Density Estimation and Combination under Model Ambiguity

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 =
nh^1/2(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) —

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

part of the proof is based mainly on Zheng (1996).

Employing the same expansion used for KI1 , where now u =

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 b_t ʌ 1      1_r /_Я/|Ч

c 2^{f f} ) ` n ∖  fτ₇(χ-j----ffx(^)— — 2 ∖ ---fτ₇(χ-j---f ' ■    In1 — 2 In2. ⁽⁸⁴⁾

Applying the mean value theorem to f_bθ (xi) we obtain:

ʃ / ʌ f ^- i'^{j dfθ}* (^xi) b fi*∖ ∣¹ b a*∖'^{d f}θ (^xi) b fi*∖.
fb ⁽xi) — fθ* ⁽xi) = —∂θ— (^θ —^θ )⁺ 2(^θ —^θ ) ∂Θ∂Θ0 (^θ —^θ );

thus,

= S1n(^bθ — θ^*) + (^bθ — θ^*)⁰S2n(^bθ — θ^*).                                 (86)

It can be noticed that the U-statistic form of S1_n is the same as that of U_n defined in theorem 2. It follows
that Sin = Op(√n).

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