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On s-convex approximations

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  01 July 2016

Michel Denuit ,
Claude Lefèvre  and
Moshe Shaked
Michel Denuit*
Affiliation:
Université Catholique de Louvain
Claude Lefèvre*
Affiliation:
Université Libre de Bruxelles
Moshe Shaked*
Affiliation:
University of Arizona
*
∗ Postal address: Institut de Statistique, Université Catholique de Louvain, Voie du Roman Pays, 20, B-1348 Louvain-la-Neuve, Belgium.
∗∗ Postal address: Institut de Statistique et de Recherche Opérationnelle, Université Libre de Bruxelles, CP 210, Boulevard du Triomphe, B-1050 Bruxelles, Belgium.
∗∗ Postal address: Department of Mathematics, University of Arizona, Tucson, Arizona 85721, USA. Email address: shaked@math.arizona.edu
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Abstract

Let ℬs([a,b];μ1,μ2,…,μs-1) be the class of all distribution functions of random variables with support in [a,b] having μ1,μ2,…,μs-1 as their first s-1 moments. In this paper we examine some aspects of the structure of ℬs([a,b];μ1,μ2,…,μs-1) and of the s-convex stochastic extrema in it. Using representation results of moment matrices à la Lindsay (1989a), we provide conditions for the admissibility of moment sequences in ℬs([a,b];μ1,μ2,…,μs-1) in terms of lower bounds on the number of support points of the corresponding distribution functions. We point out two special distributions with a minimal number of support points that are the s-convex extremal distributions. It is shown that the support points of these extrema are the roots of some polynomials, and an efficient method for the complete determination of the distribution functions of these extrema is described. A study of the goodness of fit, of the approximation of an arbitrary element in ℬs([a,b];μ1,μ2,…,μs-1) by one of the stochastic s-convex extrema, is then given. Using standard ideas from linear regression, we derive Tchebycheff-type inequalities which extend previous results of Lindsay (1989b), and we establish some limit theorems involving the moment matrices. Finally, we describe some applications in insurance theory, namely, we provide bounds 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 aid of the s-convex extrema, and are determined only by the support and the first few moments of the underlying distribution.

Keywords

Moment spacess-convex ordersstochastic extremastochastic approximationsTchebycheff-type inequalitiesrisk theoryLundberg's coefficientlife insurance

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 4 , December 2000 , pp. 994 - 1010
Copyright
Copyright © Applied Probability Trust 2000 

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