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Zbl 0674.62026
Roussas, George G.
Consistent regression estimation with fixed design points under dependence conditions.
(English)
[J] Stat. Probab. Lett. 8, No.1, 41-50 (1989). ISSN 0167-7152

Summary: For $n=1,2,..$. and i integer between 1 and n, let $x\sb{ni}$ be fixed design points in a compact subset S of ${\bbfR}\sp p$, $p\ge 1$, and let $Y\sb{ni}$ be observations taken at these points through g, an unknown continuous real-valued function defined on ${\bbfR}\sp p$, and subject to errors $\epsilon\sb{ni}$; that is, $Y\sb{ni}=g(x\sb{ni})+\epsilon\sb{ni}$. For any x in ${\bbfR}\sp p$, g(x) is estimated by $$g\sb n(x;x\sb n)=\sum\sp{n}\sb{i=1}w\sb{ni}(x;x\sb n)Y\sb{ni},$$ where $x\sb n=(x\sb{n1},...,x\sb{nn})$ and $w\sb{ni}(\cdot;\cdot)$ are suitable weights. If the errors $\epsilon\sb{ni}$ are centered at their expectations, the proposed estimate is asymptotically unbiased. It is also consistent in quadratic mean and strongly consistent, if, in addition and for each n, the random variables $\epsilon\sb{ni}$, $i\ge 1$, are coming from a strictly stationary sequence obeying any one of the four standard modes of mixing.
MSC 2000:
*62G05 Nonparametric estimation
62J02 General nonlinear regression
60F15 Strong limit theorems
60F25 Lp-limit theorems (probability)

Keywords: nonparametric estimation; regression estimation; consistency in quadratic mean; strong consistency; fixed design points; compact subset; asymptotically unbiased; strictly stationary sequence; mixing

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