Abstract
Consider ak-times differentiable unknown regression functionθ(·) of ad-dimensional measurement variable. LetT(θ) denote a derivative ofθ(·) of orderm<k and setr=(k−m)/(2k+d). Given a bivariate stationary time series of lengthn, under some appropriate conditions, a sequence of local polynomial estimators of the functionT(θ) can be chosen to achieve the optimal rate of convergencen −r inL 2 norms restricted to compacts; and the optimal rate (n −1 logn)r in theL ∞ 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.
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This work was completed while the author was visiting Mathematical Sciences Research Institute at Berkeley, California. Research was supported in part by NSF Grant DMS-8505550, NC Board of Science and Technology Development Award 90SE06 and UNC Research Council.
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Truong, Y.K. Nonparametric time series regression. Ann Inst Stat Math 46, 279–293 (1994). https://doi.org/10.1007/BF01720585
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DOI: https://doi.org/10.1007/BF01720585