New metrics for risk analysis

This paper introduces a new framework for risk analysis for distributions of finite mean, building on metrics $\mu_s$ indexed by $s\in {\mathbb R}.$ The neutral metric $\mu_0$ can be written as a simple linear combination of the mean and the cumulative entropy. The sequence $\{ \mu_n, \; n\ge 1\}$ characterizes distributions up to translation. The order derived from these metrics respects the usual stochastic order. The range of the metric is described explicitly for positive random variables and in the case of finite variance, with a unique maximizer up to affine transformation. Along the way, we obtain a characterization of the logistic and the exponential distribution by the cumulative entropy. Our metrics are then embedded in a generic risk analysis framework that entails dual properties and provides for interval screening and operators variations. Contrary to the existing literature, this framework does not parametrize risk with a quantile. This instead integrates information along all possible risk levels and assigns weight to each of them, yielding an alternative approach to risk understanding.