E(⅛>-⅛i) XXE[hκ(xjF)
∂2fθ (Xi) /∂θ∂θ0 ■
fθ* (Xi) .
(87)
μXj - Xi ∖ ∂2fθ (xi) /∂θ∂θ0 =
h h ) fa (xi)
1/■ g(^i)g(^j )d^id^j =
∂ ∂fθ ^∂2fθ (Xi) /∂θ∂θ0
(88)
I I K (u)------—g— -----g(Xi)g(Xi + hu)dXidu.
f f fa (Xi)
Similarly to Dimitriev-Tarasenko(1973), applying the Cauchy-Schwartz inequality we obtain that
Г !'if d f ∂ d fθ (Xi) /dθdθ 2l ∖r! d(ds(θfa t A 4()
lιm sup E (S2n) ≤ ---- ----g (æ)dæ = E —g(—g(x) ; (89)
n→∞ f fa (Xi) ∖ ∂θ )
further, since Efbn (X) = g(X), applying Fatou-Lebesgue theorem we have that
lim E (S2n) = E μ ds(J.,X) g(X)} . (90)
n→∞ ∂θ
Thus, we have that S2n = Op(1). Taking into account that v^(θ — θ*) = Op(1), which in turn implies that
(b — θ*) = Op(√n), it follows that Ini = S1n(b — θ*) + (b — θ*)0S2n(θ — θ*) is equal to
Ini = Op(-√n) * Op(√n) + Op(√n) * Op(1) * Op(√n) = Op( 1). (91)
Now we have to consider In2 :
In2 =1X f f - ʌ2 fn(Xi) ` (b—e-)0-1-^ XX1K μîj—Xiʌ dln fθ(Xi) dlnfθ(X) (?—«•) `
n2 n £i\ fθ* (Xi) J n(n — 1) i hf h V h J ∂θ ∂θ0 v ’
(92)
` (b — θ∙)0 nʌ XXK μ XjX1 ∖ s(θ,Xi^s(θ,Xj )0j (b — θ∙) = (b — θ∙)0S3n(b — θ∙)0. (93)
n(n — 1)h i j h
Similarly to S2n, it can be shown that S3n is Op (1) . It follows that In2
In2 = Op
(⅛)
Op (1) * Op (√n) = Op (n) .
(94)
Finally, we get that:
KI21 (fb, f'θt ) ' Ini — 7jIn2 = Op( — ) — Opp ( = Op ( ,
θ 2 n2 n n
then it follows that
(nhi/2)KI21 (fθ,fθ^) = (nhi/2)Op ɑɔ = Op(hi/2) →p 0.
(95)
31
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