Density Estimation and Combination under Model Ambiguity

So what we have to study is the asymptotic behavior of

_   ____ ,,,  2^,._, ,,,  2^,._,,..  2^,._, ,,,

√nUn = √nE(rn(x_i)) + — Vrn(Xi) — E(rn(xi)) = —= Vr(xi) — E(r(xi)) + — V∖(x_i) — E(tn(xi))
n  nn

i   ii

(51)
where r(xi) = [s(θ*, xi) — v(θ*, xi)] g(xi) and E(r(zi)) = E(E [((s(θ*, xi) — v(θ*, xi))g(xi)) /xi]) = 0 and the
last term of the above expression converges to zero in probability. Hence, the limiting distribution of √nU_n
is the same of √n Pi r(xi) = -√- Pi [s(θ*,x_i) — v(θ*,xi)] g(xi).

By Lindeberg-Levy central limit theorem, we have that

W_n(θ*) = √nU_n →^d N(0, B(θ*)) as n → ∞

B(θ*) = 4E([s(θ*,xi) — v(θ*,Xi)]² g(xi)²) = 4 / (s²(θ*,Xi) + (v(θ*,Xi))2 — 2s(θ*, x_i)v(θ*, x_i)) g(xi)³dx =

This implies that W_n(θ*) = Op (^√n) . It follows that

√n(bn — θζ) = —(An(θ))^-1Wn(θ*) → N(0,A(θ*))^-1B(θ*)A(θ*))^-1).                (54)

9.3 Proof Theorem 3:

KI can be rewritten in the following way:

f ,         , .             , . . ^ , .        ∕* ,         ,.           , . . ^ , . f , ...           , . . ^ , .

KI = ( (lnc(x)—lnfθ(x))dFn(x) = I (lnc(x)—lng(x))dFn(x)—/ (lnfg(x)—lng(x))dFn(x) = KI1—KI2.

^{x                                  x}                                                                        (55)

Similarly to Fan(1994), this representation is very helpful to examine the effect of estimating fθ_ψ by fg on
the limiting distribution of KI. From now on the index j for th model will be omitted.
c

ζn-X_χ ⁱ) j that by the Law of Large Numbers

¹E((ln fcn (x) — ln g(x)) = n^-1KI1. This first part of

the proof draws heavily upon Hall(1984) and Hong and White(2000).

Using this inequality ∣ln(1 + u) — u + 21 u²∣ ≤ |u|³ for |u| < 1 and defining u
obtain the following result:

25

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