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Distribution-free pointwise consistency of kernel regression estimate. (English) Zbl 0551.62025

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.

MSC:

62G05 Nonparametric estimation
62J02 General nonlinear regression
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