Generalized stochastic convexity and stochastic orderings of mixtures

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 is an element of N), which is obtained when this system is the family of power functions {x(0),x(1),..., x(s-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.