R E S E A R C H
Open Access
When is an integral stochastic order
generated by a poset?
María Concepción López-Díaz
1and Miguel López-Díaz
2**Correspondence: [email protected] 2Departamento de Estadística e I.O. y D.M., Universidad de Oviedo, C/Calvo Sotelo s/n, Oviedo, E-33007, Spain
Full list of author information is available at the end of the article
Abstract
Given a partial orderon a setX, one can consider the class of-preserving real functions onXcharacterized byxyimpliesf(x)≤f(y). Such a class of functions allows us the generation of a binary relationgon the set of probabilities associated
withXby means ofPgQwhen
f dP≤f dQfor all-preserving functionsf. In this paper we characterize when for an integral stochastic orderon the set of probabilities associated withX, there exists a partial orderonXsuch that the relationggenerated by the class of-preserving functions is equal to. The above
characterization is related to the maximal generator of, a result which can be applied for the search of maximal generators of stochastic orders generated by posets.
MSC: 06A06; 60E15
Keywords: poset; stochastic order;-preserving function;-comonotonic function; maximal generator
1 Introduction
Stochastic orders play a key role in many areas. They have been applied successfully in such fields as reliability theory, economics, decision theory, queueing systems, scheduling problems, medicine, genetics,etc.A stochastic order is defined as a partial order relation on a set of probabilities associated with a certain measurable space, although in some contexts the antisymmetric condition is not considered.
A method for generating binary relations on a class of probabilities associated with a set by means of partial order relations on such a set is proposed in []. That method is based on the so-called order-preserving functions. Thus, if (X,A) is a measurable space andis a partial order onX, one can take the class of all measurable-preserving real functions, that is, the set of measurable functionsf :X→Rsuch that ifx,y∈X satisfyxy, then
f(x)≤f(y). The partial ordergenerates a binary relationg on the set of probabilities on (X,A) defined by
PgQ when
f dP≤
f dQ
for all measurable-preserving functionsf.
In this paper we characterize when, for a stochastic order, there exists a partial order
onX such thatg is. The above characterization will be related to the maximal
generators of integral stochastic orders. Different examples of stochastic orders generated and not generated by partially ordered sets will be developed.
2 Preliminaries
LetX be a set. A binary relationonX which is reflexive, transitive and antisymmetric is called apartial order. The pair (X,) is said to be aposet.
A mappingf :X→Ris said to be-preservingif givenx,y∈Xwithxy, thenf(x)≤
f(y).
The reader is referred, for instance, to [] and [] for an introduction to the theory of ordered sets.
LetAbe aσ-algebra onX. It will be assumed throughout the paper that for allx∈X
we have that{x} ∈A.
LetF={f:X→R|f is measurable and -preserving}.
Let us denote byPthe set of probabilities on the measurable space (X,A). A binary relationonPis said to be astochastic orderif (P,) is a poset.
A stochastic orderonP is said to beintegralif there exists a classRof measurable mappings fromXtoRsatisfying thatPQif and only if
f dP≤
f dQ
for allf ∈Rsuch that the above integrals exist. The classRis said to be ageneratorof. It is well known that there could be different generators of the same stochastic order.
The reader is referred to [, ] and [] for a rigorous introduction to stochastic orders and integral stochastic orders.
The class of measurable-preserving functions generates a binary relation onP, de-noted byg, as follows: givenP,Q∈P, thenPgQwhen
f dP≤
f dQ
for all measurable-preserving functionsf for which both integrals exist.
3 Main results
Given a stochastic orderonP, we will obtain a sufficient and necessary condition for the existence of a partial orderonXsuch thatgandare the same order. It is immediately seen that for such a condition, the stochastic ordershould be integral. Moreover, we connect the above result with maximal generators of integral stochastic orders.
In the first place, we construct a partial order onXby means of a generator of the integral stochastic order.
Proposition Letbe an integral stochastic order onP andRbe a generator of the order.We define the binary relationRonX as follows.Let x,y∈X.Then xRy when f(x)≤f(y)for all f ∈R.It holds that(X,R)is a poset.
