Greblicki, Włodzimierz; Krzyżak, Adam; Pawlak, Mirosław Distribution-free pointwise consistency of kernel regression estimate. (English) Zbl 0551.62025 Ann. Stat. 12, 1570-1575 (1984). An estimate \(\sum^{n}_{i=1}Y_ iK((x-X_ i)/h)/\sum^{n}_{j=1}K((x-X_ j)/h)\), calculated from a sequence \((X_ 1,Y_ 1),...,(X_ n,Y_ n)\) of independent pairs of random variables distributed as a pair (X,Y) converges to the regression \(E\{Y| X=x\}\) as n tends to infinity in probability for almost all (\(\mu)\) \(x\in R^ d\), provided that \(E| Y| <\infty\), \(h\to 0\) and \(nh^ d\to \infty \quad as\quad n\to \infty.\) The result is true for all distributions \(\mu\) of X. If, moreover, \(| Y| \leq \gamma <\infty\) and \(nh^ d/\log n\to \infty\) as \(n\to \infty\), a complete convergence holds. The class of applicable kernels includes those having unbounded support. Cited in 38 Documents MSC: 62G05 Nonparametric estimation 62J02 General nonlinear regression Keywords:distribution-free pointwise consistency; kernel estimate; universal consistency PDFBibTeX XMLCite \textit{W. Greblicki} et al., Ann. Stat. 12, 1570--1575 (1984; Zbl 0551.62025) Full Text: DOI