Consistent nonparametric multiple regression: The fixed design case

Consider the nonparametric regression model Yi(n) = g(xi(n)) + [var epsilon]i(n), i = 1, ..., n, where g is an unknown function, the design points xi(n) are known and nonrandom, and [var epsilon]i(n)'s are independent random variables. The regressor is assumed to take values in A [subset of] Rp, and the regressand to be real valued. This paper studies the behavior of the general nonparametric estimate , where the weight function wni is of the form wni(x) = wni(x; xi(n), ..., xn(n)). Under suitable conditions, it is shown that the general linear smoother gn for the unknown regression function g is asymptotically pointwise unbiased, weak, mean square and complete consistent, and asymptotically normal. The results of the limit theorems can be applied to extend or improve the conditions of the estimates with various particular weights wni including all those known in the literature.