Density Estimation and Combination under Model Ambiguity



(nh1/2)KI2i(fθθ,fθ> ) p 0

the entire expression for (nh1/2)KI can be approximated in the following way

(105)


(nh1/2)


(nh1/2) 1ιNι - (nh1/2) 12N2 - 2⅛ -


Jn1


- 2 Jn2)


(106)


Thus, if h к n β with β , cn ` C(nh) 1

(nh1/2) (kI + 2eə - √1N1 - √2N2 + αC

(107)


then,

(nh1/2) Çk+ + 2eə d N (αC, 2 (σ2 - σ2))

(108)


9.4 Formula of MGF and expected utility

It can be shown that the moment generating function for the double Gamma distribution is:

MR(t)=exp(tγ)[pMGF (t)+(1- p)MGF (-t)] =
exp(tγ)[p(1
- φ1t)-ζ1 +(1- p)(1 + φ2t)-ζ2

hence E(U(R)) where t = aδ and φi = 1∕λi is given by the following expression:

Egr|s [U (W (ret, a))] = -MR(-t) = - exp(-aδγ)[p(1 - φ1aδ)-ζ1 +(1- p)(1 + φ2aδ)-ζ2]

For the Gumbel distribution we have the following expression:

MR(t) = exp(αt)Γ(1 - βt)

Egr|s [U (W (ret, a))] = -MR(-aδ)=-exp(-αaδ)Γ(1 + βaδ)

For the Normal we have the well known result:

MR(t) = exp(tμ - 2t2σ2)

Egrls [U(W(et,a))] = -Mfi(-aδ) = exp(-aδμ + 1 a2δ2σ2).

33



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