Joint weak hazard rate order under non-symmetric copulas

Abstract A weak version of the joint hazard rate order, useful to stochastically compare not independent random variables, has been recently defined and studied in [4]. In the present paper, further results on this order are proved and discussed. In particular, some statements dealing with the relationships between the jointweak hazard rate order and other stochastic orders are generalized to the case of non symmetric copulas, and its relations with some multivariate aging notions (studied in [2]) are presented. For this purpose, the new notions of Generalized Supermigrative and Generalized Submigrative copulas are defined. Other new results, examples and discussions are provided as well.


Introduction and useful notions
In the last decades, stochastic comparisons between univariate random variables have been de ned and applied in a variety of contexts (see, e.g., the monographs [16,20] and [3] for detailed descriptions and properties of the main univariate stochastic orders). It is a remarkable fact that most of the univariate stochastic orders considered in the literature are based on comparisons between the marginal distributions of the involved variables, without taking care of their mutual dependence. In fact, in many applied problems one can avoid to consider dependence among alternatives. However, in some cases one has to take it into account, and for this reason a set of alternative bivariate versions of the most well-known stochastic orders have been provided by di erent authors, like, e.g., in [4,11,22]. These versions, which allows to take into account the effects of dependence between the variables to be compared, gave rise to a new class of stochastic comparisons, commonly called joint stochastic orders.
A discussion on one of these stochastic comparisons, i.e., the joint weak hazard rate order, recently introduced and applied in [4], and in particular on its relationships with the standard hazard rate order, will be presented along this paper. Marginally, the stochastic precedence order, which is another well-known comparison that takes into account the mutual dependence among alternatives (see, e.g., [6,17] and references therein), will be considered and discussed as well.
The following is the formal de nition of all the stochastic comparisons discussed in the present paper. Here, the notation [X|A] means the variable whose distribution is the distribution of X given the event A. Also, given the bivariate random vector (X , X ), F denotes its joint survival function (i.e., F(t, s) = P[X > t, X > s]), F i denotes the survival function of X i , while f i and r i = f i /F i denote the density and the hazard rate of X i , whenever it is absolutely continuous.
We observe that from a mathematical viewpoint, survival copulas and connecting copulas turn out in any case to be copulas, and that they both describe the dependence structure of (X , X ). Also, it is well known that if the marginal distributions are continuous then the connecting copula C and the survival copula C are unique. For this reason, we will assume here, and everywhere throughout the paper, continuity of the marginal distributions for the vector (X , X ). We address the readers to the monograph [18] for further details.
The following property of bivariate copulas was rstly introduced by [2], and further studied and applied in dependence analysis in [8,9], where it has been called supermigrativity.
, and if it satis es for all u ≥ v and γ ∈ ( , ). Viceversa, we say that C is submigrative if the inequality in (1.1) is satis ed in the opposite direction. In the similar way are de ned the supermigrativity and submigrativity properties for survival copulas.

Thus, given any
i.e., whenever it assumes higher values in correspondence to points (u, v) which are near to the diagonal v = u. This means that it has its probability mass mainly concentrated close to the diagonal, and this property can be though of as a positive dependence notion. In fact, as shown in [9], the supermigrativity property satis es almost all the necessary conditions to be considered a positive dependence notion. For example, any vector (X , X ) having a supermigrative copula (or survival copula) satis es the Positive Quadrant Dependence property (PQD), i.e., it satis es Viceversa, submigrativity can be seen as a negative dependence notion.
Concerning the general meaning of stochastic dependence for a pair of random variables X and X we remind, in particular, that [12]axiomatically de ned the set of conditions that X and X should satisfy in order to be positively dependent, i.e, the property that large (respectively, small) values of X tend to go together with large (respectively, small) values of X (and the opposite for negative dependence). In any case, notions of positive dependence are typically de ned by appropriate inequalities. Corresponding notions of negative dependence can be de ned by requiring that such inequalities are reverted. Thus the analysis of both positive and negative dependence is conceptually simple in the case n = , which this paper is con ned to. The panorama about negative dependence is less clear in the case of n variables, with n > . Anyway, interesting de nitions and related results have been given in the literature; see, in particular, the basic paper [5].
The following statement has been proved in [4].
Observe that, for all t, s ≥ , Similarly, for all t, s ≥ , i.e., X ≥ hr X .

