On the L 1 convergence of kernel estimators of regression functions with applications in discrimination |
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Authors: | Luc P. Devroye T. J. Wagner |
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Affiliation: | (1) School of Computer Science, McGill University, 805 Sherbrooke W., H3A 2K6 Montreal, Canada;(2) University of Texas, 78712 Austin, Texas, USA |
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Abstract: | Summary An estimate mnof a regression function m(x)=E{Y|X=x} is weakly (strongly) consistent in L1 if ¦mn(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 |
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