In this paper, we study the estimation of a scale parameter from a sample of lifetimes of coherent systems with a fixed structure. We assume that the components are independent and identically distributed having a common distribution which belongs to a scale parameter family. Some results are obtained as well for dependent (exchangeable) components. To this end, we will use the representations for the distribution function of a coherent system based on signatures. We prove that the efficiency of the estimators depends on the structure of the system and on the scale parameter family. In the dependence case, it also depends on the baseline copula function.

This contribution shows that for certain classes of insurance risks, pricing can be based on expected values under a probability measure ℙ* amounting to quasi risk-neutral pricing. This probability measure is unique and optimal in the sense of minimizing the relative entropy with respect to the actuarial probability measure ℙ, which is a common approach in the case of incomplete markets. After expounding the key elements of this theory, an application to a set of industrial property risks is developed, assuming that the severity of losses can be modeled by “Swiss Re Exposure Curves”, as discussed by Bernegger (1997). These curves belong to a parametric family of distribution functions commonly used by pricing actuaries. The quasi risk-neutral pricing approach not only yields risk exposure specific premiums but also Risk Adjusted Capital (RAC) values on the very same level of granularity. By way of contrast, the conventional determination of RAC is typically considered on a portfolio level only.

For nonnegative random variables X and Y we write X ≤TTTY if ∫0F-1(p)(1-F(x))dx ≤ ∫0G-1(p)(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spread) order, and of other related stochastic orders, are also obtained. Applications in the statistical theory of reliability and in economics are included.