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RELATIONS BETWEEN STOCHASTIC PRECEDENCE AND

STOCHASTIC ORDERINGS

EMILIO DE SANTIS, FABIO FANTOZZI, AND FABIO SPIZZICHINO

Abstract. The concept of stochastic precedence between two real-valued ran-dom variables has often emerged in different applied frameworks, where it hasbeen compared with the usual stochastic ordering. However it cannot be seenas a notion of stochastic ordering. In this paper we consider a slightly moregeneral, and completely natural, concept of stochastic precedence and analyzeits relations with the notions of stochastic ordering. Such a study leads us tointroducing some special classes of bivariate copulas that reveal to be strictlyrelated with the analysis of non-exchangeability of copulas. Some motivationsfor our study arise in the frame of the Target-Based Approach to the field ofdecisions under risk.

Keywords. Non-Symmetric Copulas, Decision Analysis, Target Based Utili-ties, Times to Word’s Occurrences.

1. Introduction.

Let X1, X2 be two real random variables defined on a same probability space(Ω,F ,P). We will denote by FX1,X2 the joint distribution function and by G1,G2 their marginal distribution functions, respectively. For the sake of notationalsimplicity, we will initially concentrate our attention on the case whenG1, G2 belongto the class G of all the probability distribution functions on the real line, that arecontinuous and strictly increasing in the domain where positive and smaller thanone. As we shall eventually discuss, this restriction is not at all necessary, but itallows us to simplify the formulation of our results. In order to account for somecases of interest with P(X1 = X2) > 0, we will not assume that FX1,X2 is absolutelycontinuous.

The random variableX1 is said to stochastically precede X2 if P(X1 ≤ X2) ≥ 1/2,written X1 sp X2. The interest of this concept for applications has been pointedout several times in the literature (see in particular [1], [4] and [17]). Stochasticprecedence might be perceived, in some cases, as a somehow less restrictive, andmore realistic, assumption than the usual stochastic ordering X1 st X2, definedby the condition G1(t) ≥ G2(t), ∀t ∈ R.

Actually if X1, X2 are independent, it is true (see [1]) that X1 st X2 ⇒ X1 sp

X2. It is in any case easy to find several other examples of bivariate probabilitymodels where the same implication holds. For instance the condition X1 st X2

even entails P(X1 ≤ X2) = 1 when X1, X2 are comonotonic (see e.g. [19]), i.e.when

X2 = G−12 (G1(X1)) .

On the other hand, examples of stochastic dependence can also be found wherethe implication X1 st X2 ⇒ X1 sp X2 fails. In any case we recall the reader’s

1Dipartimento di Matematica, La Sapienza Universita di Roma, P.le A. Moro 5, 00185 Rome,Italy.Emilio De Santis e-mail: desantis@mat.uniroma1.it;Fabio Fantozzi e-mail: fantozzi@mat.uniroma1.it;corresponding author Fabio L. Spizzichino e-mail: fabio.spizzichino@uniroma1.it, tel.+390649913127.

1

2 RELATIONS BETWEEN STOCHASTIC PRECEDENCE AND STOCHASTIC ORDERINGS

attention on the fact that stochastic precedence does not define a stochastic orderin that, for instance, it is not transitive. Here we analyze in details the relationsthat generally exist between the properties X1 sp X2 and X1 st X2. Actually,for the sake of a more complete analysis, it is convenient to replace the conceptX1 sp X2 with the following more general

Definition 1. For given γ ∈ [0, 1], we say that X1 stochastically precedes X2 at

level γ if P(X1 ≤ X2) ≥ γ. This will be written X1 (γ)sp X2.

Let C denote the class of all bivariate copulas (see e.g. [13, 14, 19]). We say thatthe pair of random variables X1, X2, with marginals G1, G2, respectively, admitsC ∈ C as its connecting copula whenever its joint distribution function is given by

FX1,X2(x1, x2) = C(G1(x1), G2(x2)) . (1)

The theory of bivariate copulas provides us with the appropriate frame for ourstudy. Starting from (X1, X2), consider in fact the random variables X ′

1 = ϕ(X1)and X ′

2 = ϕ(X2) where ϕ : R → R is a strictly increasing function. Thus X ′1 st X

′2

if and only if X1 st X2 and X ′1 (γ)

sp X ′2 if and only if X1 (γ)

sp X2. On the otherhand the pair (X ′

1, X′2) also admits the same connecting copula C. It will turn

out in the next section that our arguments are strictly related with the topic ofnon-exchangeable copulas.

