1 Σ ln fcn(χl
n i g(xi)
Let define
fn (xi) - g(xi)
g(xi)
, 1 X ∕fn(xi) - g(xi)∖ 3 3
+ 2n ¾ g(χi) J ≤ u.
(56)
Vb1n
n ∑(
c(xi) - g(xi)
g(xi)
and
2
b 1 fn (xi) - g(xi)
v2n = n M --l x ) ) ■
By Lemma 3.1 Hong-White (2000), under assumption A1 and A2, nh4/ ln n →∞,h→ 0. Then:
KcI1 = Vb1n
13
- V2n+ + Op(n 2h 3lnn + h6).
(57)
AT 1 1 1 JlJ T^T∙ 1 T^T∙ T J 1 1 it X 1 1 —1 T xt—x — —X ZXl 1
Now we have to analyze the terms Vin and V2n. Let define f(x) = ʃh 1K( i- )g(x)dx and
h—iKh(xi
an(xi,xj ) = ----------
- xj ) - fh
iKh (xi - x)g(x)dx
g(xi)
bn(xi) = —
iKh∣χΛ-χ)f(χ)dx-g(χi)
g(xi)
Then
Vb1n
c(χi) - f (χi) + f (χi) - g(xi)
g(χi) g(χi)
n(n — ι)h Σ Σ an(xi, xj) + n Σ bn(xi) =
i j,i6=j i
(58)
= V11n +B
where Viin is a second order U-statistic and it will affect the asymptotic distribution of KIi. Similarly to
TT ιι<1nnn TJ ∙. Γ' ■ . ι 1- n ■
Hall(1984) let rewrite Viin in the following way:
Viin = -r
n(n
1y∑ ∑Hin(Xi,Xj)
i j,i6=j
1 Kh (xj
Hin(xi, χj) = 2h (
Xi) - KΚh(x - Xi)g(x)dx + Kh(xi
g(xi)
xj) - Kh(xi - x)g(x)dx
= J ] — Jn(xi,xj) + Jn(xj ,xi)
g(xi)
(59)
E(H1n (xi,xj)/xi)=0, then using Theorem 1 in Hall(1984) we can show that
^ I
Viin = < -r
n(n
π∑ ∑Hiin(Xi,Xj)
i j,i6=j

2E [H2n(χi,xj
n2
-)ɪ I →d N(0,1).
(60)
E [J‰1XJ)] ,h / Kh
xi) - RKh(xi - χ)g(χ)dχ¢2
------g2∣x )------------------g(xi)g(xj)dxidxj =
26
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