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Large-sample inference for nonparametric regression with dependent errors. (English) Zbl 0882.62039
Summary: A central limit theorem is given for certain weighted partial sums of a covariance stationary process, assuming it is linear in martingale differences, but without any restriction on its spectrum. We apply the result to kernel nonparametric fixed-design regression, giving a single central limit theorem which indicates how error spectral behavior at only zero frequency influences the asymptotic distribution and covers long-range, short-range and negative dependence. We show how the regression estimates can be Studentized in the absence of previous knowledge of which form of dependence pertains, and show also that a simpler Studentization is possible when long-range dependence can be taken for granted.

MSC:
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
60G18 Self-similar stochastic processes
60F05 Central limit and other weak theorems
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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