–403.CrossRefGoogle Scholar
[4]Breiman, L., Probability (Addison-Wesley, Reading, MA, 1968).Google Scholar
[5]Carbonez, A., Györfi, L. and Van der Meulen, E. C., ‘Partitioning estimates of regression function under random censoring’, Statist. Decisions. 13 (1995), 21–37.Google Scholar
[6]Collomb, G. and Härdle, W., ‘Strong uniform convergence rates in robust nonparametric time series analysis and prediction: Kernel regression estimation from dependent observations’, Stochastic Process. Appl. 23 (1986), 77–89.CrossRefGoogle Scholar
[7]Delecroix, M., Nogueira, M. N. and Rosa, A. C. Martins, ‘Sur l'estimation de la densité pour des observations ergodiques’, Stat. et Anal des données 16 (1991), 25–38.Google Scholar
[8]Delecroix, M. and Rosa, A. C., ‘Nonparametric estimation of a regression function and its derivatives under an ergodic hypothesis’, J. Nonparametric Statistics 6 (1996), 367–382.CrossRefGoogle Scholar
[9]Devroye, L. and Györfi, L., ‘Distribution-free exponential bound for the L 1 error of partitioning estimates of a regression function’, in: Probability and statistical decision theory (Bad Tatzmannsdorf, 1983) (Reidel, Boston, 1985) pp. 67–76.Google Scholar
[10]Diebolt, J. and Laïb, N., ‘A weak invariance principle for cumulated functionals of the regressogram estimator with dependent data’, J. Nonparametr. Statist. 4 (1994), 149–163.CrossRefGoogle Scholar
[11]Gordon, L. and Olshen, R. A., ‘Consistent nonparametric regression from recursive partitioning schemes’, J. Multivariate Anal. 10 (1980), 611–627.CrossRefGoogle Scholar
[12]Gordon, L. and Olshen, R. A., ‘Almost sure consistent nonparametric regression from recursive partitioning schemes’, J. Multivariate Anal. 15 (1984), 147–163.CrossRefGoogle Scholar
[13]Györfi, L., ‘Strong consistent density estimate from ergodic sample’, J. Multivariate Anal. 11 (1981), 81–84.CrossRefGoogle Scholar
[14]Györfi, L., ‘Universal consistencies of a regression estimate for unbounded regression function’, in: Nonparametric functional estimation and related topics (ed. Roussas, ) (Kluwer Acad. Publ., Amsterdam, 1991) pp. 329–338.CrossRefGoogle Scholar
[15]Györfi, L., Härdle, W., Sarda, P. and Vieu, P., Nonparametric curve estimation from time series, Lecture Notes in Statistics 60 (Springer, New York, 1989).CrossRefGoogle Scholar
[16]Györfi, L. and Lugosi, G., ‘Kernel density estimation from ergodic sample is not universally consistent’, Comput. Statist. Data Anal. 14 (1992), 437–442.CrossRefGoogle Scholar
[17]Györfi, L. and Masry, E., ‘The L 1 and L 2 strong consistency of recursive kernel density estimation from dependent samples’, IEEE Trans. Inform. Theory 36 (1990), 531–539.CrossRefGoogle Scholar
[18]Györfi, L., Morvai, G. and Yakowitz, S., ‘Limits to consistent on line forecasting for ergodic time series’, IEEE Trans. Inform. Theory 44 (1998), 868–892.CrossRefGoogle Scholar
[19]Hart, J. and Vieu, P., ‘Data driven bandwidth choice for density estimation based on dependent data’, Ann. Statist. 18 (1990), 837–890.CrossRefGoogle Scholar
[20]Krengel, U., Ergodic theorems, De Gruyter Studies in Mathematics 6 (Walter De Gruyter, Berlin, 1985).CrossRefGoogle Scholar
[21]Laïb, N., ‘Exponential-type inequalities for martingale difference sequences. Application to nonparametric regression estimation’, Comm. Statist. Theory Methods, to appear.Google Scholar
[22]Laïb, N., ‘Estimation non-paramétrique de la régression pour des données dépendantes. Application à la prévision’, C. R. Acad. Sci. Paris 317 (1993), 1173–1177.Google Scholar
[23]Laïb, N. and Ould-Saïd, E., ‘Estimation non-paramétrique robuste de la fonction de régression pour des observations ergodiques’, C. R. Acad. Sci. Paris 320 (1996), 271–276.Google Scholar
[24]Mack, Y. P. and Silverman, B. W., ‘Weak and strong uniform consistency of kernel regression estimates’, Z. Wahrsch. Verw. Gebiete 61 (1982), 405–415.CrossRefGoogle Scholar
[25]Ould-Saïd, E., ‘A note on ergodic processes prediction via estimation of the conditional mode function’, Scand. J. Statist. 2 (1997), 231–240.CrossRefGoogle Scholar
[26]Pham, T. D. and Tran, T., ‘Some mixing properties of time sersies models’, Stochastic Process. Appl. 19 (1985), 279–303.CrossRefGoogle Scholar
[27]Rosa, A. C., ‘Estimation non-paramétrique de la médianne conditionnelle sous hypothèse ergodique. Application à la prévision des processus’, in: XXIVe Journées de statistiques, 'Bruxelles, (1992) pp. 325–328.Google Scholar
[28]Rosenblatt, M., ‘Dependence and asymptotic independence from random processes’, M.A.A. Stud. Math. 18 (1978), 24–45.Google Scholar
[29]Roussas, G. G., ‘Nonparametric regression estimation under mixing conditions’, Stochastic Process. Appl. 36 (1990), 107–116.CrossRefGoogle Scholar
[30]Tran, L. T., ‘Nonparametric function estimation for time series by local average estimators’, Ann. Statist. 21 (1993), 1040–1057.CrossRefGoogle Scholar
[31]Truong, Y. K., ‘Robust nonparametric regression in time series’, J. Multivariate Anal. 41 (1992), 163–177.CrossRefGoogle Scholar
[32]Truong, Y. K. and Stone, C. J., ‘Nonparametric function estimation involving time series’, Ann. Statist. 20 (1992), 77–97.CrossRefGoogle Scholar
[33]Withers, C. S., ‘Central limit theorems for dependent variables’, Z. Wahrsch. Verw. Gebiete 57 (1981b), 509–534.CrossRefGoogle Scholar
[34]Yakowitz, S., ‘Nonparametric density and regression estimateion for Markov sequences without mixing assumptions’, J. Multivariate Anal. 30 (1989), 124–136.CrossRefGoogle Scholar