Consistent nonparametric multiple regression: The fixed design case

AbstractConsider the nonparametric regression model Yi(n) = g(xi(n)) + εi(n), i = 1, …, n, where g is an unknown function, the design points xi(n) are known and nonrandom, and εi(n)'s are independent random variables. The regressor is assumed to take values in A ⊂ Rp, and the regressand to be real valued. This paper studies the behavior of the general nonparametric estimate gx(x)=∑i=1nwni(x)Yi(n), 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