We show that the asymptotic mean of the log-likelihood ratio in a misspecified model is a differential geometric quantity that is related to the exponential curvature of Efron (1978), Amari (1982), and the preferred point geometry of Critchley et al. (1993, 1994). The mean is invariant with...

Stochastically ordered random variables with given marginal distributions are combined into a joint distribution preserving the ordering and the marginals using a maximum entropy formulation. A closed-form expression is obtained. An application is in default estimation for different portfolio...