Density Estimation and Combination under Model Ambiguity



1             I Kl K-x f κ(xx- - x) - f Kh(xi - xj )g(xi )dxj↑ f( (xi) - g(xi) -,      f,7,7×

H3ni / j an(xi, xj)bn(xi) h J                    g(x )                    y g(x )     d dx (77)

Under assumptions A1 and A2 and given that EH23ni =0, by Lemma1 in Hall(1984) we have that 2Vb23n
is asymptotically normally distributed with zero mean and variance given by:

2 ' 2 -1h4 2

σ3n ` 2n h μ2


I ,ζ⅛)'0 dxi - (/ (g(2^50 dx)


(78)


which can be easily seen if we consider that f (xi) g(xi) = h μ2g—(x)and that
g(xi)            g(xi)

EH32ni = h4μ2


I          dxi- (J (gβ,<x<)) dxi)


Also this term will affect the asymptotic distribution of κI1 .

To summarize all previous steps, we can rewrite the expansion of κI1 in the following way:

κI 1 = VzIln + Bbn - 2 ^½1n + V22n + 2V23nJ -                       (79)

N(02n)+h-μ2 / g(2)(x)dx+o(h2)-1 fσ^n + Op(n 3/2h 1) + 2N(02n) + h-μ2 [ (g  (x))

dx + o(h4) + 2N (023n) .


1n    2 2                        2 n                               2n     4 2      g(x)

Once more, following Hall(1984), from the definition of Vb21n and the fact that nh → ∞, we have that the
difference between
n(n-i) Pi Pj=i αn(xi, xj) and σgl is negligible w.r.t. 2U2n, hence the previous expression

can be rewritten as follows:


KKI1 - (nh1/2)-11N1 - (nh1^)-12N2 - n-1¾23N3 + Bbn - |cn

(80)


where N1 ,N2 and N3 are asymptotically normal N(0,1); and

σ1


κ2(u)d


u, σ2 =      κ (u)κ (u + v)du


dv and σ3 = μ2


I           dxi - μj (g(2)(xi))    1


and cn


h4

= (nh)-1   K 2(u)du + -^j-μ2


[ μ g⅛) )2 dx+0(n-1ft-1
g(x)


+ h4).


(81)


It is important to notice that Bbn, which is Op (n-1/2h2), will asymptotically cancel out with n-1/2h2 √2σ3N3,
since they are of the same order of magnitude.

Thus, we have the following results: asn →∞,h0, nh →∞ and nh5 0

nh1/2(jcI 1 + 2Cn) d ιNι - √3N2.

(82)


29



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