Distribution-Free Consistency Results in Nonparametric Discrimination and Regression Function Estimation

Abstract

Let (X, Y) be an Rd × R-valued random vector and let (X1, Y1), ⋯, (Xn, Yn) be a random sample drawn from its distribution. We study the consistency properties of the kernel estimate mn(x) of the regression function m(x) = E{Y∣ X = x} that is defined by mn(x) = Σn i=1 Yik((Xi - x)/hn)/Σn i=1k((Xi - x)/hn) where k is a bounded nonnegative function on Rd with compact support and {hn} is a sequence of positive numbers satisfying hn →n0, nhd n →n∞. It is shown that E{∫|mn(x) - m(x)|pμ(dx)} →n 0 whenever $E\{|Y|^p\} < \infty(p \geqslant 1)$ . No other restrictions are placed on the distribution of (X, Y). The result is applied to verify the Bayes risk consistency of the corresponding discrimination rules.