Convex extrema for nonincreasing discrete distributions: Effects of convexity constraints

Abstract

In risk management, the distribution of underlying random variables is not always entirely known. Sometimes, only the mean value and some shape information on the distribution (decreasingness, convexity . . .) are available. The present paper provides discrete convex extrema for several situations of this type. The starting point is the class of discrete distributions whose probability mass functions are nonincreasing on a support D n ≡ {0, 1, . . . , n}. Convex extrema in that class of distributions are well-known. Our purpose is to point out how additional shape constraints of convexity type modify these extrema. Two cases are considered: the p.m.f. is globally convex on N, or it is convex only from a given point m ≥ 1. The corresponding convex extrema are derived by using simple crossing properties between two distributions. Several applications to insurance problems are briefly presented. The results provide a complement to recent works by Lefèvre and Loisel [9,10].

Testing k-monotonicity of a discrete distribution. Application to the estimation of the number of classes in a population

“…Such approach may be well adapted in some situations because it combines the straightforwardness of parametric models (no choice of parameter is left to the user) and the great flexibility of nonparametric estimation. Moreover shape constraint arises naturally in many frameworks such as insurance [24], reliability studies [29], epidemiology [3] or ecology [14,15].…”