Page 79 - Textos de Matemática Vol. 47

REGRESSION ESTIMATION FOR FUNCTIONAL DATA 69
This is a usual assumption in almost all literature on regression estimation in a functional framework.
In order to have some control on the asymptotics we need a more precise
description of the decrease rate of F when h goes to 0. The control of this
decrease rate appeared for the first time in Ferraty, Mas, and Vieu [6] through
the function ⌧0(x, z) = limh!0 F (x,hz) , x 2 S, z 2 [0, 1]. To include in the asyp- F (x,h)
totic description of the mean square error some knowledge of the distribution of the functional variable we need some information about the convergence rate towards ⌧0. This is better achieved through a density like variant of ⌧0, that we introduce in the following assumption.
(SB2) There exists F0 , defined in S ⇥ [0, 1], such that
( x , z , h ) = h @ F ( x , z h ) F 0 ( x , z ) ! 0 , w h e n h ! 0 .
F(x,h) @h
It is easily seen seen that if ⌧0 is di↵erentiable with respect to the second
variable then F0(x, z) = @⌧0 (x, z). Thus (SB2) is a translation into density @z
like functions of H3 in [6]. As it will become apparent later (see Example 2.1 (3) below), the two approaches are not equivalent. Indeed, Ferraty, Mas, and Vieu [6] assumption H3, treating distribution functions, regularizes everything thus allowing to apply the Dominated Convergence Theorem, while our den- sity like functions give raise to di culties on handling the asymptotics of the integrals.
We will be interested in kernel estimaXtion so, define, for each x 2 S, 1n
fb ( x ) = K ( kx X ik ) , n nF(x,h) h
i=1 1 Xn
gb n ( x ) = Y i K ( k x X i k ) , nF (x, h) h
i=1
where K is a real valued function, and h the bandwidth. Further, let
gb b ( x ) rbn(x)=n .
fn (x)
This estimator is the Nadaraya-Watson estimator studied by several authors.
Consider the following assumption on the kernel function. (K) K is bounded nonnegative with support [0, 1].
Remark that we do not require K Zto be a probability density. Define now
(K, x, h) = K(z) (x, z, h) dz. [0,1]