Ordering univariate distributions by entropy and variance

This paper examines the role of variance and entropy in ordering distributions and random prospects. There is no universal relation between entropy and variance orderings of distributions. But we place their relationship in the context of a stronger ordering relation known as dispersion ordering. Further, some conditions are identified under which variance and entropy order similarly when continuous variables are transformed. We also analyze parametric changes which do not disturb the agreement between these rankings. The results are conveniently tabulated in terms of distribution parameters.