The behavioral meaning of the median
We extend the notion of statistical preference to the general framework of imprecise probabilities, by proposing a new notion of desirability of gambles called sign-desirability, different from the usual desirability notion in Walleys framework. We axiomatically characterize coherent families of sign-desirable gambles. We furthermore prove that the pair of lower and upper previsions of a gamble, according to this new desirability notion, coincides with the pair of bounds (infimum and supremum) of the set of medians associated to a coherent family of linear previsions. Thus, a general notion of median is naturally derived, and provided with a behavioral meaning. As a consequence of these results, the connection between statistical preference in classical Probability Theory, and the sign of the median of the difference of two random variables is laid bare.
median statistical preference imprecise probability stochastic ordering desirable gamble