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 theta(·) of a d-dimensional measurement variable. Let T(theta) denote a derivative of theta(·) 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(theta) can be chosen to achieve the optimal rate of convergence n-r in L2 norms restricted to compacts; and the optimal rate (n-1 log n)r in the L\infty 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.

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