We will use the notation

A := (x1, x2) ∈ R2 : x1 ≤ x2 , (2)

so that we write

P(X1 ≤ X2) =

∫

A

dFX1,X2(x1, x2) =

∫

R2

1A(x1, x2) dFX1,X2(x1, x2) . (3)

For given G1, G2 ∈ G and C ∈ C we also set

η(C,G1, G2) := P(X1 ≤ X2) , (4)

ξ(C,G1, G2) := P(X1 = X2) , (5)

where X1 and X2 are random variables with marginals G1, G2 respectively, and

connecting copula C. Thus the condition X1 (γ)sp X2 can be written

η(C,G1, G2) ≥ γ .

More specifically this note will be devoted to analyzing different aspects of thespecial classes of bivariate copulas, defined as follows.

Definition 2. For γ ∈ [0, 1], we denote by Lγ the class of all copulas C ∈ C such

that

η(C,G1, G2) ≥ γ (6)

for all G1, G2 ∈ G with G1 st G2.

Related with these definitions, we will see below how certain inequalities on theprobabilities P (X1 ≤ X2) will come out as natural. Furthermore we will see howthe quantities η(C,G1, G2) are tied with non-exchangeability of copulas.

One initial motivation of ours for this analysis had arisen in some apparentparadoxes concerning stochastic comparisons among waiting times to occurrencesof words in random sequences of letters from an alphabet (see e.g. [7, 9, 10, 11]).The analysis of some examples in this direction, related with the classes Lγ , will bedeferred to a different paper. The frame of the Target-Based Approach to single-attribute utility functions (see [5] and [6]), as we will briefly discuss below, providesa further motivation. Our study can, in fact, be of interest in the discussion of

RELATIONS BETWEEN STOCHASTIC PRECEDENCE AND STOCHASTIC ORDERINGS 3

some critical aspects emerging in the extension of such an approach to the casewhen stochastic dependence is admitted among targets and prospects.

More precisely the structure of the paper is as follows. In Section 2 we analyzethe main aspects of the class Lγ and present a related characterization. Somefurther basic properties will be detailed in Section 3. Section 4 presents a fewexplanatory examples. Finally, in Section 5, we will review the basic aspects of theTarget-Based Approach and point out some related connections with our work.

We remark that, in some applications, we may be interested in comparing twodifferent scenarios determined by conditioning upon two different statistical obser-vations. An example in this direction emerged in [8] where stochastic orderings andstochastic precedence are considered for instants of change-points.

2. A characterization of the class Lγ .

This Section will be devoted to providing a characterization of the class Lγ (seeTheorem 5 below) along with related discussions. We start by detailing a few basicproperties of the quantities η(C,G1, G2), for G1, G2 ∈ G and C ∈ C. In view of thecondition G1, G2 ∈ G we can use the change of variables u = G1(x1), v = G2(x2).Thus we can rewrite the integral in (3) according to the following

Proposition 1. For given G1, G2 ∈ G and C ∈ C, one has

η(C,G1, G2) =

∫

[0,1]21A(G

−11 (u), G−1

2 (v)) dC(u, v). (7)

The use of the next Proposition is two-fold: it will be useful for characterizingthe class Lγ and for giving lower and upper bounds for the quantity η(C,G1, G2).

Proposition 2. Let G1, G′1, G2, G

′2 ∈ G. Then

G2 st G′2 ⇒ η(C,G1, G2) ≤ η(C,G1, G

′2) ;

G1 st G′1 ⇒ η(C,G1, G2) ≥ η(C,G′

1, G2) .

Proof. We prove only the first relation of Proposition 2, since the proof for thesecond one is analogous. By hypothesis, for each x ∈ (0, 1), one has

G−12 (x) ≤ G′−1

2 (x).

Therefore

(G−11 (x), G−1

2 (x)) ∈ A ⇒ (G−11 (x), G′−1

2 (x)) ∈ A .

Hence, the proof can be concluded by recalling (3) and (4).

From Proposition 2, in particular we get

η(C,G,G) ≤ η(C,G′, G) and η(C,G,G) ≤ η(C,G,G′′) , (8)

for any choice of G,G′, G′′ such that G′ st G st G′′.

A basic fact in the analysis of the classes Lγ is that the quantities of the formη(C,G,G) and ξ(C,G,G) only depend on the copula C. More formally we statethe following result.

