Page 80 - Textos de Matemática Vol. 47

70
and
f0(x) = Z
P. E. OLIVEIRA
K(z)F0(x, z) dz, f1(x) = Z
[0,1]
[0,1]
The behaviour of the functions and depends on the geometry and smooth-
ness of the distribution of the random process X. Typically, one expects that
limh!0 (K, x, h) = 0. Now, this follows from (SB2) if a Dominated Con-
vergence Theorem is applicable, but this is not necessarily true, as see in
the examples below. To refer to Ferraty, Mas, and Vieu [6] approach, as-
suming oRf course the di↵erentiability of ⌧0 and K, it is easily verified that
f0(x) = 1 K0(z)@⌧0 (x,z)dz. Moreover, in [6] it is assumed, besides the distri-
0 @z
bution function version of (SB2), that the kernel satisfied (K) and also that
it is decreasing with K(1) > 0. Thus, kernels with weights decreasing smoothly to 0, such as K(z) = 1 z✓, ✓ > 0, are not allowed under the assumptions in [6]. The need for such extra asumptions on K is due to the peculiarities of the control of the asymptotics, made through distribution functions and not needing the convergence rate towards ⌧0. However, kernel functions satisfying the assumptions of [6] leave us with an estimator the resambles the regresso- gram, as there is a sudden cut-o↵ on the weights once the observation becames to far from the reference point x.
Examples 2.1. We shall next describe in more detail the functions and in a few significative examples. The models (1)–(3) are also discussed in Ferraty, Mas, and Vieu [6] describing the characterizations adapted to their assumptions.
(1) Let F(x,h) = f(x)h , > 0. If 2 N this model includes the case of X being a -dimensional random vector. For general > 0, the model corresponds to fractal type processes of order (see Ferraty and Vieu [3, 5]). It is easily verified that F0(x, z) = z 1, (x, z, h) = 0, thus, also (K, x, h) = 0.
(2) With respect to the previous example, we may relax the concentration
rate of the probability mass around x assuming F (x,h) = f (x)h |log h|k ,
, k > 0. Then, F0(x,z) = z 1, (x,z,h) = kz 1 , and
K2(z)F0(z) dz.
|log h| kZ1
(K, x, h) = |log h| K(z)z 1 dz ! 0, 0
if this last integral is finite. Thus (K,x,h) does converge to 0, but
with a rather slow rate.
(3) Assume F(x,h) = f(x)h e h k, , k > 0. This model for the distri-
bution of X includes di↵usion processes, fractional Brownian motions, fractional Brownian sheets or fractional Ornstein-Uhlenbeck processes.