Proof Obviously,Ris reflexive and transitive. Letx,y∈XwithxRyandyRx. The first condition implies thatf(x)≤f(y) for allf ∈R, or equivalently,
f dPx≤
where Pw∈P denotes the probability distribution degenerated at the pointw∈X. As a consequence, we obtain thatPxPy. In a similar way, we have thatPyPx. Sinceis antisymmetric, it holds thatPx=Py. Now, note that{x} ∈A, which implies thatx=y.
Now, we show that the relation defined in Proposition does not depend on a particular generator of the order, but on the order itself.
Proposition Letbe an integral stochastic order onPandRandRbe generators of
the order.ThenRandR are the same partial order.
Proof Let us suppose thatxRy. Thus,f(x)≤f(y) for allf ∈R, or equivalently,
f dPx≤
f dPy for allf∈R,
that is,PxPy. SinceRis a generator of, we have that
g dPx≤
g dPy for allg∈R.
Therefore, g(x)≤g(y) for all g ∈R, which means that xR y. So, xRy implies
xR y. Reasoning in a similar way, we obtain thatxR yimpliesxRy, and so the
result is proved.
Givenan integral stochastic order, we will denote bythe partial order onX deter-mined by, that is,xywhenf(x)≤f(y) for all mappings of a generator of. Proposi-tion says that such an order is well defined.
Next, we prove thatPgQimpliesPQ.
Proposition Letbe an integral stochastic order onP.Letbe the partial order onX
determined by.Then PgQ implies PQ.
Proof LetP,Q∈P withPgQ, that is,
f dP≤
f dQ
for all measurable-preserving functionsf. LetRbe a generator of. In accordance with the definition of the partial order, any mappingg∈Ris-preserving; as a consequence,
g dP≤
g dQ
for anyg∈R, that is,PQ.
Example Consider the measurable space (R,B),B being the usual Borelσ-algebra onR. Letbe the integral stochastic order given by the generator
R={f :R→R|f is convex}
usually referred to as the convex order. Thus, the partial orderonRdetermined by is given byxywhenf(x)≤f(y) for allf ∈R. It is seen that{z∈R|xz}={x}for any
x∈R. Therefore, any measurable maph:R→Ris-preserving, and sog andare not the same order. In fact,PgQif and only ifP=Q.
The next proposition shows that if there exists a partial order which generates the stochastic order, then the partial orderonXdetermined byalso generates.
Proposition Letbe an integral stochastic order onPandbe the partial order onX determined by.Letbe a partial order onX such thatg andare equal.Theng
andare the same stochastic order.
Proof We have thatF is a generator of. In accordance with Proposition , it holds thatxywhenf(x)≤f(y) for allf ∈F.
Let us suppose thatxy. It implies thatf(x)≤f(y) for allf ∈F and soxy. Now, note that iff ∈Fandxy, thenxyand sof(x)≤f(y), that is,f ∈F. As a conse-quence, we have thatF⊂F, which derives thatPgQimpliesPgQ. Now,
Propo-sition provides the result.
We introduce the following definition which will be key for our purposes.
Definition Let (X,) be a poset. The mappings f,g : X → R are said to be
-comonotonicif there are no two elementsx,x∈X with x x, f(x) <f(x) and
g(x) >g(x).
Now, we obtain sufficient and necessary conditions to guarantee thatg andare the same stochastic order.
Theorem Letbe an integral stochastic order onP andbe the partial order onX determined by.Thenandg are the same order if and only if there exists a
gener-atorRofsatisfying that if g:X→Ris a measurable mapping such that g and f are -comonotonic for all f ∈R,then g∈R.
Proof In the first place, let us see thatandgare the same order under the existence of a generator satisfying the above condition.
In Proposition we have shown that Pg Q impliesPQ. Now, let P,Q∈P with
PQ. Let g be a measurable -preserving function and f ∈R. Clearly, f and g are
-comonotonic, thusg∈R, and so
g dP≤
g dQ
Let us prove the converse. Ifg andare the same order, we have thatR={f :X →
R|f is measurable and -preserving}is a generator of. Let us see that this generator satisfies the condition of the statement.