Proposition 3. For any pair of distribution functions G′, G′′ ∈ G, one has

η(C,G′, G′) = η(C,G′′, G′′) , (9)

ξ(C,G′, G′) = ξ(C,G′′, G′′) . (10)

4 RELATIONS BETWEEN STOCHASTIC PRECEDENCE AND STOCHASTIC ORDERINGS

Proof. Recalling (4) one obtains∫

[0,1]21A(G

′−1(u), G′−1(v)) dC(u, v) =

∫

[0,1]21A(G

′′−1(u), G′′−1(v)) dC(u, v)

(11)because 1A(G

′−1(u), G′−1(v)) = 1A(G′′−1(u), G′′−1(v)) = 1A(u, v), so equality in

(9) is proved. For what concerns (10), the steps are similar, one just needs toreplace the set A with B = (x, y) ∈ R

2 : x = y.

As a consequence of Proposition 3 we can introduce the symbols

η(C) := η(C,G,G) , (12)

ξ(C) := ξ(C,G,G) , (13)

for G ∈ G. From Proposition 2 and from the inequalities (8), we obtain

Proposition 4.

G1 st G2 ⇒ η(C) ≤ η(C,G1, G2) .

We then see that the quantity η(C) characterizes the class Lγ , 0 ≤ γ ≤ 1, in factwe can state the following

Theorem 5. C ∈ Lγ if and only if η(C) ≥ γ.

We then haveLγ = C ∈ C : η(C) ≥ γ (14)

and we can also write

η(C) = infG1,G2∈G

η(C,G1, G2) : G1 st G2 , (15)

namely the infimum is a minimum and it is attained when G1 = G2. We noticefurthermore that the definition of η(C,G1, G2) can be extended to the case whenG1, G2 ∈ D(R) the space of distribution functions on R. The class G has howevera special role in the present setting, as it is shown in the following result.

Theorem 6. Let C ∈ C, G,H ∈ D(R) with G st H, then η(C,G,H) ≥ η(C).

Proof. Consider two sequences (Gn : n ∈ N), (Hn : n ∈ N) such that Gn, Hn ∈ Gand Gn

w→ G, Hnw→ H . Applying the result in [21], we obtain that C(Gn, Hn)

w→C(G,H).

Take the new sequence (Hn : n ∈ N) where Hn(x) := minGn(x), Hn(x). We

notice that Hn ∈ G, moreoverGn st Hn and Hnw→ H . This implies C(Gn, Hn)

w→C(G,H).

Now using the standard characterization of weak convergence on separable spaces(see [3] p. 67 Theorem 6.3)

lim supn→∞

∫

B

dC(Gn, Hn) ≤∫

B

dC(G,H) ,

for any B closed set. Taking the closed set A defined in (2) one has

η(C) ≤ lim supn→∞

∫

A∩[0,1]2dC(Gn, Hn) ≤

∫

A∩[0,1]2dC(G,H) = η(C,G,H). (16)

Remark 2.1. Theorem 6 shows that the minimum of η(C,G,H), for G,H ∈ D(R),is attained at (C,G,G), for any G ∈ G ⊂ D(R). This result allows us to replace the

class G with D(R) in the expression of Lγ given in (15). We notice furthermore

that one can have η(C,G′, G′) 6= η(C,G′′, G′′) when G′, G′′ are in D(R).

RELATIONS BETWEEN STOCHASTIC PRECEDENCE AND STOCHASTIC ORDERINGS 5

Concerning the classes Lγ , we also define Bγ := C ∈ C | η(C) = γ, so that

Lγ =⋃

γ′≥γ

Bγ′.

We now show that the classes Bγ , γ ∈ [0, 1], are all non empty. Several nat-ural examples might be produced on this purpose. We fix attention on a simple

example built in terms of the random variables X1, X(γ)2 defined as follows. On

the probability space ([0, 1],B[0, 1], λ), where λ denotes the Lebesgue measure, wetake X1(ω) = ω, and

X(γ)2 (ω) =

ω + 1− γ if ω ∈ [0, γ] ,ω − γ if ω ∈ (γ, 1] .

(17)

As it happens for X1, also the marginal distribution of X(γ)2 is uniform in [0, 1]

for any γ. We denote by Cγ the (unique) connecting copula of (X1, X(γ)2 ).

Proposition 7. For any γ ∈ (0, 1], one has

(i) Cγ ∈ Bγ.

(ii) Cγ(u, v) = minu, v,maxu− γ, 0+maxv + γ − 1, 0.

Proof. (i) First we notice that P(X1 ≤ X(γ)2 ) = γ. In fact

P(X1 ≤ X(γ)2 ) = P(X1 ≤ X1 + 1− γ,X1 ≤ γ) + P(X1 ≤ X1 − γ,X1 > γ) = γ.

