NONPARAMETRIC TIME SERIES REGRESSION

YOUNG K. TRUONG

Department of Biostatistics, University of North Carolina, Chapel Hill, NC 27514, U.S.A.

(Received February 25, 1992; revised February 2, 1993)

Abstract. Consider a k-times differentiable unknown regression function θ(·) of a d-dimensional measurement variable. Let T (θ) denote a derivative of θ(·) of order m < k and set r = (k − m)/(2k + d). Given a bivariate stationary time series of length n, under some appropriate conditions, a sequence of local polynomial estimators of the function T (θ) can be chosen to achieve the optimal rate of convergence n^{−r} in L_{2} norms restricted to compacts; and the optimal rate (n^{−1} log n)^{r} in the L_{∞} norms on compacts. These results generalize those by Stone (1982, Ann. Statist., 10, 1040–1053) which deals with nonparametric regression estimation for random (i.i.d.) samples. Applications of these results to nonlinear time series problems will also be discussed.

Key words and phrases: Nonparametric regression, kernel estimator, local polynomials, optimal rates of convergence.