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Zbl 0596.62041
Georgiev, Alexander A.; Greblicki, Włodzimierz
Nonparametric function recovering from noisy observations.
(English)
[J] J. Stat. Plann. Inference 13, 1-14 (1986). ISSN 0378-3758

The authors consider the nonparametric regression model $Y\sb i=g(x\sb i)+\zeta\sb i$, where g is a bounded function over the interval [0,1] which is to be estimated, $x\sb i's$ are nonrandom and $\zeta\sb i's$ are independent identically distributed random variables with $E(\zeta\sb i)=0$. They study the behavior of the general family of nonparametric estimates $g\sb n(x)=\sum\sp{n}\sb{i=1}Y\sb iw\sb{ni}(x)$, where the weight functions $\{w\sb{ni}\}$ are of the form $w\sb{ni}(x)=w\sb{ni}(x;x\sb 1,...,x\sb n)$, $i=1,...,n$. Sufficient conditions for mean square and complete convergence are derived. Also proposed is a class of new nearest neighbor estimates of g. A simulation experiment demonstrates the success of the nearest neighbor technique with bandwidth depending on the local density of the design points.
[V.P.Gupta]
MSC 2000:
*62G05 Nonparametric estimation
62J02 General nonlinear regression
60F15 Strong limit theorems

Keywords: kernel estimate; consistency; curve fitting; regression function; strong pointwise convergence; weight functions; complete convergence; nearest neighbor estimates

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