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, a
2 + 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)

ln g(xi))dFbn(xi).


We start examining the limiting distribution of

11

KI2 = ^n∑ (log fθ (xi) log fθ* i)) 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

1E(ln fbθ (x) ln g(x)). This


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

fb(xi)~fθ* (xi) :
fθ-(xi)      ■


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

1 X μ fθ (xi) fθ* (xi) λ2

2n ⅛V    fθ* (xi)    √ ,


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* (xi2 bt ʌ 1      1r /Я/|Ч

c 2f f ) ` n ∖  fτ7-j----ffx()2 ∖ ---fτ7-j---f ' ■    In1 2 In2. (84)

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

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

thus,

In1 =


1 X fn(xi)
n i=ι fθ* (xi)


ffθ (Xi) fθ* (Xi)) `


(85)


fn(xi) ∂fθ (xi)
fθ* (Xi)    ∂θ


(b θ*) + 2n X(b θ*)0
i


fn(xi) ∂2fθ* (Xi)
fθ* (Xi) ∂θ∂θ0


(bθ θ*)=


_1_XX 1K

n(n 1)        h

+(b θ*)0 —ɪ
2 2n(n


Xj Xi

I-J


1)∑ ∑h к
ij


∂fθ* (Xi) /∂θ b

(θ θ )+

ʃθ* (Xi)

XxjXi^ 2fθ (Xi) /∂θ∂θ' b

:    ThX-)    (θ θ ) =


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

It can be noticed that the U-statistic form of S1n is the same as that of Un defined in theorem 2. It follows
that S
in = Op(n).

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