Necessary and Sufficient Conditions for the Pointwise Convergence of Nearest Neighbor Regression Function Estimates

where (v,1, ..., v,,) is a given probability vector, and (Xt(x), Yl(x)), ..., (X.(x), Y,(x)) is a permutation of (X1, I71) . . . . , (X, , Y,) according to increasing values of IlXi-x[I, x e R a. When [IXi-xll = H X j x l [ but i < j , X i is said to be closer to x than X~. The consistency properties of m, for special choices of the weight vector (v,,, ..., v,,,) are discussed in Cover (1968), Stone (1977), Devroye (1978) and Collomb (1979, 1980). For an analysis of the bias and variance with rate of convergence results, see Lai (1977) and Mack (1981). See also the survey by Collomb (1981). In this paper we give necessary and sufficient conditions on the weight vector for weak, strong and complete pointwise convergence of m, to m under no assumptions whatsoever on the probability measure # of X. Any Borel measurable function of x and the data will be called a regression function estimate. We let d be the collection of all random vectors (X, Y) taking values in R a x [ c , c] for some integer d > 1 and some constant c>0.