Consistency of a recursive nearest neighbor regression function estimate

Let (X, Y) be an ^d × -valued random vector and let (X₁, Y₁),…,(X_N, Y_N) be a random sample drawn from its distribution. Divide the data sequence into disjoint blocks of length l₁, …, l_n, find the nearest neighbor to X in each block and call the corresponding couple (X_i^*, Y_i^*). It is shown that the estimate m_n(X) = Σ_{i = 1}ⁿ w_niY_i^*/Σ_{i = 1}ⁿ w_ni of m(X) = E{Y|X} satisfies E{|m_n(X) − m(X)|^p} 0 (p ≥ 1) whenever E{|Y|^p} < ∞, l_n∞, and the triangular array of positive weights {w_ni} satisfies sup_{i ≤ n}w_ni/Σ_{i = 1}ⁿ w_ni 0. No other restrictions are put on the distribution of (X, Y). Also, some distribution-free results for the strong convergence of E{|m_n(X) − m(X)|^p|X₁, Y₁,…, X_N, Y_N} to zero are included. Finally, an application to the discrimination problem is considered, and a discrimination rule is exhibited and shown to be strongly Bayes risk consistent for all distributions.