Density Estimation and Combination under Model Ambiguity

(nh^1/2)KI2i(f_θθ,fθ> ) →^p 0

the entire expression for (nh^1/2)KI can be approximated in the following way

Thus, if h к n ^β with β > ∣, c_n ` C(nh) ¹

(nh¹/²) (kI + 2eə - √2σ1N1 - √2σ2N2 + αC

then,

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

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 φ_i = 1∕λ_i is given by the following expression:

Eg_r|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)

Eg_r|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)

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

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