Nadaraya-Watson estimation of a nonparametric autoregressive model

Ben Célestin KOUASSI; Ouagnina Hili; Edoh KATCHEKPELE

doi:10.26637/mjm904/009

Abstract

We investigate the asymptotic behavior of the Nadaraya-Watson (NW) estimator of the regression function of a $\tau$−mixing process. We prove the strong consistency and the asymptotic normality of this estimator and we illustrate these two properties using simulated data.

Keywords:

Nonparametric autoregression, Nonparametric estimation, Asymptotic normality, Nadaraya-Watson estimator, $\tau$−mixing

Mathematics Subject Classification:

62E20 , 62G05 , 62G08 , 62G20

Authors Imprint References Funding Related Articles Downloads

Ben Célestin KOUASSI UMRI Mathematiques et Nouvelles Technologies de l'Information , Institut National Polytechnique Felix Houphouet-Boigny, BP 1093 Yamoussoukro, Cote d'Ivoire.
Ouagnina Hili UMRI Mathematiques et Nouvelles Technologies de l'Information , Institut National Polytechnique Felix Houphouet-Boigny, BP 1093 Yamoussoukro, Cote d'Ivoire.
Edoh KATCHEKPELE Département de Mathematiques, Faculte des Sciences et Techniques, Universite de Kara, Kara, TOGO. https://orcid.org/0000-0001-9029-6420

Pages: 251-258
Date Published: 01-10-2021
Vol. 9 No. 04 (2021): Malaya Journal of Matematik (MJM)

D.W.K. ANDREws, Nonstrong mixing autoregressive processes, Journal of Applied Probability, 21(4)(1984), 930—934.

B. Bercu, T. M. N. Nguyen and J. Saracco, On the asymptotic behaviour of the Nadaraya-Watson estimator associated with the recursive sliced inverse regression method, A journal of Theoretical and Applied Statistics, 49(3)(2015), 660-679.

P. Billingsley, Probability and Measure : Third Edition, WILEY SERIES IN PROBABILITY AND MATHEMATICAL STATISTICS , 1995.

Z. CAI, Weighted Nadaraya-Watson regression estimation. Statistics and Probability Letters, 51(3)(2001), $307-318$.

G. Collomb And W. Hardle, Strong uniform convergence rates in robust nonparametric time series analysis and prediction: kernel regression estimation from dependent observations. Stochastic Processes and their Applications, 23(1)(1986), 77-89.

J. Dedecker And C. Prieur , Coupling for $tau$-Dependent Sequences and Applications. Journal of Theoretical Probability, 17(4)(2004), 861-885.

H. Dette, J. C. Pardo-Fernández and I. Van Keilegom, Goodness-of-Fit Tests for Multiplicative Models with Dependent Data. Scandinavian Journal of Statistics, 36(4) (2009), 782—799.

L. P. Devroye, The uniform convergence of the Nadaraya-Watson regression function estimate. The Canadian Journal of Statistics. 6(2) (1978), 179-191.

P. Doukhan And O. Wintenberger, Weakly dependent chains with infinite memory. Stochastic Processes and their Applications, 118(11)(2008), 1997-2013.

J. FAN AND Q. YAO, Nonlinear Time Series : Nonparametric and Parametric Methods, Springer-Verlag New York Inc. 2003.

F. Ferraty and P. Vieu, Nonparametric Functional Data Analysis: Theory and Practice, Springer, 2006.

L. GyÖRFI, W. HÄRdLE, P. SARDA AND VIEU, Nonparametric Curve Estimation From Time Series,SpringerVerlag New York, 60(1) 1989.

S. Y. Hong AND O. Linton, Nonparametric estimation of infinite order regression and its application to the risk-return tradeoff. Journal of Econometrics, 219(2) (2020), 389-424.

U. Krengel , Ergodic Theorems, De Gruyter, Berlin, 1985.

P. Li, X. Li AND L. ChEN, The asymptotic normality of internal estimator for nonparametric regression, Journal of Inequalities and Applications 2018, 231.

Y. P. Mack and B. W. Silverman, Weak and strong uniform consistency of kernel regression estimates, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 61(3) (1982), 405-415.

E. A. NAdaraya, On estimating Regression. Theory of Probability and its Applications, 9(1) (1964), 141142 .

K. NODA , Estimation of a regression function by the Parzen kernel-type density estimators. Annals of the Institute of Statistical Mathematics 28(1) (1976), 221-234.

E. PARZEN, On estimation of a probability density function and mode. Annals of Mathematical Statistics 33(3) (1962), 1065-1076.

M. Rosenblatt , Remarks on some non-parametric estimates of a density function. Annals of Mathematical Statistics, 27(3) (1956), 832-837.

E. F. SChUSTER, Joint asymptotic distribution of the estimated regression function at a finite number of distinct points, Annals of Mathematical Statistics 43(1) (1972), 84-88.

G. S. WAtson, Smooth Regression Analysis. Sankhya Series A 26 (1964) 359-372.

T. ZHU AND D. N. Politis, Kernel estimates of nonparametric functional autoregression models and their bootstrap approximation, Electronic Journal of Statistics 11 (2) (2017), 2876-2906.