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

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 X = x\ that is defined by mₙ(x) = Σⁿᵢ₌₁ Yᵢk((Xᵢ - x)/hₙ)/Σⁿᵢ₌₁k((Xᵢ - x)/hₙ) where k is a bounded nonnegative function on Rᵈ with compact support and ₙ\ is a sequence of positive numbers satisfying hₙ →ₙ0, nhᵈₙ →ₙ∞. It is shown that E\∫|mₙ(x) - m(x)|ᵖμ(dx)\ →ₙ 0 whenever E\|Y|ᵖ\ < ∞(p 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.