Density Estimation and Combination under Model Ambiguity

ki 22(fθ* ,g) =

1 XX μ fθ* (xi) - g(xi) v b (_x )
n M   g(xi)     f

1 X <fθ* (xi) - g(xi)A² b, ʌ ₁     1 _τ

^- 2n Σ —g(χi)— f ^fn(xi) = Jn^1- 2J∙2∙

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E Jin(fθ* ,g)) = E

K(u) (fθ* (x) — g(x)) g(x + hu)dxdu.

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Applying the same steps used for S2n we can show that

lim sup E
n→∞

(Jin(^fθ*^,g)) ≤ У

(fθ* (x) — g(x)) g(x)dx = E (fθ* (x) — g(x))

=E

(^fθ* (xi) ^- g(xi))²

g(xi)

f_n(xi)dxi I = (( K(u)    ( ) g^ g( g(x + hu)dxdu,

g(x)

lim sup E (J₂n(fθ* ,g)) ≤ ( (fθ∙ (x) — g(x))² dx

n→∞

lim E (J2n(fθ* ,g))= ( (fθ* (x) — g(x))² dx > 0.

n→∞

Then also J2n(fθ* ,g) = Op(1). This implies that KI22(fθ* ,g) = Jni — 2 Jn2 = Op(1).
Then it is clear that given assumptions A1-A4, if h → 0,nh→∞then

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KI22(fθ* ,g) →^p E (fθ* (x) — g(x)) — 2 У (fθ* (x) — g(x))² dx = E [lnfθ* — lng],

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this implies that nh^1/2 KTI 22 →^p ∞, hence we need to rescale it by d_n = n^-1h^-1/2 where d_n → 0 as
This is embodied in assumption A6:

n →∞.

Finally we can put all terms together:

KI =

(ln

x

KTI22 ` ah^1/2Cn

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—~~                           —~~                 —~~

fTn(x) — lnfb_θ(x))fTn(x)dx =⁼ KTI1

—-~.

— KTI2

(nh¹/²) ¹√2σιNι — (nh¹/²) ¹√2σ2N2 — 2 Cn

^KI21(^fb^{, f}θ* ) ^{+ KI}22(^fθ* ^, g)J ^,

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