Ordered random vectors and equality in distribution

Ka Chun Cheung, Jan Dhaene, Alexander Kukush and Daniël Linders

Scandinavian Actuarial Journal, 2015, vol. 2015, issue 3, 221-244

Abstract: In this paper we show that under appropriate moment conditions, the supermodular ordered random vectors X¯=(X1,X2,…,Xn)$ \underline{X}=\left( X_{1},X_{2},\ldots ,X_{n}\right) $ and Y¯=(Y1,Y2,…,Yn)$ \underline{Y}=\left( Y_{1},Y_{2},\ldots ,Y_{n}\right) $ with equal expected utilities (or distorted expectations) of the sums X1+X2+…+Xn$ X_{1}+X_{2}+\ldots +X_{n} $ and Y1+Y2+…+Yn$ Y_{1}+Y_{2}+\ldots +Y_{n} $ for an appropriate utility (or distortion) function, must necessarily be equal in distribution, that is X¯=dY¯$ \underline{X}\overset{\text{ d}}{=}\underline{Y} $. The results in this paper can be considered as generalizations of some recent results on comonotonicity, where necessary conditions related to the distribution of X1+Xn+…+Xn$ X_1 + X_n+ \ldots + X_n $ are presented for the random vector X¯=(X1,X2,…,Xn)$ \underline{X}=(X_1, X_2,\ldots ,X_n) $ to be comonotonic.

Date: 2015
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DOI: 10.1080/03461238.2013.807470

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