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Zbl 0637.62044
Georgiev, Alexander A.
Consistent nonparametric multiple regression: the fixed design case.
(English)
[J] J. Multivariate Anal. 25, No. 1, 100-110 (1988). ISSN 0047-259X

Consider the nonparametric regression model $Y\sb i\sp{(n)}=g(x\sb i\sp{(n)})+\epsilon\sb i\sp{(n)}$, $i=1,...,n$, where $g$ is an unknown function, the design points $x\sb i\sp{(n)}$ are known and nonrandom, and $\epsilon\sb i\sp{(n)}$'s are independent random variables. The regressor is assumed to take values in $A\subset \Bbb R\sp p$, and the regressand to be real valued. This paper studies the behavior of the general nonparametric estimate $$g\sb n(x)=\sum\sp{n}\sb{i=1}w\sb{ni}(x)Y\sb i\sp{(n)},$$ where the weight function $w\sb{ni}$ is of the form $w\sb{ni}(x)=w\sb{ni}(x;x\sb i\sp{(n)},\dots,x\sb n\sp{(n)})$. Under suitable conditions, it is shown that the general linear smoother $g\sb n$ for the unknown rendent variable.
MSC 2000:
*62G05 Nonparametric estimation
62J02 General nonlinear regression
62F15 Bayesian inference

Keywords: kernel estimate; nearest neighbor estimate; orthogonal series estimate; splines; law of large numbers; central limit theorem; consistency; general linear smoother

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