Necessary and sufficient conditions for Bayes risk consistency of a recursive kernel classification rule (Corresp.)

It is shown that, for a nonparametric recursive kernel classification rule, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∑ⁿᵢ₌₁hᵈ(i)I_{ \{h(i) > ε \} } / ∑ⁿⱼ₌₁ hᵈ (j) → 0 { as} n → ∞,</tex> all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ε > 0</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∑∞ᵢ₌₁hᵈ(i)= ∞</tex> constitute a set of conditions which are not only sufficient but also necessary for weak and strong Bayes risk consistency of the rule. In this way, weak and strong consistencies are shown to be equivalent.