Asymptotic properties of the multivariate Nadaraya-Watson regression function estimate: The fixed design case (Q1113582)

Let \(\{x_ i^{(n)},Y_ i^{(n)}\}^ n_{i=1}\) be a sequence of independent observations from the model \(Y_ i^{(n)}=g(x_ i^{(n)}) + error\), where the regression function g is unknown and defined on a compact set in \(R^ d\). We show that a smoothed Nadaraya-Watson estimate of the function g(x) is asymptotically weak, mean square, strong, and completely consistent and asymptotically normal. The class of applicable kernels includes those with noncompact support for which \(\int^{\infty}_{0}z^{d-1}\sup_{\| x\| \geq z}K(x)dz<\infty\) and \(K(x)\geq cI_{(\| x\| \leq r)}(x)\), \(c,r>0\).