Supermodular Stochastic Orders and Positive Dependence of Random Vectors

Abstract

The supermodular and the symmetric supermodular stochastic orders have been cursorily studied in previous literature. In this paper we study these orders more thoroughly. First we obtain some basic properties of these orders. We then apply these results in order to obtain comparisons of random vectors with common values, but with different levels of multiplicity. Specifically, we show that if the vectors of the levels of multiplicity are ordered in the majorization order, then the associated random vectors are ordered in the symmetric supermodular stochastic order. In the non-symmetric case we obtain bounds (in the supermodular stochastic order sense) on such random vectors. Finally, we apply the results to problems of optimal assembly of reliability systems, of optimal allocation of minimal repair efforts, and of optimal allocation of reliability items. Academic Press

The Impact of Lehmann’s Work in Applied Probability

“…Some authors considered defining a positive dependence comparison of random vectors by comparing probabilities of lower or upper orthants in IRn; see, for example, Joe [11]. Others, such as Bauerle [2] and Shaked and Shanthikumar [25], studied a stronger integral stochastic order, which compares the expectations of supermodular functions of the compared random vectors.…”

Comparison Results for Branching Processes in Random Environments

“…Joe (1997), Shaked and Shanthikumar (1997), or Bäuerle and Rieder (1997)). In this paper, we consider two of these dependence orders, whose definitions are given here.…”

When Are On–off Sources Sis?: Conditions and Applications

“…Several stochastic orderings have been found well suited for comparing the dependence structures of random vectors. Here we rely on the supermodular ordering which has recently been used in several queueing and reliability applications [3,4,16]. We begin by introducing the class of functions associated with this ordering.…”

“…Additional information on the sm ordering can be found in [3,4,11,12,16,20]. In Sections 7 and 8 we shall need the fact that the sm ordering is closed under convolution.…”

“…On the theoretical side, when considering an associated input stream [Definition 4.1], the effective bandwidth calculations [5] [6] lead naturally to an asymptotic version of this fact. Recently, this "folk theorem" has received a more formal grounding with the help of ideas from the theory of multivariate stochastic orderings established by Meester and Shanthikumar [11] and by Shaked and Shanthikumar [16] where input sequences to the Lindley recursion (1) are compared in the supermodular (sm) ordering [Definition 3.3] and the buffer contents in the increasing convex (icx) ordering [Definition 3.2]. The sm ordering is well suited to capture positive dependence in the components of a random vector, while the icx ordering formalizes comparability in terms of variability and size.…”