Nonparametric multiple function fitting

Consider the d-dimensional unit cube [0,1]d and portion it into n regions, A1,..., An. Select and fix a point in each one of these regions so we have x1,..., xn. Consider observable variables Yi, i = 1,..., n, satisfying the multivariate regression model Yi = g(xi) + [var epsilon]i, where g is an unknown real-valued function defined on [0,1]d and [var epsilon]i, i = 1,..., n, are independent random variables with E[var epsilon]i = 0. This paper studies gn(x) = [Sigma]ni=1YiKh(x - xi)[lambda](Ai) as an estimate of g(x), where Kh(·) = h-dK(·/h) is a known function, h = h(n) is a sequence of positive reals converging to zero as n --> [infinity], and [lambda](Ai) is the Lebesgue measure of region Ai. Under suitable conditions, it is shown that gn is asymptotically pointwise weakly, mean square and completely consistent. Asymptotic normality of the estimate is also established. The class of applicable kernel functions {K(·)} includes those discontinuous kernels and kernels with unbounded support. Finally, some global results are stated.