The pointwise rate of convergence of the kernel regression estimate (Q1821449)

Let \((X,Y),(X_ 1,Y_ 1),...,(X_ n,Y_ n)\) be independent, identically distributed random vectors from \({\mathbb{R}}^ d\times {\mathbb{R}}\), and \(E| Y|^ s<\infty\), \(s>1\). We study the rate of weak and strong convergence of Parzen kernel estimate of a regression function \(m(x)=E\{Y| X=x\}\). For Lipschitz(\(\alpha)\) m(x) and all distributions of X the weak rate is \(O(n^{-\alpha /(2\alpha +d)})\) provided that \(E| Y|^ s<\infty\), \(s\geq 2\), and the kernel has compact support. The rate is optimal in Stone's sense. Similarly for \(E| Y|^ s<\infty\), \(s>1\), the strong rate of convergence is \(O(n^{-\alpha (s-1)/s(2\alpha +d)}(\log n)^{1/2})\).