GENERALIZED STOCHASTIC CONVEXITY AND STOCHASTIC ORDERINGS OF MIXTURES

  • Authors:

  • Michel Denuit;Claude Lefèvre;Sergey Utev

  • Affiliations:

  • Institut de Statistique et de Recherche Opérationnelle, Université Libre de Bruxelles, CP 210, Boulevard du Triomphe, B-1050 Bruxelles, Belgium, mdenuit@ulb.ac.be, clefevre@ulb.ac.be;Institut de Statistique et de Recherche Opérationnelle, Université Libre de Bruxelles, CP 210, Boulevard du Triomphe, B-1050 Bruxelles, Belgium, mdenuit@ulb.ac.be, clefevre@ulb.ac.be;Department of Mathematics and Statistics, La Trobe University, Bundoora, Victoria 3083, Australia, stasu@luff.latrobe.edu.au

  • Venue:
  • Probability in the Engineering and Informational Sciences
  • Year:
  • 1999

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Abstract

In this paper, a new concept called generalized stochastic convexity is introduced as an extension of the classic notion of stochastic convexity. It relies on the well-known concept of generalized convex functions and corresponds to a stochastic convexity with respect to some Tchebycheff system of functions. A special case discussed in detail is the notion of stochastic s-convexity (s ∈ [real number symbol]), which is obtained when this system is the family of power functions {x0, x1,..., xs−1}. The analysis is made, first for totally positive families of distributions and then for families that do not enjoy that property. Further, integral stochastic orderings, said of Tchebycheff-type, are introduced that are induced by cones of generalized convex functions. For s-convex functions, they reduce to the s-convex stochastic orderings studied recently. These orderings are then used for comparing mixtures and compound sums, with some illustrations in epidemic theory and actuarial sciences.