Realizable monotonicity and inverse probability transform
Abstract
A system (P_a: a in A) of probability measures on a common state space S indexed by another index set A can be ``realized'' by a system (X_a: a in A) of S-valued random variables on some probability space in such a way that each X_a is distributed as P_a. Assuming that A and S are both partially ordered, we may ask when the system (P_a: a in A) can be realized by a system (X_a: a in A) with the monotonicity property that X_a <= X_b almost surely whenever a <= b. When such a realization is possible, we call the system (P_a: a in A) ``realizably monotone.'' Such a system necessarily is stochastically monotone, that is, satisfies P_a <= P_b 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 (X_a: a in A) realizing the monotonicity of a stochastically monotone system when the two notions of monotonicity are equivalent.
Cite
@article{arxiv.math/0010026,
title = {Realizable monotonicity and inverse probability transform},
author = {James Allen Fill and Motoya Machida},
journal= {arXiv preprint arXiv:math/0010026},
year = {2007}
}
Comments
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/