Moreover, it holds
and, similarly,

Clearly,
for every t, s > , i.e., X ≥ hr:wj X . In fact, the vector (X , X ) has a Clayton survival copula, de ned as which satis es the supermigrative property, thus the last inequality is also proved by inequality (1.2) and Theorem 1.1 (a).
In general, one can thus observe that the existence of positive dependence strengthens the transition from standard hazard rate order to the corresponding joint one. As a main purpose of this article, in the next section we will treat the case of non-symmetric copulas, providing a generalization of Theorem 1.1 to nonexchangeable cases. Also, coming back to the case of exchangeable survival copulas, in Section 3 we will point out some basic relations between the joint weak hazard rate order and concepts of bivariate ageing for exchangeable lifetimes, that had been presented in [2].

Relationships among stochastic orders
In this section we aim to analyze some aspects of the joint weak hazard rate order for pairs of lifetimes (X , X ) whose survival copula C is generally non-symmetric. On this purpose, the following notation will be useful. Let us denote by S the class of ordered pairs (X , X ) such that X ≥ hr X , and with S( C) the subclass of S of the pairs having survival copula C. We furthermore denote by M the class of ordered pairs (X , X ) such that X ≥ hr:wj X , and with M( C) the subclass of M of the pairs having survival copula C. By using this notation in the case of a symmetric copula, for instance, implication (a) of Theorem 1.1 reads As already noticed in Section 1, we have that, when X , X are independent, namely when C is the product copula, then S( C) = M( C). We shall in particular see that there exist other copulas C for which the same identity holds (see subsequent Remark 2.1). Further properties of the class M( C) will be demonstrated below.
We start by observing that, trivially, the following equivalence holds. In the following statement, and everywhere along the paper, the notation = st means equality in law.
Recalling the assumption of continuity of the marginal distributions (needed to guarantee unicity of connecting and survival copulas, see Section 1), we can assert there exist t, s ≥ such that u = F (t + s) = F (t + s) and v = F (t) = F (t). Thus Examples of survival copulas C satisfying the assumption of Theorem 2.1 will be provided later (see, e.g., Example 2.2 and Example 2.3).
The main reason of interest in Theorem 2.1 is in the property required on the survival copula, i.e., the inequality C(u, v) ≥ C(v, u) for all ≤ u ≤ v ≤ . As shown later, this property will play a fundamental role in the analysis of relationships between joint weak hazard rate order and other stochastic orders in case of non-symmetric copulas. A rst example in this direction is given by the following statement, dealing with relationship with the stochastic precedence order.
Proof. To simplify the notation we give here the proof assuming that the survival copulaĈ connecting the pair (X , X ) admits a density c. The proof can be easily generalized to not absolutely continuous survival copulas.
Let X = st X . Then, by Theorem 2.1, where the last equivalence, which holds true whenever X = st X , follows from Theorem 5 in [6].
It should be remarked here that implication X ≥ hr:wj X ⇒ X ≥sp X holds even in case X and X have di erent distributions, as one can prove by using a proof similar to the one above, just replacing the survival copula C with the joint survival function F.
The following example shows that the opposite implication can fail. In particular, this conclusion can be achieved by showing that there exist survival copulas It is easy to observe that, for e = , it holds A d C(u, v) = / and C(u, v) ≥ C(v, u) for all ≥ u ≥ v ≥ , thus both X ≥ hr:wj X and X ≥sp X are satis ed when X = st X .
Let now < e < / . It is easy to observe that in this case thus X ≥sp X holds for X = st X . However, xing u = / − e and v = u + d, with d ∈ ( , / ), it holds Thus, the inequality C(u, v) ≥ C(v, u) is no more satis ed for all ≥ u ≥ v ≥ , and inequality X ≥ hr:wj X does not hold.
We now drop the assumption X = st X , and look for conditions for X ≥ hr:wj X , without necessarily requiring that C is symmetric. For this purpose we provide the following de nitions, which extend the notions of supermigrativity and submigrativity to non-symmetric copulas and survival copulas .
Viceversa, we say that C is generalized submigrative if inequality (b) is satis ed in the opposite direction. Obviously, symmetric supermigrative (submigrative) copulas are also generalized supermigrative (submigrative) copulas. Examples of non-symmetric generalized supermigrative or generalized submigrative copulas (or survival copulas) will be given later (see Example 2.2 and Example 2.3). Theorem 1.1 can be now generalized to non-symmetric copulas. More precisely, we have the following statement (which includes Theorem 1.1 as special case). Theorem 2.3. Let (X , X ) be any couple of lifetimes, and let C denote its survival copula.
Proof. We give here only the proof of statement (a), the other being similar. Thus, let us assume that X ≥ hr X . Let F i denotes the survival function of X i , for i = , . From X ≥ hr X it follows X ≥ st X , thus also, for s ≤ t, Moreover, from X ≥ hr X also follows, for s ≤ t, Assume that (2.1) holds. Then, for s ≤ t, Then where the rst inequality follows from generalized supermigrativity of C and by (2.1), while the second one from (2.3). Assume now that (2.2) holds. Then, for s ≤ t, Then where the second inequality follows from from generalized supermigrativity of C and inequality (2.2), while the third from (2.3).
In both cases we have F(t, s) ≥ F(s, t) for all s ≤ t, thus X ≥ hr:wj X .
We now provide some examples of non-symmetric survival copulas satisfying the generalized supermigrative property. A rst example is given by non-symmetric Marshall-Olkin copulas, described below.

