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



More intriguing information

1. The Impact of Financial Openness on Economic Integration: Evidence from the Europe and the Cis
2. The name is absent
3. Foreign Direct Investment and Unequal Regional Economic Growth in China
4. Tax systems and tax reforms in Europe: Rationale and open issue for more radical reforms
5. AN IMPROVED 2D OPTICAL FLOW SENSOR FOR MOTION SEGMENTATION
6. The name is absent
7. The name is absent
8. The Mathematical Components of Engineering
9. Financial Market Volatility and Primary Placements
10. The name is absent