Page 84 - Textos de Matemática Vol. 47

74 P. E. OLIVEIRA
We will now go through each term above, characterizing their asymptotics. The first result describes the mean square error of fb (x) and gb (x), handling the
two first terms. It is an immediate consequence of Lemmas 3.1 and 3.2.
Lemma 4.1. Assume that (SB1), (SB2) and (K) hold. Then
and
nF (x, h)
E (gbn(x) r(x)f0(x))2 = (s(x) + ws(x, h))(f1(x) + (K2, x, h))
E ( fb ( x ) f ( x ) ) 2 = f 1 + ( K 2 , x , h ) + 2 ( K , x , h ) .
n0
nF (x, h) +r(x) 2(K, x, h) + wr2(x, h)(f0(x) + (K, x, h))2
+2r(x) (K,x,h)wr(x,h)(f0(x)+ (K,x,h))+o✓ 1 ◆. nF (x, h)
nn
Lemma 4.2. Let r be continuous and assume that (SB1), (SB2) and (K) hold. Then
E ⇣(fb (x) f (x))(gb (x) r(x)f (x))⌘ n0n0
= r(x) (f1(x) + (K2, x, h)) + r(x) 2(K, x, h) nF (x, h)
+wr(x,h)(f0(x)+ (K,x,h)) (K,x,h)+o✓ 1 ◆. nF (x, h)
term follows from Lemmas 3.1 and 3.2. On the other hand
Probof. Write E (fb (x) f (x))(gb (x) r(x)f (x)) = Cov(fb (x), gb (x)) + n0n0nn
b
E (fn(x) f0(x)) E (gbn(x) r(x)f0(x)) . The characterization of the last
C o v ( f n ( x ) , gb n ( x ) ) ✓
= 1 E YK(kx Xk) +o 1
◆
= 1 E (r(x)+r(X) r(x))K(kx Xk) +o✓ 1 ◆
n2F 2(x, h) h nF (x, h)
n2F 2(x, h) h nF (x, h)
= r(x) f1(x)+ (K2,x,h) +o✓ 1 ◆. nF (x, h) nF (x, h)
The conclusion now follows imediately. ⇤