Density Estimation and Combination under Model Ambiguity



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 fd 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*)0S2nθ*) 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) ` (be-)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, ft ) ' 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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