Since aN (0, 1) + bN (0, 1) can be written as partial sum of martingale difference array, and it can be
proved to be asymptotically normal N(0, a2 + b2) (see Hall(84) p.10), then we have that ∙∖∕2σιNι — √2σ3N2 =
nh1/2(KIι + 1 cn) → √2(σι — σ3)N(0,1).
Let now examine the term
KI2 = J^(ln fθ(x) — ln g(x))dFn(x) = ʃ(ln fb (xi) — log fθ* (xi)+logfθ* (xi) —
ln g(xi))dFbn(xi).
We start examining the limiting distribution of
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KI2 = ^n∑ (log fθ (xi) — log fθ* (æi)) fn(xi) + n Σ(lθg fθ* (xi ) — log ^(^i)) fn(xi) = c 21 + KI22, (83)
i=1 i=1
that similarly of KI1 by the LLN, can be considered a good approximation of n
1E(ln fbθ (x) — ln g(x)). This
part of the proof is based mainly on Zheng (1996).
fb(xi)~fθ* (xi) :
fθ-(xi) ■
Employing the same expansion used for KI1 , where now u =
— 1 X μ fθ (xi) — fθ* (xi) λ2
2n ⅛V fθ* (xi) √ ,
1 Vl f fθ (xi) ʌ 1 ∖ fθ (xi) — fθ* (xi)
n ∑log IfHxi) J ` n ∑ f.-(xi)
we can rewrite KI21 in the following way:
C If 1 ∖ 1 f fb (xi ) — fθ* (xi)A b( 1 1 f f^θ (xi) — fθ* (xi)λ 2 bt ʌ 1 1r /Я/|Ч
c 2f f ) ` n ∖ fτ7(χ-j----ffx()— — 2 ∖ ---fτ7(χ-j---f ' ■ In1 — 2 In2. (84)
Applying the mean value theorem to fbθ (xi) we obtain:
ʃ / ʌ f ^- i'j dfθ* (xi) b fi*∖ ∣1 b a*∖'d fθ (xi) b fi*∖.
fb (xi) — fθ* (xi) = —∂θ— (θ —θ )+ 2(θ —θ ) ∂Θ∂Θ0 (θ —θ );
thus,
In1 =
1 X fn(xi)
n i=ι fθ* (xi)
ffθ (Xi) — fθ* (Xi)) `
(85)
fn(xi) ∂fθ∙ (xi)
fθ* (Xi) ∂θ
(b — θ*) + 2n X(b — θ*)0
i
fn(xi) ∂2fθ* (Xi)
fθ* (Xi) ∂θ∂θ0
(bθ — θ*)=
_1_XX 1K
n(n — 1) h
+(b — θ*)0 —ɪ
2 2n(n
Xj — Xi
I-J
1)∑ ∑h к
ij
∂fθ* (Xi) /∂θ b
(θ — θ )+
ʃθ* (Xi)
Xxj — Xi^ ∂2fθ (Xi) /∂θ∂θ' b
: ThX-) (θ —θ ) =
= S1n(bθ — θ*) + (bθ — θ*)0S2n(bθ — θ*). (86)
It can be noticed that the U-statistic form of S1n is the same as that of Un defined in theorem 2. It follows
that Sin = Op(√n).
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