Density Estimation and Combination under Model Ambiguity

∂ ∂fθ ^∂²fθ (Xi) /∂θ∂θ⁰

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 f}θ ^(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 Ef^bn (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 S2_n = O_p(1). Taking into account that _v^(θ — θ*) = O_p(1), which in turn implies that
(b — θ*) = O_p(√n), it follows that I_ni = S_1n(b — θ*) + (b — θ*)⁰S_2n(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—<sub>e</sub>-)0-1-^ XX<sup>1</sup>K μîj—Xiʌ <sup>dln fθ(Xi) dlnfθ</sup>(X) (?—«•) `

ⁿ²   n £i\    f_θ* (X_i)     J                   n(n — 1)  i hf h   V h J ∂θ ∂θ^{0 v} ’

(92)

` (b — θ∙)0 nʌ XXK μ X^jX₁ ∖ 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

Finally, we get that:

^KI21 ^(fb^{, f}'θ^t ) ' Ini ^— 7jIn2 = Op⁽ — ) ^— Opp (    = Op (    ^,

^θ           2        n2 n       n

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