On the L 1 convergence of kernel estimators of regression functions with applications in discrimination

Summary An estimate m_nof a regression function m(x)=E{Y|X=x} is weakly (strongly) consistent in L₁ if ¦m_n(x)-m(x)¦(dx) converges to 0 in probability (w.p. 1) as the sample size grows large ( is the probability measure of X).We show that the well-known kernel estimate (Nadaraya, Watson) and several recursive modifications of it are weakly (strongly) consistent in L₁ under no conditions on (X, Y) other than the boundedness of Y and the absolute continuity of . No continuity restrictions are put on the density corresponding to . We further notice that several kernel-type discrimination rules are weakly (strongly) Bayes risk consistent whenever X has a density.Research of both authors was sponsored by AFOSR Grant 77-3385