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Abstract:
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The usual order (< ) on the real numbers is both additive and multiplicative. These properties are preserved when comparing these constant random variables in the usual stochastic order. However, the statements for the non-constant real-valued random variables have still been missing in the literature. To examine the existential conditions of order preserving of additivity and multiplication of stochastic orders on the real-valued random variables a set of six stochastic orders including the usual, hazard rate, moment, Laplace transform, convolution and increasing convex order were considered; and, under independence assumption their order preserving property statuses were discussed. The results indicated that while the usual, the moment and the Laplace transform order are both additive and multiplicative, the hazard rate and the increasing convex order preserve these properties partially and the convolution order is only additive. As a conclusion, additivity and multiplication order preserving status of stochastic orders over real-valued random variables vary by the type of the order.
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