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NONPARAMETRIC TIME SERIES REGRESSION

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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 *theta*(·) of a *d*-dimensional measurement
variable. Let *T*(*theta*) denote a derivative of *theta*(·) of
order *m* < *k* and set *r* = (*k*-*m*)/(2*k*+*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
*L*_{2} 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.

**Source**
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