1 I ∖Kl K-x f ∖κ(xx- - x) - f Kh(xi - xj )g(xi )dxj↑ f( (xi) - g(xi)∖ -, f,7,7×
H3ni / j an(xi, xj)bn(xi) h J g(x ) y g(x ) d dx (77)
Under assumptions A1 and A2 and given that EH23ni =0, by Lemma1 in Hall(1984) we have that 2Vb23n
is asymptotically normally distributed with zero mean and variance given by:
2 ' 2 -1h4 2
σ3n ` 2n h μ2
I ,ζ⅛)'0 dxi - (/ (g(2^50 dx)
(78)
which can be easily seen if we consider that f (xi) g(xi) = h μ2g—(x)and that
g(xi) g(xi)
EH32ni = h4μ2
I dxi- (J (gβ,<x<)) dxi)
Also this term will affect the asymptotic distribution of κI1 .
To summarize all previous steps, we can rewrite the expansion of κI1 in the following way:
κI 1 = VzIln + Bbn - 2 ^½1n + V22n + 2V23nJ - (79)
N(0,σ2n)+h-μ2 / g(2)(x)dx+o(h2)-1 fσ^n + Op(n 3/2h 1) + 2N(0,σ2n) + h-μ2 [ (g (x))
dx + o(h4) + 2N (0,σ23n) .
1n 2 2 2 n 2n 4 2 g(x)
Once more, following Hall(1984), from the definition of Vb21n and the fact that nh → ∞, we have that the
difference between n(n-i) Pi Pj=i αn(xi, xj) and σgl is negligible w.r.t. 2U2n, hence the previous expression
can be rewritten as follows:
KKI1 - (nh1/2)-1√2σ1N1 - (nh1^)-1√2σ2N2 - n-1¾2√2σ3N3 + Bbn - |cn
(80)
where N1 ,N2 and N3 are asymptotically normal N(0,1); and
σ1
κ2(u)d
u, σ2 = κ (u)κ (u + v)du
dv and σ3 = μ2
I dxi - μj (g(2)(xi)) 1
and cn
h4
= (nh)-1 K 2(u)du + -^j-μ2
[ μ g⅛) )2 dx+0(n-1ft-1
g(x)
+ h4).
(81)
It is important to notice that Bbn, which is Op (n-1/2h2), will asymptotically cancel out with n-1/2h2 √2σ3N3,
since they are of the same order of magnitude.
Thus, we have the following results: asn →∞,h→ 0, nh →∞ and nh5 → 0
nh1/2(jcI 1 + 2Cn) →d √2σιNι - √2σ3N2.
(82)
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