Density Estimation and Combination under Model Ambiguity

2
_b 1       fn (xi) - g(xi)

^v2ⁿ = n M   --^l x )    ) ■

By Lemma 3.1 Hong-White (2000), under assumption A1 and A2, nh⁴/ ln n →∞,h→ 0. Then:

AT          1       1         1      JlJ         T^T∙        1 T^T∙    T J 1 1 it X      1 1 —1 T xt—x — —X ZXl       1

Now we have to analyze the terms Vin and V2n. Let define f(x) = ʃh ¹K( ⁱ- )g(x)dx and

iKh∣^χΛ-^χ)f(^χ)dx-g(^χi)
g(xi)

Then

= V11n +B

where Viin is a second order U-statistic and it will affect the asymptotic distribution of KIi. Similarly to

TT ιι<₁nnn TJ        ∙. Γ' ■   . ι 1- n ■

Hall(1984) let rewrite Vii_n in the following way:

xj) - Kh(xi - x)g(x)dx

= J                  ] — Jn(^xi^,xj) + Jn(^xj ^,xi)

g(xi)

(59)

E(H1_n (xi,xj)/xi)=0, then using Theorem 1 in Hall(1984) we can show that

_π∑ ∑Hiin(Xi,Xj)
i j,i6=j

2E [H2n(^χi^,xj

xi) ^- RKh(xi ^{- χ})g^(χ)^dχ¢²

------g₂∣_x )------------------g(xi)g(xj)dxidxj =

26

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