A _ _ 1 T г» TT 11∕-∣ no n J 1 r∙ , , r τ^k ∙ ∙ 1
As n → ∞,by Lemma 2 Hall(1984) the first term of V21n is given by
1
n(n — 1)2
a2n(xi,xj) = σ2n + Op(n-3/2h-1),
i j,i6=j
(70)
where σ2l = 21nσ2n.
2Ub2l =
2
n(n — 1)
i i6=j
al(xj, x)al(xi, x)g(x)dx =
n(n — 1) XX H2n(xi,xj),
i i6=j
(71)
1 K ʌ 1 1 [Kh(x
H2n(xi,Xj)= h h2 -----
— χj ) — ʃ Κh(χi — χj )g(χj )dχj
g(χi)
Κh(xi-Xz)-J^Κh(xi-Xz)g(xz)dXz
g(χi)
g(xi)dxi.
E Hn(χi,xj )] = h4 E
xi —
xi — xj ) — ʃ Kh (xi
g(xi)
xj) — ʃ Kh(xi — xj)g(xj)dxj ∖ KKh(xi
g(xi) g к
— xj)g(xj)dxj∖ μKh(xi — xz) — ʃKh(xi
g к g(xi)
xz) — ʃ Kh(xi — xz)g(xz)dxz λ ^(x )d
g(xi) g g i '
— xz)g(xz)dxz 2
------------------ I g(xi)dxi
2
''T^∙ .
xi
g(xj)g(xz)dxj dx
=⅛/KPk
xi — xj )Kh(χi
g2(xi)
xz ) l ∖rl
----g(xi)dxi
g(xj )g(xz )dXj dxz + o( ɪ ) =
K (u)K (u + v)
----,------—-—du
g(xj + hu)
g(xj )g(xj +hu-hz)dxj hdv+o(-) = — f
hh
g2(xj)
2
K(u)K(u + v)du
g2(xj)dxjdv =
' h-1
K K K(u)K(u + v)du dv + o(^)∙
(72)
By Lemma 3 in Hall(84), then Ub2l is asymptotically Normally distributed N (0, σ22l), where
σ22l ' 2n
2h-1
K (u)K (u + v)du
dv∙
(73)
Hence finally we have that
V2in ~ σ2l + Op(n-3/2h-1) + 2σ2nN(0,1).
(74)
Vλn — 1 P h f (χi)-g(χi) i2
22n n Z-^i g(xi)
= l Pi b2n(xi), which is a purely deterministic Bias-squared term, and it will
affect the mean of the asymptotic distribution∙ That is,
1X b2 _ h4 2 f (g(2)(x)¢ 2d , fh4>. r75j
nÇbn 4 μ2J g(x) dx+ o(h )∙ (75)
Finally we can analyze Vb23l :
2V23n = 2 χf c(xi) — f (xi) ! μf (xi) - g(xi) ʌ = 2 X H3ni, (76)
n i g(xi) g(xi) n(n — 1) i
similarly to Hall(1984) define
28
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