Page 78 - Textos de Matemática Vol. 47

68 P. E. OLIVEIRA
Ferraty and Vieu [3, 4], Masry [14] or Ferraty, Mas, and Vieu [6] for regression estimation, followed by many other authors. A good account of essential results of the theory may be found in the monography by Ferraty and Vieu [5].
A crucial assumption for the convergence of the estimators concerns the local behaviour of the distribution of the functional variable, known as the small-ball probabilities. Some kind of regularity on these probabilities is always required to prove any convergence result for the kernel estimator. The bandwidth choice, which is well known to be essential for this family of estimators, is usually only indirectly addressed. The classical way to discuss this choice, based on the as- ymptotic description of the mean square error of the regression estimator, has been studied in Ferraty, Mas, and Vieu [6]. Assuming the di↵erentiability of a variant of the regression function and kernels that are compactly supported and bounded away from 0, these authors give a rather compact asymptotic descritp- tion of the mean square error. This characterization is easily used to describe the bandwidth optimal choice. We want to consider kernel functions that are still compactly supported but allow the weights to approach 0 and also avoid di↵erentiability assumptions on the regression function. This choice drives us to make assumptions on an infinite dimensional equivalent of the traditional density functions, giving raise to some integrability issues to be handled. More- over, we may have some knowledge of the distribution of the functional variable. Thus, including this knowledge into the bandwith description leads to a better adapted estimator. With this in mind we give an alternative characterization of the mean square error of the kernel estimator for the regression function. The expression obtained, described in Theorem 4.9 below, is rather long and somewhat intricate, but highlights the role of the geometry and smoothness of the distribution of the functional variable.
On the sequel (Xi,Yi), i 1, are equally distributed and independent ran- dom elements, where Yi are random variables, and Xi take values in some normed space S. Our goal is to estimate r(x) = E(Y|X = x), x 2 S. We will denote the conditional second order moment by s(x) = E (Y 2 |X = x), x 2 S. Given a function q : S ! R, define the local modulus of continuity as
wq(x,h)= sup |q(y) q(x)|, x2S,h>0. ky xkh
Of course, if q is continuous at x then limh!0 wq (x, h) = 0.
2. Assumptions on small-ball probabilities
We introduce a first assumption on the behaviour of small-ball probabilities.
(SB1) Let F(x,h) = P(kx Xk  h), and assume that, for each x 2 S, limh!0 F (x, h) = 0 and F is partially di↵erentiable with respect to h.