On s-convex approximations

Let B-s([a, b]; mu (1), mu (2),...,mu (s-1)) be the class of all distributi on functions of random variables with support in [a, b] having mu (1), mu ( 2),..., mu (s-1) as their first s - 1 moments. In this paper we examine som e aspects of the structure of B-s ([a, b]; mu (1), mu (2),..., mu (s-1)) an d of the s -convex stochastic extrema in it. Using representation results o f moment matrices Li la Lindsay (1989a), we provide conditions for the admi ssibility of moment sequences in B-s ([a, b]; mu (1), mu (2),..., mu (s-1)) in terms of lower bounds on the number of support points of the correspond ing distribution functions. We point out two special distributions with a m inimal number of support points that are the s-convex extremal distribution s. It is shown that the support points of these extrema are the roots of so me polynomials, and an efficient method for the complete determination of t he distribution functions of these extrema is described. A study of the goo dness of fit, of the approximation of an arbitrary element in B-s ([a, b]; mu (1), mu (2),..., mu (s-1)) by One of the stochastic s-convex extrema, is then given. Using standard ideas from linear regression, we derive Tchebyc heff-type inequalities which extend previous results of Lindsay (1989b), an d we establish some limit theorems involving the moment matrices. Finally, we describe some applications in insurance theory, namely, we provide bound s on Lundberg's coefficient in risk theory, and on the actual interest rate relating to a life insurance policy. These bounds are obtained with the ai d of the s-convex extrema, and are determined only by the support and the f irst few moments of the underlying distribution.