Guia informativa sobre el sexismo y la violencia
5.3. El reconocimiento de los derechos de la mujer en las Naciones Unidas
The following two observations for the generic HARA utilities closely matches Jose et al. [2008]’s results for LRT utilities;
§A.2 HARA and the power and pseudo-spherical scores and divergences 90
happens to be affinely related to weighted Pseudo-spherical Score.
u(W−N+xk) =u I(λqpk k) = β−11 pka qkλ β−1 −1 = β−11 (b+βaW) β−1 β p k/qk (Σkqk(pk/qk)β)1/β β−1 −1 , (by using λ a β = Σkqk(pk/qk)β b+βaW ) = β−11 (b+βaW) β−1 β [(β−1)SS β(p,q,k) +1]−1 = (b+βaW) β−1 β SS β(p,q,k) +u(W)
2)The constrained maximum expected utility (MEU) is affinely related to the pseudo- spherical divergence, which in turn is monotonically related to the power divergence (oral pha-divergence)
UW(p||q) =Σkpku(W−N+xk) = β−11 (b+βaw) β−1 β Σ kqk(pk/qk)β 1/β −1 = β−11 (b+βaW) β−1 β [(β−1)DS β(p,q) +1]−1 = (b+βaW) β−1 β DS β(p,q) +u(W)
Further we can show that the u∗λ-divergence is also affinely related to the pseudo- spherical divergence. u∗λ(p||q)=Σkpku∗(λqpk k)(Or=UW(p||q)−λW) =Σkpk " λqk apk 1−β β(β−1) + λqkb βapk − 1 β−1 # , (usingu∗(y) = (y/a)1−β β(β−1) + yb βa− β−11) =Σkpk " apk λqk β−1 β(β−1) # + λbβa − β−11 ... ... = ( (b+aW) b+βaW)1/β 1 (β−1) Σkqk(pk/qk)β 1/β − β−11, (by using λ a β = Σkqk(pk/qk)β b+βaW ) = ( (b+aW) b+βaW)1/β 1 (β−1)[(β−1)D S β(p,q) +1]− 1 β−1 = ( (b+aW) b+βaW)1/βD S β(p,q) + 1 β−1 h ( b+aW) (b+βaW)1/β −1 i
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