Necessary and sufficient consistency conditions for a recursive kernel regression estimate
A recursive kernel estimate \(\sum^{n}_{i=1}Y_ iK((x-X_ i)/h_ i)/\sum^{n}_{j=1}K((x-X_ j)/h_ j)\) of a regression \(m(x)=E\{Y| X=x\}\) calculated from independent observations \((X_ 1,Y_ 1),...,(X_ n,Y_ n)\) of a pair (X,Y) of random variables is examined. For \(E| Y| <\infty\), the estimate is weakly pointwise consistent for almost all (\(\mu)\) \(x\in R^ d\), \(\mu\) is the probability measure of X, if and only if \[ \sum^{n}_{i=1}h^ d_ iI_{\{h_ i>\epsilon \}}/\sum^{n}_{j=1}h^ d_ j\to 0\quad as\quad n\to \infty, \] all \(\epsilon >0\), and \(\sum^{\infty}_{i=1}h^ d_ i=\infty\), d is the dimension of X. For \(E| Y|^{1+\delta}<\infty\), \(\delta >0\), the estimate is strongly pointwise consistent for almost all (\(\mu)\) \(x\in R^ d\), if and only if the same conditions hold. For \(E| Y|^{1+\delta}<\infty\), \(\delta >0\), weak and strong consistency are equivalent. Similar results are given for complete convergence.
scientific article; zbMATH DE number 4020218