Nonparametric function recovering from noisy observations (Q1078952)

The authors consider the nonparametric regression model \(Y_ i=g(x_ i)+\zeta_ i\), where g is a bounded function over the interval [0,1] which is to be estimated, \(x_ i's\) are nonrandom and \(\zeta_ i's\) are independent identically distributed random variables with \(E(\zeta_ i)=0\). They study the behavior of the general family of nonparametric estimates \(g_ n(x)=\sum^{n}_{i=1}Y_ iw_{ni}(x)\), 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.