E (Hn (xi ,xj
))=1/
K(u) [v(θ*,Xi)
— v(θ*, Xi + hu)] g(xi)g(xi + hu)dxihdu =
= K (u)du [v(θ*,Xi)
— v(θ*,Xi)] g2(xi)dxi = 0.
(50)
So what we have to study is the asymptotic behavior of
_ ____ ,,, 2^,._, ,,, 2^,._,,.. 2^,._, ,,,
√nUn = √nE(rn(xi)) + — Vrn(Xi) — E(rn(xi)) = —= Vr(xi) — E(r(xi)) + — V∖(xi) — E(tn(xi))
n nn
i ii
(51)
where r(xi) = [s(θ*, xi) — v(θ*, xi)] g(xi) and E(r(zi)) = E(E [((s(θ*, xi) — v(θ*, xi))g(xi)) /xi]) = 0 and the
last term of the above expression converges to zero in probability. Hence, the limiting distribution of √nUn
is the same of √n Pi r(xi) = -√- Pi [s(θ*,xi) — v(θ*,xi)] g(xi).
By Lindeberg-Levy central limit theorem, we have that
Wn(θ*) = √nUn →d N(0, B(θ*)) as n → ∞
(52)
B(θ*) = 4E([s(θ*,xi) — v(θ*,Xi)]2 g(xi)2) = 4 / (s2(θ*,Xi) + (v(θ*,Xi))2 — 2s(θ*, xi)v(θ*, xi)) g(xi)3dx =
У (v2(θ*,Xi) — (v(θ*,x ))2j g(xi)3dxi
У var(s(θ*,Xi))g(xi)3dxi = E(var(s(θ*, Xi))g(xi)2).
(53)
This implies that Wn(θ*) = Op (^√n) . It follows that
√n(bn — θζ) = —(An(θ))-1Wn(θ*) → N(0,A(θ*))-1B(θ*)A(θ*))-1). (54)
9.3 Proof Theorem 3:
KI can be rewritten in the following way:
f , , . , . . ^ , . ∕* , ,. , . . ^ , . f , ... , . . ^ , .
KI = ( (lnc(x)—lnfθ(x))dFn(x) = I (lnc(x)—lng(x))dFn(x)—/ (lnfg(x)—lng(x))dFn(x) = KI1—KI2.
x x (55)
Similarly to Fan(1994), this representation is very helpful to examine the effect of estimating fθψ by fg on
the limiting distribution of KI. From now on the index j for th model will be omitted.
c
ζn-Xχ i) j that by the Law of Large Numbers
(LLN) can be considered a good approximation of n
1E((ln fcn (x) — ln g(x)) = n-1KI1. This first part of
the proof draws heavily upon Hall(1984) and Hong and White(2000).
fn-g(x) = fn
g(x) g(x)
— 1 we
Using this inequality ∣ln(1 + u) — u + 21 u2∣ ≤ |u|3 for |u| < 1 and defining u
obtain the following result:
25
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