Whence, η(Cγ) = P(X1 ≤ X(γ)2 ) = γ. In fact X1, X

(γ)2 have the same marginal

distribution, and the latter belongs to G.(ii) For x1, x2 ∈ [0, 1] we can write

FX1,X

(γ)2

(x1, x2) = P(X1 ≤ x1, X(γ)2 ≤ x2) = P(X1 ≤ x1, X1 + 1− γ ≤ x2, X1 ≤ γ)

+ P(X1 ≤ x1, X1 ≤ x2 + γ, X1 > γ)

= P(X1 ≤ minx1, x2 + γ − 1, γ) + P(γ < X1 ≤ minx1, x2 + γ)= maxminx1, x2 + γ − 1, γ, 0+maxminx1, x2 + γ − γ, 0= minx1, x2,maxx1 − γ, 0+maxx2 + γ − 1, 0 .

Since both the marginal distributions of X1 and X(γ)2 are uniform, it follows that

Cγ(u, v) = minu, v,maxu− γ, 0+maxv + γ − 1, 0.

We point out that the identity η(Cγ) = γ (for γ ∈ (0, 1]) could also be obtainedfrom formula (7), by applying a computation procedure as in [19].

As an immediate consequence of Proposition 7 we have that Lγ′ is strictly con-tained in Lγ for any 0 ≤ γ < γ′ ≤ 1. We notice furthermore that L0 = C andL1 6= ∅. In fact any copula C ∈ C with

∫A∩[0,1]2

dC = 1 belongs to L1.

For the special case γ = 1/3, the copula Cγ had been introduced in [20] andwas obtained as a shuffle of M (see [19]), with M denoting the maximal copulaM(u, v) = minu, v. In this particular case the underlying copula Cγ providesan example of maximal non-exchangeable copula, namely a copula with maximaldegree of asymmetry (for further details see below and [12]). The class Cγ has alsobeen studied in [12], for γ spanning the interval [0, 1/3], and was used to constructexamples with prescribed values of asymmetry (for a related argument see also[15]). As was noticed in [23], the Definition in (7) can be actually considered forany γ ∈ [0, 1] and such extension has a specific interest in our setting.

Graphs of Cγ for different values of γ are provided in Figure 1.

6 RELATIONS BETWEEN STOCHASTIC PRECEDENCE AND STOCHASTIC ORDERINGS

Figure 1. Copulas from the family Cγ with γ = 0.3, 0.5, 0.8 respectively

3. Basic properties of Lγ

In this Section we analyze further properties of the classes Lγ that can also shed

light on the relations between (γ)sp and stochastic orderings.

First we notice that the previous Definition 2 has been formulated in terms of theusual stochastic ordering st. It is well known however that several other conceptsof stochastic ordering have been considered in the literature such as the hazard rate,the likelihood ratio, and the mean residual life orderings (see [22]).

Let us consider in fact a stochastic ordering ∗ different from st. Definition2 can then be modified by replacing therein st with ∗ and this operation leads

us to a new class of copulas that we can denote by L(∗)γ . For given γ ∈ (0, 1), one

might wonder about possible relations between L(∗)γ and Lγ . Actually one has the

following result.

Proposition 8. Let ∗ be a stochastic ordering with the following properties

(a) for any G ∈ D(R) one has G ∗ G;

(b) for G1, G2 ∈ D(R) with G1 ∗ G2 one has G1 st G2.

Then Lγ = L(∗)γ .

Proof. Similarly to (14) and (15) we can write

L(∗)γ = C ∈ C : η∗(C) ≥ γ (18)

where

η∗(C) := infG1,G2∈G

η(C,G1, G2) : G1 ∗ G2. (19)

In view of (b), one has that η(C) ≤ η∗(C). In fact both the quantities η(C) andη∗(C) are obtained as an infimum of a same functional and, compared with η, η∗

is an infimum computed on a smaller set.Due to (a), however, η(C) and η∗(C) are both obtained, in (15) and (19) re-

spectively, as minima attained on a same point (G,G). We can then conclude that

L(∗)γ = Lγ .

Concerning Proposition 8 we notice that, for example, the stochastic orderingslr and hr satisfy both the conditions (a) and (b).

For the sake of notational simplicity we come back to considering the usualstochastic ordering st and the class Lγ .

Proposition 9. For γ ∈ [0, 1], the classes Lγ, Lcγ, and Bγ are convex.

RELATIONS BETWEEN STOCHASTIC PRECEDENCE AND STOCHASTIC ORDERINGS 7

Proof. We consider two bivariate copulas C1, C2 ∈ Lγ and a convex combination ofthem, i.e. take α ∈ (0, 1) and C := αC1+(1−α)C2. We show that C ∈ Lγ , indeed

η(C) =

∫

A

dC(u, v) = α

∫

A

dC1(u, v)+(1−α)

∫

A

dC2(u, v) = αη(C1)+(1−α)η(C2).

