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Nonparametric regression under long-range dependent normal errors. (English) Zbl 0852.62035
Summary: We consider the fixed-design regression model with long-range dependent normal errors and show that the finite-dimensional distributions of the properly normalized Gasser-Müller [T. Gasser and H. G. Müller, Lect. Notes Math. 757, 23-68 (1979; Zbl 0418.62033)] and Priestley-Chao [M. B. Priestley and M. T. Chao, J. R. Stat. Soc., Ser. B 34, 385-392 (1972; Zbl 0263.62044)] estimators of the regression function converge to those of a white noise process. Furthermore, the distributions of the suitably renormalized maximal deviations over an increasingly finer grid converge to the Gumbel distribution.
These results contrast with our previous findings [see Ann. Stat. 23, No. 3, 990-999 (1995; Zbl 0843.62037)] for the corresponding problem of estimating the marginal density of long-range dependent stationary sequences.

MSC:
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62M99 Inference from stochastic processes
60F17 Functional limit theorems; invariance principles
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