Letg:X→Rbe a measurable mapping satisfying thatgandf are-comonotonic for allf∈R. Let us see that givenx,y∈Xwithxy(x=y) there exists a mappingl∈Rwith
l(x) <l(y), which implies thatg(x)≤g(y), and sogis-preserving, that is,g∈R.
If the above result is false, there existx,y∈X withx=ysuch thatxyandl(x) =l(y) for alll∈R. This implies that
l dPx=
l dPy for alll∈R.
The antisymmetric property ofimplies thatPx=Py, which is a contradiction withx=y
since{x} ∈A.
Let us consider examples of integral stochastic orders which are in fact generated by posets.
Example Consider the measurable space (R,B). Letbe the usual stochastic order, that is,PQwhen
I(t,∞)dP≤
I(t,∞)dQ
for anyt∈R, whereIAstands for the indicator function of the setA. According to Propo-sition , the partial orderonRdetermined byis given byxywhenx≤y.
It is known that the class of non-decreasing functions is a generator of . LetRbe that class. Ifgis measurable and-comonotonic with any map ofR, theng∈R, and in accordance with Theorem ,gandare the same order.
Example Consider the space (R,B). Letbe the bidirectional order (see [] and []). It is known thatPQif and only if
I(t,∞)dP≤
I(t,∞)dQ and
I(–∞,–t)dP≤
I(–∞,–t)dQ
for anyt∈(,∞). The definition of the order provides the partial orderonRdetermined by. We obtain thatxywhen
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x≤y ifx> ,
y≤x ifx< ,
y∈R ifx= .
We will analyze some examples of multivariate stochastic orders after Theorem . Note that, in general, if R is a generator of a stochastic order and g and f are
-comonotonic functions for allf ∈R,g is not necessarily an element ofR. Consider the generator of Example , there are comonotonic functions with the maps of the class of convex functions which are not convex. In fact, any mappingg:R→Ris-comonotonic with all functions ofR={f:R→R|f is convex}.
Now we are proving that under an appropriate framework, a generator satisfying the condition of Theorem is the maximal generator of the order.
We briefly describe the concept of a maximal generator of an integral stochastic order (see [] or []). Such a concept is associated with the so-calledweight function. That func-tion is a measurable mappingb:X→[,∞). The weight function determines the space of mappings in which we are looking for the maximal generator. Such a space will be the class of measurable mappings with boundedb-norm, where theb-normof a mappingf :X→R is
fb=sup x∈X
|f(x)|
b(x).
For our purpose, it is sufficient to considerb= . In this way, theb-norm of any mapping
f :X→Ris equal to
fb=sup x∈X
f(x) .
So, ifBdenotes the set of measurable functionsf :X→Rwith the finiteb-norm,Bis the set of all measurable and bounded mappings.
Themaximal generatorof an integral stochastic orderis the set of all functionsf∈B
such that
PQ implies
Xf dP≤
Xf dQ.
We will assume that all functions in the following results belong to the classB.
Theorem Letbe an integral stochastic order onP.LetRbe a generator ofsatisfying
that if g:X →Ris a measurable mapping such that g and f are-comonotonic for all f ∈R,then g∈R.It holds thatRis the maximal generator of.
Proof Let us see thatRis a convex cone containing the constant functions and closed under pointwise convergence.
Let f ∈R and λ> . Suppose that there exists h∈R such that h and λf are not
-comonotonic. Then there existx,x∈Xwithxx,h(x) <h(x) andλf(x) >λf(x). This implies thatf(x) >f(x) which is a contradiction withxxsincef ∈R. Therefore,
λf andhare-comonotonic for allh∈Rand soλf∈R. As a consequence,Ris a cone. Now, considerf,f∈Randλ∈[, ]. Suppose that there existsh∈Rsuch thathand
λf+ ( –λ)fare not-comonotonic. We have that there existx,x∈X withxx,
On the other hand, any constant function and any element ofRare-comonotonic. So, any constant function is inR.