Since η(C1), η(C2) are larger or equal than γ then η(C) ≥ γ, so Lγ is convex. Nowone can use the same argument in order to show that Lc

γ and Bγ are also convex.

Many of the most well-known bivariate copulas are symmetric, namely theysatisfy

C(u, v) = C(v, u), ∀(u, v) ∈ [0, 1]2.

Fix now G ∈ G and let X1, X2 be random variables with a symmetric connectingcopula C and both marginal distribution functions coinciding with G. Then theirjoint distribution function FX1,X2 is exchangeable and P(X1 < X2) = P(X2 <X1) = (1− ξ(C))/2. Thus

η(C) = P(X1 < X2) + ξ(C) =1 + ξ(C)

2≥ 1

2.

We have η(C) = 1/2 when ξ(C) = 2η(C)− 1 = 0. As an immediate consequenceof Theorem 5, we then get that any symmetric copula belongs to Lγ for any γ ≤ 1/2.

In the literature about copulas has been often considered the topic of transfor-mations. It can then be interesting to analyze how the quantity η(C) behaves undertransformations of C. We devote the remaining part of this Section to the case ofsurvival copulas and transpose copulas.

Especially in the case of pairs of non-negative random variables, it comes as

natural considering the notion of survival copula. We say that a copula C is thesurvival copula of the pair (X1, X2) when their joint survival function has the form

FX1,X2(x1, x2) = C[G1(x1), G2(x2)

], (20)

with G1 and G2 respectively denoting the marginal survival functions:

G1(x1) = P(X1 > x1), G2(x2) = P(X2 > x2) .

The relationship between the survival copula C of (X1, X2) and the connectingcopula C is given by (see [19])

C(u, v) = u+ v − 1 + C(1− u, 1− v) . (21)

Starting from C ∈ C, it has also interest to consider Ct, the transposed copula ofC, defined by

Ct(u, v) := C(v, u) (22)

The following result shows the relations tying the different quantities η(C), η(C),η(Ct).

Proposition 10. Let C ∈ C. The following relation holds:

η(C) = η(Ct) = 1− η(C) + ξ(C) .

For a given C ∈ C, let us now consider the quantity η(C) − η(Ct). The aboveresult immediately shows that η(C) − η(Ct) = 2η(C) − 1 − ξ(C). We will denoteby ν(C) the modulus of η(C) − η(Ct), i.e.

ν(C) := |η(C) − η(Ct)| = |2η(C)− 1− ξ(C)| . (23)

ν(C) can, in a sense, be seen as an index of how much C is far from being symmetric.In this respect, we refer to the definition ofmeasure of non-exchangeability proposedin [12] (Definition 2). One can check that ν(C) satisfies all the properties in such

8 RELATIONS BETWEEN STOCHASTIC PRECEDENCE AND STOCHASTIC ORDERINGS

Definition except the equivalence between symmetry of the underlying copula andvanishing of the measure of non-exchangeability. In our case, provided ξ(C) = 0,we have that C symmetric implies ν(C) = 0, but the opposite implication does notnecessarily hold.

In the special case of copulas Cγ (see Proposition 7) the equivalence holds, forγ ∈ [0, 1]. In this case we have

ν(Cγ) =

|2γ − 1| for γ ∈ (0, 1),0 for γ = 0, 1.

ξ(Cγ) =

0 for γ ∈ (0, 1),1 for γ = 0, 1.

4. Examples and special cases.

Some aspects of the definitions and results presented in the previous Sectionswill be illustrated here by presenting the following examples. The first examplewill be devoted to bivariate gaussian models, i.e. to a relevant case of symmetriccopulas.

Example 1. Gaussian Copulas.

The family of bivariate gaussian copulas (see e.g. [13] and [19]) is parameter-ized by the correlation coefficient ρ ∈ (−1, 1). The corresponding copula C(ρ) isabsolutely continuous and symmetric, and η(C(ρ)) = 1/2 and, thus, it does notdepend on ρ. For fixed pairs of marginal distributions G1, G2, on the contrary, thequantity η(C(ρ), G1, G2) does actually depend on ρ, besides G1 and G2. In the casewhen G1, G2 are gaussian we have an expression for η(C(ρ), G1, G2) that, besidescontaining the parameters of G1, G2, is in terms of ρ and of the standard gauss-ian distribution function Φ. Let X1, X2 denote gaussian random variables withconnecting copula C(ρ) and parameters µ1, µ2, σ

21 , σ

22 . Since the random variable

Z = X1−X2 is distributed according to N (µ1−µ2, σ21 +σ2

2 −2ρσ1σ2) we can write

η(C(ρ), G1, G2) = P(Z ≤ 0) = Φ

(µ2 − µ1√

σ21 + σ2

2 − 2ρσ1σ2

). (24)

If G1, G2 are not gaussian, one can give, by using Proposition 2, inequalities forη(C(ρ), G1, G2) in terms of (24), provided G1, G2 are suitably comparable in thest sense with gaussian distributions.