Example 2.2.
Let the survival copula C be a Marshall-Olkin copula, i.e., be de ned as Observe that C is not absolutely continuous, having a singularity on the curve v = u α/β . Let us rst prove that C(u, v) ≤ C(v, u) holds for all u ≥ v (i.e., for all points (u, v) below the diagonal of the unit square, thus also below the singularity). For it, we should consider two cases. i) Assume that (v, u) is such that u β > v α , i.e., that (v, u) is above the singularity. Then C(v, u) = vu −β and C(u, v) = u −α v, and the inequality is satis ed by β ≥ α. ii) Assume that (v, u) is such that u β ≤ u α , i.e., that (v, u) is below the singularity. Then C(v, u) = v −α u and C(u, v) = u −α v, and the inequality is satis ed by u ≥ v.
Actually, this assertion can be also directly proved and strengthen just observing that, for this vector (X , X ), it holds for all s, t ≥ (see [14]). Thus the inequalities X ≥ st X , X ≥ hr X and X ≥ hr:wj X become equivalent, even if X and X are not independent. In other words, this example shows that the standard hazard rate order and the joint weak hazard order can both hold true even for survival copulas C di erent than the copula of independence.

for details). Even in this case the inequalities
and i.e., X ≥ hr X and X ≥ hr:wj X , become equivalent. Thus, in this case it holds S( C) = M( C) even if X and X are not independent. However, now the equivalence between the standard hazard rate, the joint weak hazard rate and the usual stochastic (≥ st ) orders no longer applies, since the two conditional probabilities in (2.5) are no more equal to P[X i > s], as it is in (2.4).
The following is a further example of copula satisfying generalized supermigrativity. [19], i.e., let C be a generalization of the FGM copula de ned as

Example 2.3. Let the survival copula C be a member of the the copulas de ned in
with β i , α i ≥ , i = , , and ≤ ρ ≤ . In case β ≤ β and α ≥ α then generalized supermigrativity holds.

In fact, for inequality (a) in De nition 2.1 it is easy to observe that C(u, v) ≤ C(v, u) is satis ed i u β
and the latter follows from which are satis ed when γ ≤ and u ≥ v.
An example of application of Theorem 2.3 is now given.