Moreover, let{fn}n⊂Rsuch that{fn}ntends tof :X→Rpointwise. Leth∈R. Suppose that handf are not-comonotonic. We have that there existx,x∈X withxx, h(x) <h(x) andf(x) >f(x). Then fornlarge enough, we have thatfn(x) >fn(x), which contradicts thatfn∈R. Thus,Ris closed under pointwise convergence.
Now, the result is a consequence of Corollary .. in [] which says that a generator which is a convex cone containing the constant functions and is closed under pointwise convergence is the maximal generator of the order.
Example Consider (Rd,B
d) withBdthe usual Borelσ-algebra onRd. Letstand for the usual multivariate stochastic order. Thus, two probabilitiesPandQon the measurable space (Rd,Bd) are ordered in the usual multivariate stochastic order if
f dP≤
f dQ
for any bounded increasing mappingf :Rd→R, where by increasing we mean thatf(x)≤
f(y) for anyx,y∈Rdwithxcwy, andcwdenotes the usual componentwise order onRd. LetRbe the above class of mappings which is a generator of. Such a generator allows to obtain the partial orderonRddetermined by. It can be seen that this order is the ordercw.
To analyze if the integral stochastic orderis generated by, we apply Theorem , but taking into account Theorem ; that is, if there exists a generator satisfying the condition of Theorem , such a generator should be the maximal generator of. It is well known that the classRis the maximal generator of.
It is not hard to prove that any bounded measurable mapping which is-comonotonic with all the mappings ofRbelongs to it, and so we conclude that the usual multivariate stochastic order is generated by the componentwise order.
Example Now letstand for the upper orthant order on the class of probabilities as-sociated with the measurable space (Rd,B
d). Two probabilitiesPandQare ordered in the upper orthant order if
P(t,∞)≤Q(t,∞)
for any t∈Rd, where (t,∞) denotes the set (t,∞)×(t,∞)× · · · ×(td,∞) witht= (t, . . . ,td).
Thus, the setR={I(t,∞)|t∈Rd}is a generator of. This generator leads to the partial orderonRddetermined by. It can be seen that this order iscw.
As an immediate consequence, we obtain that there does not exist a partial order gener-ating the upper orthant order, since we have seen in the above example thatcwgenerates the usual multivariate stochastic order.
A mappingf :Rd→Ris said to be-monotone if for every subsetJ={i,i, . . . ,i k} ⊂
{, , . . . ,d}and for everyε,ε, . . . ,εk> , it holds that
ε i
ε i· · ·
εk
ikf(x)≥
for allx∈Rd, where
εif(x) =f(x+εei) –f(x),
eibeing theith unit vector.
In [] it is proved that the classLof all bounded-monotone functions is the maximal generator of.
Let us see that there are-comonotonic mappings with all the mappings ofL, which do not belong to such a class.
Considerd= and the mappingg:R→Rwith
g(x,x) =
⎧ ⎨ ⎩
if (x,x)∈ {(z,z)|(z≥,z≥) or (z≥,z≥)},
in any other case.
The mappinggis-comonotonic with all the elements ofL, butgis not-monotone.
To conclude we should point out that Theorem could be applied to obtain maximal generators of integral stochastic orders which have been generated by means of partially ordered sets, since the maximal generator is the unique generator which satisfies the con-dition required in Theorem .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
MCL-D and ML-D have worked together to obtain the results in this manuscript.
Author details
1Departamento de Matemáticas, Universidad de Oviedo, C/Calvo Sotelo s/n, Oviedo, E-33007, Spain.2Departamento de Estadística e I.O. y D.M., Universidad de Oviedo, C/Calvo Sotelo s/n, Oviedo, E-33007, Spain.
Acknowledgements
We would like to thank the editor and the referees for their key comments and suggestions. The authors are indebted to the Spanish Ministry of Science and Innovation since this research is financed by Grants MTM2010-18370 and MTM2011-22993.
Received: 17 February 2012 Accepted: 29 October 2012 Published: 14 November 2012
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doi:10.1186/1029-242X-2012-265