We recall that, when Xi ∼ N (µi, σ2i ) for i = 1, 2, the necessary and sufficient

condition for X1 st X2 is µ1 ≤ µ2 and σ1 = σ2 (see e.g. [1]). By using the formulain (24), with σ1 = σ2 = σ, we have

η(C(ρ), G1, G2) = Φ

(µ2 − µ1

σ√

2(1− ρ)

). (25)

Thus G1 st G2 ⇒ η(C(ρ), G1, G2) ≥ 1/2, as shown by Proposition 4. Moreprecisely G1 st G2 means that X1 sp X2 and σ1 = σ2.

In the cases when ξ(C) > 0, we should obviously distinguish between computa-tions of P(X1 ≤ X2) and P(X1 < X2), where C is the connecting copula of X1, X2.A remarkable case when this circumstance happens is considered in the followingexample.

Example 2. Marshall-Olkin Models

In order to introduce the Marshall-Olkin copula (see e.g [14, 16, 19]), we considerthree random variables V , W and Z, independent and with exponential distribu-tions of parameters µ1, µ2 and µ, respectively. The new random variables

X1 := V ∧ Z, X2 := W ∧ Z ,

RELATIONS BETWEEN STOCHASTIC PRECEDENCE AND STOCHASTIC ORDERINGS 9

have marginal survival functions

G(αi)

i (xi) := 1−G(αi)i (xi) := exp(−µixi) (26)

where αi = µ/(µ+ µi), 0 < αi < 1, for i = 1, 2. Their joint survival function is

FX1,X2(x1, x2) = P(V ∧ Z > x1, W ∧ Z > x2) = exp(−µ1x1 − µ2x2 − µ(x1 ∨ x2)).(27)

By combining (26) and (27) we obtain that the survival copula C(α1,α2) is

C(α1,α2)(u, v) := minv u

µ1µ1+µ , u v

µ2µ2+µ

= u v minu−α1 , v−α2

(see [16] for details). Notice that the Marshall-Olkin copula has a singular partthat is concentrated on the curve uα1 = vα2 (see also Figure 2). The singularitydepends only on the fact that the probability P(X1 = X2) is larger than zero. Moreprecisely

ξ(C(α1,α2), G(α1)1 , G

(α2)2 ) = P(X1 = X2) =

µ

µ1 + µ2 + µ=

α1α2

α1 + α2 − α1α2.

Furthermore, from

P(X1 < X2) = P(V ∧ Z < W ∧ Z) = P(V < W < Z) + P(V < Z < W )

= P(V < W ∧ Z) ,

we can derive

P(X1 < X2) =µ1

µ1 + µ2 + µ=

(1 − α1)α2

α1 + α2 − α1α2,

and finally

η(C(α1,α2), G(α1)1 , G

(α2)2 ) =

µ+ µ1

µ1 + µ2 + µ=

α2

α1 + α2 − α1α2.

We notice that the above arguments for the evaluation of η(C(α1,α2), G(α1)1 , G

(α2)2 ),

cannot be used to evaluate η(C(α1,α2), G1, G2) for pairs of marginal distributions

different from (G(α1)1 , G

(α2)2 ). In particular they cannot be used for determining

η(C(α1,α2)) = η(C(α1,α2), G,G) (with G ∈ G). As to the computation of η(C(α1,α2)),we should refer to the integral expression in (7), letting G1 = G2 = G. In sucha computation, it is convenient to distinguish among three different cases, namelyα1 > α2, α1 = α2, and α1 < α2. This procedure in fact allows us to reduce inte-

gration to domains where C(α1,α2) is absolutely continuous, i.e. to domains whichdo not contain the curve uα1 = vα2 . In conclusion we obtain

η(C(α1,α2)) =1

2− α1 ∧ α2

(1− (α1 − α1 ∧ α2)(α1 ∧ α2)

α1 − α2

).

Finally, the evaluation of ν(C) is straightforward and we obtain

ν(C(α1,α2)) =α1 ∧ α2

2− α1 ∧ α2.

