- 224 Downloads
We have presented in Chapter 1 a collection of univariate orderings which are commonly used in the literature. There is also a large collection of orderings used in more dimensional spaces. For a general treatment we refer the reader to books by Marshall and Olkin (1979), Tong (1980), Dharmadhikari and Joag-Dev (1988), Shaked and Shanthikumar (1994), and references therein. In this chapter we concentrate ourselves on multivariate strong stochastic orderings because, it is possible to provide for them some coupling constructions, and they still play a dominant role in applications. We start with Strassen’s theorem which is in a sense fundamental for strong stochastic orderings. This theorem asserts the existence of almost surely comparable versions of two vectors which are strongly stochastically ordered. Unfortunately the proof is carried out in a not effective way, not giving us a method how to construct such vectors. Such constructions are possible under some sufficient conditions, and we provide in this chapter two detailed coupling constructions of almost surely comparable random vectors which lead naturally to strong stochastic ordering. These constructions give us a method for proving strong stochastic ordering in applied probability models. A similar approach will be used then to point processes.
KeywordsPoint Process Renewal Process Counting Process Replacement Policy Continuous Time Markov Chain
Unable to display preview. Download preview PDF.