Page 87 - Textos de Matemática Vol. 47

REGRESSION ESTIMATION FOR FUNCTIONAL DATA 77
Proof. Taking into account the discussion above, it is enough to control ↵2 (E (fb (x) f (x))4)1/2(P(|gb (x) E gb (x)| > " ))1/2
nn0nnn ↵n2✓1 (K,x,h)2 ◆✓1◆
= " O nF(x,h)+n1/2F1/2(x,h)+ (K,x,h) O n1/2F1/2(x,h) . n
Nowchoose↵, >0suchthat2↵+ <1 andput↵n=(nF(x,h))↵and"n=
2
= (nF(x,h))2↵+ 1/2 ! 0, so the term
Lemma 4.7. Assume that wr(x,·), ws(x,·), (K,x.·) and (K2,x,·) are bounded, (SB1), (SB2), (K) and (3.2) hold. If, for some ↵ > 0,
(nF(x,h)) . Then
↵2n
"n n1/2 F 1/2 (x,h)
⌘
+ 2(K, x, h) . The term corresponding
⌘
⇣
to P fb (x) E fb (x) > (E fb (x))2
above is an o
1
+ (K,x,h) nF (x,h) n1/2 F 1/2 (x,h)
⇣
is treated analogously. ⇤ Remark 4.6. If we choose > ↵ then ↵n"n ! 0.
"n n
n n 4 Egbn(x)
then
4
E(rb(x)I n
↵ ) !0, { rb n ( x ) > ( n F ( x , h ) ) }
+ (K,x,h) + 2(K,x,h)◆. n1/2 F 1/2 (x, h)
(4.2)
Cn =o✓
Proof. Put ↵n = (nF (x, h))↵ ! +1 and write
1
nF (x, h)
⇣bb⌘
222
Cn = E (rbn(x) ren(x)(x))(fn(x) f0(x)) I{rbn(x)>↵n}
⇣
 E (fn (x) f0 (x)) E (rbn (x)I{rbn (x)>↵n } )
b 4⌘1/2 4 1/2 ⇣ 4⌘1/2 2 1/2
E (fn(x) f0(x)) ren(x) (P(rbn(x) > ↵n))
.
⇤
+
The result is now obvious taking into account (4.2).
To complete the characterization of the mean square error we still have to control the final term in (4.1). The asymptotic behaviour of this last term depends more directly on the geometry and smoothness the distribution of X.