We notice that, also in the present Marshall-Olkin case, the index ν defined in (23)perfectly fits with the Definition of measure of non-exchangeability given in [12].In fact one has that ν(C) = 0 only in the case α1 = α2 = 0, that corresponds toC(u, v) = uv, the independence copula.

In the previous two examples the quantity η(C,G1, G2) can be easily computedfor special cases of marginal distributions G1 and G2. This task can be directlyaccomplished by considering probabilistic arguments which cannot be extended togeneral choices of G1, G2.

We now conclude this Section with an example showing an extreme case in thisdirection.

10 RELATIONS BETWEEN STOCHASTIC PRECEDENCE AND STOCHASTIC ORDERINGS

Figure 2. Marshall-Olkin Copula with α1 = 0.2, α2 = 0.4 (left) andα1 = 0.4, α2 = 0.2 (right)

Example 3. Copulas of order statistics.

Let A,B be two i.i.d. random variables with distribution function G ∈ G anddenote by X1, X2 their order statistics, namely X1 = minA,B, X2 = maxA,B.The marginal distributions of X1, X2 depend on G and are respectively given by

F(G)1 (x1) = P(minX1, X2 ≤ x1) = 2G(x1)−G(x1)

2 ,

F(G)2 (x2) = P(maxX1, X2 ≤ x2) = G(x2)

2 .

The connecting copula of (X1, X2) (see [2]) is given by

K(u, v) =

2(1− (1− u)1/2)v1/2 − (1− (1− u)1/2)2 if v ≥ (1− (1− u)1/2)2 ,v otherwise .

We have, by definition,

η(K,F(G)1 , F

(G)2 ) = 1 ,

and it does not depend on G. We notice, on the other side, that the computationof η(K) = η(K,G,G), with G ∈ G, is to be carried out explicitly, since the pair(G,G) can never appear as the pair of marginal distributions of order statistic. Byrecalling (7) one can obtain

η(K) =

∫

[0,1]2

1A(u, v)

2√v√1− u

dv du = 2− π

2<

1

2,

and ν(K) = |2η(K) − 1| = π − 3. We can extend this example to the case whenthe connecting copula of A,B is a copula D different from the product copula Π,but still A and B are identically distributed according to a marginal distributionfunction G. In this case the connecting copula K of X1, X2 depends on D, butagain it does not depend on G (see [18] page 478).

Figure 3. Ordered Statistic Copula K

RELATIONS BETWEEN STOCHASTIC PRECEDENCE AND STOCHASTIC ORDERINGS 11

5. The classes Lγ and the Target-Based Approach to Utility

Functions.

In this Section we look at the arguments of the previous Sections in the per-spective of the Target-Based Approach (TBA) to the theory of utility and decisionsunder risk. In such a theory, as very well-known, the Expected Utility Principle

prescribes to assign a utility function U and, then, to evaluate a risky prospect X(with its probability distribution FX) in terms of the expected-utility

E(U(X)) =

∫

R

U(x)FX (dx) .

Here, we consider in particular one-attribute decision problems, where X is ascalar random variable. Let U : R → R be a utility function. It is then naturalto assume U to be increasing and we also assume it to be bounded and right-continuous. This entails that, by means of normalization, it is possible to takeU : R → [0, 1] and to regard U as a probability distribution function on the realline.

The TBA amounts to look at U(x) as the distribution function of a scalar randomvariable T with the meaning of a Target. More explicitly we can say that, in theevaluation of a risky prospectX , the decision criterion amounts to choosing a targetT and to evaluatingX in terms of the probability P(T ≤ X). The first formalizationof such an approach was given by Bordley and Li Calzi and Castagnoli and Li Calzi(see [5, 6]). Some related ideas were already around in the economic literaturein the past and other interesting developments appeared in the subsequent years,especially for what concerns a multi-attribute setting. The TBA can be brieflydetailed as follows. Let X be a risky prospect with a probability distributionFX and let T be a (generally random) target with a probability distribution FT .Assume X and T to be stochastically independent and suppose that we evaluateX in terms of the probability P(T ≤ X). We can write, in view of independence,

P(T ≤ X) =

∫

R

P(T ≤ x|X = x)FX (dx)

=

∫

R

P(T ≤ x)FX(dx) =

∫

R

FT (x)FX(dx).

Whence P(T ≤ X) can be seen as the expected value of a utility: by consideringthe utility function U = FT we have

E(U(X)) =

∫

R

U(x)FX(dx) =

∫

R

FT (x)FX (dx) = P(T ≤ X) .

