Density Estimation and Combination under Model Ambiguity

Hn(xi,Xj ) = h K ( x^j

x

-ⁱ)[s(θ∙,xi) — s(θ∙,xj∙)].

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First, we need to verify that E ∣j∣H_n(xi,Xj)∣∣²j = o(n). Let define v²(θ∙,x) = E(s²(θ∙, x)/x) and v(θ*,x) =
E(s(θ^∙, x)/x),

Hn(xi,xj)∣∣²∣ = E E ^∣Hn(xi,xj)∣² /xi,xj∙)] =

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1     ^°    xj

= h2   K⁽ -j

— xi
h

°₂

)   [v²(θ*,xi) + v²(θ∙,xj) — 2v(θ∙,xi)v(θ*,xj)] g(xi)g(xj)dxidxj =

now using the change of variable from (xi, xj ) to (xi, u = —

—) we obtain

= — ʃ ∣∣K(u)∣∣² £v²(θ*, xi) + v²(θ*, xi + hu) — 2v(θ∙, xi)v(θ*, xi + hu)] g(xi_ι)g(xi + hu)dxihdu =

= O(—) = O(n(nh) ¹ ) = o(n) since nh → ∞.
h

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This implies that ʌ/-(U_n — U_n) = o_p(1). Thus, we need just to study the behavior of U_n which is given by

2

Un = E(rn(xi)) + - 2^rn(xi) — Ε(rn(xi)).

i

Let compute rn (xi) which is defined in the following way:

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where

and

= — K K(u)[s(θ∙,xi) — v(θ∙,xi + hu)] g(xi + hu)hdu = r(xi) + t_n(xi)
^h

r(xi) = [s(θ^∙, xi) — v(θ^∙, xi)] g(xi)

^tn(^xi) =       ,θ θ. ^u2K (^u)d^u = h

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This last expression has been obtained applying the mean value theorem to v(θ*, Xi + hu) and g(xi + hu),
whichyields v(θ*,Xi) + huv⁰(xi, θ*) and g(θ*,xi) + hug⁰(xi,θ*) where xi_t lies in [xi,xi + hu].

Further, we need to compute E(r_n (xi)) = E(H_n(xi, xj))

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