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Zbl 0698.62040
Fan, Y.
Consistent nonparametric multiple regression for dependent heterogeneous processes: the fixed design case.
(English)
[J] J. Multivariate Anal. 33, No.1, 72-88 (1990). ISSN 0047-259X

Summary: Consider the nonparametric regression model $$Y\sb i\sp{(n)}=g(x\sb i\sp{(n)})+\epsilon\sb i\sp{(n)},\quad i=1,...,n,$$ where $g$ is an unknown regression function and assumed to be bounded and real valued on $A\subset \Bbb R\sp p$, $x\sb i\sp{(n)}$'s are known and fixed design points and $\epsilon\sb i\sp{(n)}$'s are assumed to be both dependent and non-identically distributed random variables. \par This paper investigates the asymptotic properties of the general nonparametric regression estimator $$g\sb n(x)=\sum\sp{n}\sb{i=1}W\sb{ni}(x)Y\sb i\sp{(n)},$$ where the weight function $W\sb{ni}(x)$ is of the form $W\sb{ni}(x)=W\sb{ni}(x;x\sb 1\sp{(n)},x\sb 2\sp{(n)},\dots,x\sb n\sp{(n)})$. The estimator $g\sb n(x)$ is shown to be weak, mean square error, and universal consistent under very general conditions on the temporal dependence and heterogeneity of $\epsilon\sb i\sp{(n)}$'s. Asymptotic distribution of the estimator is also considered.
MSC 2000:
*62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics
60G44 Martingales with continuous parameter
60F05 Weak limit theorems

Keywords: weak consistency; universal consistency; mixingale; near epoch dependent; mixing sequence; martingale difference sequence; multivariate central limit theorem; non-identically distributed random variables; asymptotic properties; nonparametric regression estimator; weight function; mean square error; dependence; heterogeneity

Cited in: Zbl 1109.62315

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