Stochastic Convexity and Concavity

Abstract

In this chapter we study stochastic monotonicities of parametric families of distributions with respect to various stochastic orders. We have already encountered stochastic monotonicities earlier in this book. For example, condition (1.A.13) in Theorem 1.A.6, condition (3.A.47) in Theorem 3.A.21, and condition (4.A.17) in Theorem 4.A.18 describe such monotonicities. In this chapter a systematic study of such stochastic monotonicities is given. Various notions of stochastic convexity and concavity are reviewed. A multivariate extension of the notion of stochastic convexity, namely, stochastic directional convexity, is investigated in this chapter as well.