It often arises in practice that, although the first few moments of a distribution are known, the density of the distribution cannot be determined in closed form. In such cases, Gram-Charlier and Edgeworth series are commonly used to analytically approximate the unknown density in terms of the known moments. Although convenient, these series contain polynomial factors, and can hence lead to density approximations taking negative values or becoming multimodal in general. Consequently it is of interest to determine the set of moments for which the corresponding density approximations are positive definite and unimodal. In contrast to the existing literature that determines the boundaries of such sets numerically, explicit analytic expressions for the two boundaries are given in this paper for the Gram-Charlier series. Moreover, a method for projecting a given set of moments onto the boundaries of the two regions in order to minimizes the Kolmogorov-Smirnov statistic of corresponding density approximations is also provided.

We develop a scoring rule tailored to a decision maker who makes simultaneous bets on events that occur at times that require bets to be placed together. The rule proposed captures the economic benefit to a well-defined bettor who acts on one set of probabilities p against a baseline or rival set q. To allow for simultaneous bets, we assume a myopic power utility function with a risk aversion parameter tailored to suit the given user or application. Our score function is “proper” in the usual sense of motivating honesty. Apart from a special case of power utility, namely, log utility, the score is not “local,” which we excuse because a local scoring rule cannot capture the economic result that our score reflects. An interesting property of our rule is that the individual scores from individual events are multiplicative, rather than additive. Probability scores are often added to give a measure of aggregate performance over a set of trials. Our rule is unique in that scores must be multiplied to reach a meaningful aggregate.