Technical Note—Ranking Distributions When Only Means and Variances Are Known.

In "Technical Note—Ranking Distributions When Only Means and Variances Are Known," Müller, Scarsini, Tsetlin, and Winkler address the question of ranking distributions when only the first two moments—that is, means and variances—are known. This is important in decision making under uncertainty, with potential applications in economics, finance, statistics, and other areas. Previous results require some assumptions about the shape of the distributions, while this paper's approach is to impose bounds on how much marginal utility can change, thus constraining risk preferences. Such a ranking is consistent with almost stochastic dominance and provides a new connection between the Sharpe and Omega ratios from finance. Consider a choice between two random variables, for which only means and variances are known. Is it possible to rank them by putting some constraints on risk preferences? We provide such a ranking by bounding how much marginal utility can change. Such bounds enable us to rank all distributions with given means and variances by first-order almost-stochastic dominance. We show how our results can be used to compare a risky project and a sure payoff and also provide a new connection between the Sharpe and Omega ratios from finance. [ABSTRACT FROM AUTHOR]