Density Estimation and Combination under Model Ambiguity



as n -→ ∞, fn - g -→p 0, then D21 -→ 0

(33)


and since   i=1 ln f(θ, xi)g(xi) = ln f(θ, x)g(x)dx, then D22 -→ 0.

We can conclude that KIjn (θ) -→p KIj (θ), hence it is a contrast relative to the contrast function KIj (θ)
according to the Definitions 3 and 4 in Dhrymes (1998).

Further since KIjn (θ) can be rewritten as Hn (θ) - Hn(fn), where

nn

Hn(fcn) = -     lnfcn(xi) fcn(xi) and Hn(θ) = -    ln f (θ, xi)fcn(xi)

then


(34)

(35)


Hn1) - Hn2) = [Hn1) -Hn(fcn)] - [Hn 2) -Hn(fcn)].

It follows that

Hn1) - Hn2) p KIj1) - KIj2).                               (36)

By the continuity of Kullback-Leibler Information and by A3, assumption (iii) of Theorem 1 in Dhrymes

MΛΛ∩', ∙   ∙ j ∙ Γ∙ 1 ГГ11      Jl            ∙ J            PjI Л /Г/ɔ j∙      J ^Λ^      C 11        ∙          T J 1   1 Jl ∙             Jl

(1998) is justified. Then the consistency of the MC estimator θMj follows immediately by this same theorem.

9.2 Proof Theorem 2:

By the mean value theorem around the parameter θ*

0 = VKI(fn, fb) ` VKI(fn, fb)θ. +V2KI(fn, fθ)θ (bn - θn)                  (37)

(bn - θ) ' -(V2KI(fn, fθ)θ)-1 ∙ VKI(fn, fθ)θ.                          (38)

,b   fl∙A ('d2 log f,xi) bl Λ ('d log f *,xi) bl Λ

(bn - θ) ' -(∑   ∂θi∂θj   fn(xiζl   ∙(Σ-----∂θ-----fn(xiζl

Г b a∙A 11 X d2 logf (θ,xi) bl Λ 1 1 X d logf *,xi) bl Λ        ∕9λ

nn - θ ) ' - (n^   ∂θi∂θj   fn(xi)J   (√n∑ -----∂θ-----fn(x,)J .       (39)

Let us define s(θ,x) = d logfθθ,xi) = dff(⅛)

n(bn - θ* ) ' -


/1 X θ   fn( 1f√s X s(θ.,χi)fn(χ.A

n ∂θ              n


-(An(θ))-1Wn*).


(40)


Rewriting Wn . ) as a second order U-statistic of the form

n n                                 -1 n-1 n

Un = S⅛-Γ)Σ∑hk(xj-x)s(θ,χi)= n)  ∑∑hK(xj-xi)[s(θi)-s(θj)] (41)

n n -

i=1 j=1                                     i=1 j=i+1

j6=i

23



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