A strong law of large numbers for nonparametric regression
Suppose \((X_ 1,Y_ 1),...,(X_ n,Y_ n)\) are i.i.d. \(R^ d\times R\)-valued random variables with finite conditional means \(m(x)=E[Y_ 1| X_ 1=x]\). Let \(\| \cdot \|\) denote the norm in \(R^ d\). For fixed \(x_ 0\in R^ d\) and for each \(n\geq 1\) let \(\{X_{1n},...,X_{nn}\}\) denote the order statistics of \(\{X_ 1,...,X_ n\}\), ordered by the distances \(\{\| x_ 0-X_ i\| \}\). Let \(\{Y_{1n},...,Y_{nn}\}\) denote the corresponding induced order statistics of \(\{Y_ 1,...,Y_ n\}\) and let \(S_{jn}=\sum^{j}_{i=1}Y_{in}\), \(1\leq j\leq n.\) Under the finite first moment condition for \(Y_ 1-m(X_ 1)\) given \(X_ 1=x\) for all x in some neighborhood of \(x_ 0\) and the smoothness condition of m(x), the author proves the strong law of large numbers for the nonparametric regression estimators of \(m(x_ 0)\) of the form \(m_ n(x_ 0)=S_{k_ nn}/k_ n\) where \(\{k_ n\}\) is a sequence of positive integers with \(k_ n\to \infty\) and \(k_ n/n\to 0.\) He applies these results to the k-nearest neighbor procedure and the moving window or uniform kernel procedure. Reviewer's comments: \(U_{n^ 2}=\sum^{n}_{i=n^ 2-n+1}Z_{in}/n\) [line 5 from bottom on p. 18] should be read as \(U_{n^ 2}=\sum^{n^ 2}_{i=n^ 2-n+1}Z_{in^ 2}/n\), \(m_ n(x_ 0)\to E m_ n(x_ 0)\quad a.s.\) Theorems 3.1, 3.3, and 4.3 should be read as \(m_ n(x_ 0)-E m_ n(x_ 0)\to 0\) a.s.
scientific article; zbMATH DE number 4106051
scientific article