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Nonparametric function recovering from noisy observations. (English) Zbl 0596.62041
The authors consider the nonparametric regression model Y i =g(x i )+ζ i , where g is a bounded function over the interval [0,1] which is to be estimated, x i ' s are nonrandom and ζ i ' s are independent identically distributed random variables with E(ζ i )=0. They study the behavior of the general family of nonparametric estimates g n (x)= i=1 n Y i w ni (x), where the weight functions {w ni } are of the form w ni (x)=w ni (x;x 1 ,···,x 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.
Reviewer: V.P.Gupta
62G05Nonparametric estimation
62J02General nonlinear regression
60F15Strong limit theorems