Page 83 - Textos de Matemática Vol. 47

REGRESSION ESTIMATION FOR FUNCTIONAL DATA 73
Theorem 3.4. Let r be continuous and assume that (SB1), (SB2) and (K) hold. Then rbn(x) is asymptotically unbiased.
Proof. Indeed, let ren(x) = E gbn(x) . It follows from the above characterizations b
Efn(x)
that ren(x) = r(x) + wr(x, h) ! r(x), as r is continuous. ⇤
We now state convergence results about the estimator. The proofs are straight- forward adaptations of the proof of Theorems 3.2, 3.3 and 3.4 in Oliveira [16], so we will not include them here.
Theorem 3.5. Assume s is continuous, (SB1), (SB2) and (K) hold. If there exists M1 >0 such that, for every ` 2, E(Y`|X =x)M1``!s(x) and
nF (x, h) ! +1, (3.3) log n
then rbn(x) E rbn(x) converges almost completely to zero with a convergence rate of order ⇣ log n ⌘1/2.
nF (x,h)
Theorem 3.6. Assume r is continuous, (SB1), (SB2) and (K) hold. Then
p 0 f1(x)+ lim (K2, x, h) 1 d @ h!0 2 A
nF(x,h)(rbn(x) Erbn(x)) !N 0, f0(x)+lim (K,x,h) (s(x) r (x)) . h!0
4. The mean square error
We want to characterize the asymptotic behaviour of E (rbn(x) r(x))2. For this, as done in Bosq and Cheze [2] and later explored by Oliveira [15] and Bensa¨ıd and Fabre [1], we consider the decomposition
E(rb (x) r(x))2 = r2(x)E(fb (x) f (x))2 n f02(x)n0
+ 1 E ( gb n b( x ) r ( x ) f 0 ( x ) ) 2 f 02 ( x )
2r(x) E ⇣(fn(x) f0(x))(gbn(x) r(x)f0(x))⌘ f 02 ( x )
(4.1)
⇣2 2 bb 2⌘ f02(x)n n0
1
+ E (rb (x) r (x))(f (x) f (x))
2 E ⇣(rbn(x) r(x))(fn(x) f0(x))(gbn(x) r(x)f(x))⌘ . f 02 ( x )