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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) = ⁿ₈=₁ Yᵢk ( (Xᵢ - x) /hₙ) /ⁿ₈=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, nhᵈₙ ₙ. It is shown that E\|mₙ (x) - m (x) |ᵖ (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.
Devroye et al. (Sat,) studied this question.