Page 85 - Textos de Matemática Vol. 47

REGRESSION ESTIMATION FOR FUNCTIONAL DATA 75
We will need the characterization of the asymptotics of E (fb (x) f (x))4 . n0
Lemma 4.3. Assume that wr(x,·), ws(x,·), (K,x.·) and (K2,x,·) are bounded, and (SB1), (SB2), (K) hold. Then
E(fb(x) f (x))4 =O✓ 1 + 2(K,x,h)+ 4(K,x,h)◆.
n0
Proof. Write ⇣ ⌘4
n2F 2(x, h) nF (x, h) E(fb(x) f (x))4 =E (fb(x) Efb(x))+(Efb(x) f (x))
n0nnn0
=E(fb(x) Efb(x))4 +4E(fb(x) Efb(x))3(Efb(x) f (x)) nnnnn0
+6E(fb(x) Efb(x))2(Efb(x) f (x))2 +(Efb(x) f (x))4. nnn0n0
We shall now characterize each term of the previous expansion. Obviously (Efb(x) f (x))4 = 4(K,x,h).
n0 bbb2b2
E(fn(x) Efn(x)) (Efn(x) f0(x)) =Varfn(x) 2(K,x,h)=O✓ 2(K,x,h) + 4(K,x,h)◆.
nF (x, h)
For the second term in the expansion, we have, expanding again and taking
into account that the terms are centered and independent,
E(fb (x) Efb (x))3 = 1 E ⇣K(kx Xk) EK(kx Xk)⌘3
n n n2F3(x,h) h h
= 1 ⇣EK3(kx Xk) 3EK2(kx Xk)EK(kx Xk)+2(EK(kx Xk))3⌘
n2F3(x,h) h h h h =O✓ 1 ◆.
n2F 2(x, h)
Finally, expanding once more, we find that E (fb (x) E fb (x))4 = O 1 .
n n n2F2(x,h) Summing up all the characterizations, the result follows.
⇤
To continue with the analysis of the terms in (4.1), write first
⇣2 2 b 2⌘ E (rb (x) r (x))(f (x) f (x))
nn0
⇣2 2 b 2⌘2 2 b 2
=E (rb (x) re (x))(f (x) f (x)) +(re (x) r (x))E(f (x) f (x)) . nnn0n n0