Density Estimation and Combination under Model Ambiguity

as n -→ ∞, f_n - g -→^p 0, then D21 -→ 0

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

We can conclude that KIj_n (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 KIj_n (θ) can be rewritten as H_n (θ) - H_n(f_n), where

nn

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

Hn(θ1) - Hn(θ2) = [Hn(θ1) -Hn(fcn)] - [Hn (θ2) -Hn(fcn)].

It follows that

Hn(θ1) - Hn(θ2) →^p KIj(θ1) - KIj(θ2).                               (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 θM_j 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 (θ^,xi) bl Λ (∖'^d log ^f (θ*^,xi) bl Λ

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

Г b a∙A 1¹ X ^d2 log^f (^θ,xi) bl Λ ^{1 1} X ^d log^f (θ*^,xi) b_l Λ        ∕₉∩_λ

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

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

Rewriting W_n (θ^. ) as a second order U-statistic of the form

n n                                 -1 n-1 n

Un = S⅛-Γ)Σ∑_h^k(xj-x)s(θ∙^,χi)= n)  ∑∑_hK(^xj-xⁱ)[s(θ∙^,χi)-s(θ∙^,χj)] (41)

n n -

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

j6=i

23

<<   21   22   23   24   25   26   27   >>

More intriguing information