2000/10/03 by James Allen Fill, Fill, James Allen, Motoya Machida +1
Mathematics · #05C05 #05C38 (secondary) #06A06 #60E05 (primary) #60J10 #Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #math.CO #math.PR #msc:05C05 #msc:05C38 #msc:06A06 #msc:60E05 #msc:60J10
paper · pdf · doi:10.48550/arxiv.math/0010026
Presented at "Distributions with Given Marginals and Statistical Modelling," Barcelona, Spain, July 17-20, 2000. See also http://www.mts.jhu.edu/~fill/ and http://www.math.usu.edu/~machida/
arxiv created 2000/10/03 · arxiv updated 2009/11/30
A system (Pa: a in A) of probability measures on a common state space S indexed by another index set A can be ``realized'' by a system (Xa: a in A) of S-valued random variables on some probability space in such a way that each Xa is distributed as Pa. Assuming that A and S are both partially ordered, we may ask when the system (Pa: a in A) can be realized by a system (Xa: a in A) with the monotonicity property that Xa <= Xb almost surely whenever a <= b. When such a realization is possible, we call the system (Pa: a in A) ``realizably monotone.'' Such a system necessarily is stochastically monotone, that is, satisfies Pa <= Pb in stochastic ordering whenever a <= b. In general, stochastic monotonicity is not sufficient for realizable monotonicity. However, for some particular choices of partial orderings in a finite state setting, these two notions of monotonicity are equivalent. We develop an inverse probability transform for a certain broad class of posets S, and use it to explicitly construct a system (Xa: a in A) realizing the monotonicity of a stochastically monotone system when the two notions of monotonicity are equivalent.