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Realizable monotonocity and inverse probability transform. (English) Zbl 1139.60304
Cuadras, Carles M. (ed.) et al., Distributions with given marginals and statistical modelling. Papers presented at the meeting, Barcelona, Spain, July 17–20, 2000. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0914-3/hbk). 63-71 (2002).
A system $$(P_{\alpha}$$: \alpha \in A)$$of probability measures on a common state space$$S$$indexed by another index set$$A$$can be realized'' by a system$$(X_{\alpha} : \alpha \in A)$$of$$S$$-valued random variables on some probability space in such a way that each$$X_{\alpha}$$is distributed as$$P_{\alpha}$$. Assuming that$$A$$and$$S$$are both partially ordered, we may ask when the system$$(P_{\alpha}: \alpha \in A)$$can be realized by a system$$(X_{\alpha}: \alpha \in A)$$with the monotonicity property that$$X_{\alpha}\leq X_{\beta}$$almost surely whenever$$\alpha \leq \beta$$. When such a realization is possible, we call the system$$(P_{\alpha}: \alpha \in A)$$realizably monotone.'' Such a system necessarily is stochastically monotone, that is, satisfies$$P_{\alpha}\leq P_{\beta}$$in stochastic ordering whenever$$\alpha \leq \beta$$. In general, stochastic monotonicity is not sufficient for realizable monotonicity. However, for some particular choices of partial orderings in a finite state setting, these two notions of monotonicity are equivalent. We develop an inverse probability transform for a certain broad class of posets$$S$$, and use it to explicitly construct a system$$(X_{\alpha} : \alpha \in A)$$realizing the monotonicity of a stochastically monotone system when the two notions of monotonicity are equivalent.$$
For the entire collection see [Zbl 1054.62002].

##### MSC:
 6e+16 Inequalities; stochastic orderings