Summary: Suppose for a distribution function F the first n moments are known. The authors [ibid. 4, 99-111 (1985; Zbl 0559.62086)] derived bounds for integrals \(\int^{\xi}_{-\infty}f(x)dF(x)\) for arbitrary values of \(\xi\) and non-negative functions f having n non-negative derivatives, like \((x_+)^{n+1}\) or \(e^ x\) or 1.
Here we refine the techniques used in that paper to obtain bounds in case the spectrum of F is required to be in [0,b] or in \({\mathbb{R}}^+\), which is often realistic in insurance applications. It is shown that the bounds obtained are precise, in the sense that extremal distributions are given attaining these bounds. These extremal distributions are (limits of) distribution functions satisfying the moment and spectrum constraints.
As an application we compare extreme values of some distributions to the value of the appropriate NP2 approximation.
Some comments by G. C. Taylor follow this paper.