Statistica Sinica 11(2001), 705-722

ROBUST LOCAL POLYNOMIAL REGRESSION

FOR DEPENDENT DATA

Jiancheng Jiang and Y. P. Mack

Peking University and University of California

Abstract: Let $(X_j,Y_j)_{j=1}^{n}$ be a realization of a bivariate jointly strictly stationary process. We consider a robust estimator of the regression function $m(x)=E(Y\vert X=x)$ by using local polynomial regression techniques. The estimator is a local M-estimator weighted by a kernel function. Under mixing conditions satisfied by many time series models, together with other appropriate conditions, consistency and asymptotic normality results are established. One-step local M-estimators are introduced to reduce computational burden. In addition, we give a data-driven choice for minimizing the scale factor involving the $\psi$-function in the asymptotic covariance expression, by drawing a parallel with the class of Huber's $\psi$-functions. The method is illustrated via two examples.

Key words and phrases: Data-driven, local M-estimator, local polynomial regression, mixing condition, one-step, robustness.