Page 89 - Textos de Matemática Vol. 47

REGRESSION ESTIMATION FOR FUNCTIONAL DATA 79
The first and second terms of this decomposition are controled as in Lemmas 4.4
⇣ 1 b 2 (K,x,h) ⌘
and 4.5 to obtain an o nF(x,h) + (K,x,h)+ n1/2F1/2(x,h) . Finally
E ⇣(rbn(x) ren(x))2(fn(x) bf0(x))2I{rbn(x)>↵n}⌘ ⇣222⌘
2E (rb (x)+re (x))(f (x) f (x)) I
n n n 0 { rb n ( x ) > ↵ n }
= o ✓ 1 + 2(K, x, h) + (K, x, h) ◆ , nF (x, h) n1/2 F 1/2 (x, h)
taking into account (4.2). Collecting these upper bounds, the result follows. ⇤ To describe the mean square error for the regression estimator it remains to
put together the characterizations just obtained for each term of (4.1). Theorem 4.9. Assume that r is continuous, ws(x, ·), (K, x.·) and (K2, x, ·)
are bounded, (SB1), (SB2), (K), (3.2) and (4.2), for some ↵ 2 (0, 1 ), hold.
Then,
E(rbn(x) r(x))2
= f1(x) + (K2, x, h) ✓ r2(x) + s(x) + ws(x) 2r(x) ◆
6
nF (x, h) f02 (x) f02 (x) f02 (x) +wr2(x,h)(f0(x)+ (K,x,h))2 2r(x)wr(x,h) (K,x,h)(f0(x)+ (K,x,h))
+o
f02 (x) f02 (x)
1 + 1/2(K,x,h) + (K,x,h)+ 1/2(K,x,h)+wr(x,h)
nF (x, h) n3/4F 3/4(x, h) n1/2F 1/2(x, h) + 3/2(K,x,h)+ 1/2(K,x,h)wr(x,h)+ (K,x,h)
n1/4 F 1/4 (x, h) + 2(K,x,h)+ 3/2(K,x,h)+ (K,x,h)wr(x,h)!.
Let us get back to the case where limh!0 (K, x, h) = 0. We may trace the calculations above to find that, under this additional assumption and the