We thus see, in conclusion, that any bounded right-continuous utility functioncan be seen as the distribution function of a target T , and vice-versa. Such anapproach gives rise to easily-understandable and practically useful interpretationsof several notions of utility theory.

For any decision-maker I, the choice of T (with the corresponding subjectivedistribution function FT ) can be seen as a way to describe I’s attitude with respectto risk. Consider, for instance, the comparison between two decision-makers I ′, I ′′,owning a same capital and respectively choosing targets T ′ and T ′′. We can sayin some sense that I ′ is more “risk-adverse” than I ′′ whenever T ′ is stochasticallysmaller than T ′′.

Concentrating our attention on a fixed decision-maker I we can think of T , orof FT , as being fixed once for ever as a criterion to evaluate and compare all thepossible prospects X , stochastically independent of T .

TBA can however be rendered, in a sense, more general than the expected utilityapproach by allowing for stochastic dependence between targets and prospects. We

12 RELATIONS BETWEEN STOCHASTIC PRECEDENCE AND STOCHASTIC ORDERINGS

see in fact that, if we admit the possibility of some correlation between the targetand the prospect, then the TBA considers more general decision rules. Let usconcentrate attention on the jointly absolutely continuous case where X admits amarginal probability density fX(x) and where we can consider the function

υ(x) := P(T ≤ x|X = x). (28)

Taking into account (28), we can write

P(T ≤ X) =

∫

R

υ(x)fX(x) dx. (29)

IfX,T are not independent, υ(x) does not coincide with the distribution functionFT (x). Moreover υ(x) may reveal to be non-everywhere increasing, this time, andit thus generally fails to coincide with a bona-fide utility function.

Remark 5.1. The function υ(x) may in particular turn out to be non-increasing,

in some cases of strong correlation between T and X.

Concerning the form of the conditional distribution of T given a prospect X ,we observe that it may be too general, however, to allow stochastic dependence tochange with the choice of X . This level of generality would add special difficultyto the problem of comparing, in terms of corresponding values for P(T ≤ X), thedifferent prospects X .

On the other hand, the condition of stochastic independence between any singleprospect X and the fixed target T can be formalized by saying that the choice isrestricted only to prospects X ’s for which the connecting copula of (T,X) remainsequal to the product copula CT (u, v) = Π(u, v) = u · v.

For these reasons we can argue that, as an appropriate generalization, the con-dition of independence can be extended by introducing the following

Definition 3. We have a Fixed-Copula Target-Based Model when, for any fixed

target T and, for any considered prospect X, the connecting copula CT,X(u, v) of

(T,X) remains equal to a fixed copula CT (u, v).

Different reasons of interest for such a definition can be found in the frame of theTBA. Actually Fixed-Copula Target-Based Models can be natural e.g. in micro-economic scenarios, where stochastic dependence between X ’s and T is induced byunderlying variables, with macro-economic meaning (such as inflation, sovereignwealth funds, volatility indexes of a market,...) which are in common to any pair(T,X).

The results in Section 2 and Section 3 admit a direct interpretation in the TBAcontext. In particular Proposition 2 above can be rephrased as follows. Considerrandom variables T,X ′, X ′′ with marginal distributions FT , FX′ , and FX′′ respec-tively and let the pairs (T,X ′), (T,X ′′) share the same connecting copula C.

Proposition 11. X ′ st X′′ implies

P(T ≤ X ′) ≤ P(T ≤ X ′′) . (30)

In other words stochastic comparison between two prospects X ′ and X ′′ im-plies that X ′′ is preferred to X ′ for any target T , provided (T,X ′) and (T,X ′′)share the same connecting copula C. We note that, on the contrary, the implica-tion in the above Proposition does not necessarily hold in the case when the pairs(T,X ′), (T,X ′′) manifest two different connecting copulas. To see this it is suffi-cient to think of the case when T is stochastically increasing both in X ′ and in X ′′,but

P(T ≤ x|X ′ = x) ≥ P(T ≤ x|X ′′ = x) ∀x.

RELATIONS BETWEEN STOCHASTIC PRECEDENCE AND STOCHASTIC ORDERINGS 13

In the cases of correlation, under the condition of a fixed-Copula Model however,the conceptual possibility of simultaneous appearance of the conditions T st X

and X (γ)sp T constitutes a warning for a thoughtful choice of the copula CT . In

particular we see that the copula CT should belong to Lγ with γ large enough (or,in other terms, CT should manifest an adequate level of non-exchangeability). Inthis respect we see the interest of Theorem 5. It guarantees in fact that, in order tocheck the condition C ∈ Lγ , we only need an appropriate lower bound for P(T ≤ X)with T and X identically distributed.

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