On the Uniform Strong Consistency of Local Polynomial Regression Under Dependence Conditions

Abstract

In this article, nonparametric estimators of the regression function, and its derivatives, obtained by means of weighted local polynomial fitting are studied. Consider the fixed regression model where the error random variables are coming from a stationary stochastic process satisfying a mixing condition. Uniform strong consistency, along with rates, are established for these estimators. Furthermore, when the errors follow an AR(1) correlation structure, strong consistency properties are also derived for a modified version of the local polynomial estimators proposed by Vilar-Fernández and Francisco-Fernández (Vilar-Fernández, J. M., Francisco-Fernández, M. (2002). Local polynomial regression smoothers with AR-error structure. TEST 11(2):439–464).

Keywords:

• Nonparametric regression estimation
• Local polynomial fitting
• Dependent data
• Strong consistency

Acknowledgments

This work was partially supported by Grants PGIDT01PXI10505PR, PB98-0182-C02-01 and MCyT Grant BFM2002-00265 (European FEDER support included and PGIDIT03PXIC10505PN). The authors are pleased to acknowledge the helpful comments of the referee.