Example 2.4. Consider two components having dependent lifetimes X and X , respectively, and assume identical exponential distribution with rate λ for both. Assume that a task should be performed by both components up to a xed time T, and then continued by just one of them. Having the same hazard rate, i.e., being
that is, the residual lifetimes of the components at T, given that both components work at T, can be di erent. Moreover, in case the inequality in (2.8) is not satis ed, nothing can be asserted in general on the direction of the corresponding inequality. Assume for example, that the survival copula C of (X , X ) is the copula de ned in (2.6), with ρ = , β = α = and β = α = (so that condition β ≤ β and α ≥ α fail). Then, as it can be veri ed with a direct computation (see Figure 2)  Similar examples can be provided in other applicative contexts, like in insurance, letting X and X be two dependent risks, and having to compare truncated random claims (where T is the truncation level), or in medicine, letting X and X be two competing risks.
Concerning Theorem 1.1, it is then rather interesting to understand why positive dependence plays in favor of S( C) ⊆ M( C), i.e., in favor of the implication between inequality X ≥ hr X and inequality X ≥ hr:wj X (or the viceversa in case of negative dependence). The heuristic interpretation of this fact is given here.
Recalling the de nitions of hazard rate order and joint weak hazard rate order, and the intuitive meaning of positive dependence, one can try to write a chain of stochastic inequalities showing the assertion, as follows. Let t ∈ R, then where the rst inequality follows by the positive dependence existing between X and X (informally, because large values of X tend to go together with large values of X ), the second one by assumption X ≥ hr X , and the third one again by positive dependence. This chain of inequalities can not be used to prove the assertion that X ≤ hr:wj X , since the last one is in the wrong direction. However, by comparing the inequalities and one can argue that under some suitable conditions of positive dependence the rst one is in some sense stronger than the second one, being the additional conditional event {X > t} stronger, in some stochastic sense, than {X > t}, since X ≥ hr X . This is actually the case if the structure of dependence among X and X , i.e., the survival copula C, satisfy the generalized supermigrativity property: under this assumption, the last inequality is counterbalanced by the rst one.
Because of Theorem 2.3, and its heuristic interpretation, one can conjecture that there exist conditions of positive dependence strong enough to let the inequality X ≥ hr:wj X be satis ed under conditions weaker than X ≥ hr X . In the following two examples, we in particular present special models of dependence for which the relation X ≥ hr:wj X is equivalent to X ≥ st X , and where X ≥ hr X can fails. In the rst example we analyze the case of maximal positive dependence (i.e. comonotonicity). where h = ϕ − is the inverse of ϕ.

Let ϕ(t) ≤ t for every t ≥ (thus also X ≥ st X ). Under this conditions we have t ≤ h(t) and s ≤ h(s) ≤ h(t) for all ≤ s ≤ t, thus also max(t, h(s)) ≤ h(t) = max(h(t), s). It follows
i.e., X ≥ hr:wj X .
On the other hand, still assuming X = ϕ(X ) for an increasing ϕ, stronger conditions on ϕ are required for the inequality X ≥ hr X . Denoted with r i the hazard rate of X i , it holds r (t) = r (h(t))h (t), where h is the inverse ϕ − . Thus, X ≥ hr X i r (t) ≤ r (h(t))h (t) for all t ≥ . Now, if X is exponentially distributed with rate λ, such inequality is satis ed only for h (t) ≥ , i.e., for ϕ (t) ≤ for all t. This is actually possible, but it is a stronger condition than ϕ(t) ≤ t, which is the one required for X ≥ st X and X ≥ hr:wj X . Thus, under maximal positive dependence both X ≥ hr:wj X and X ≥ hr X can hold true, but, in some cases, it can be satis ed only the inequality X ≥ hr:wj X .
We present next a further example where the relation X ≥ st X is equivalent to X ≥ hr:wj X . It is interesting to remark in this respect that we do not hinge here on conditions of positive dependence.

Example 2.6. Load-sharing models with time-homogeneous failure parameters
In the case of absolute continuity, the joint probability law of n non-negative random variables X , . . . , Xn can be described by means of the family of its Multivariate Conditional Hazard Rates (MCHR) functions. Such a description is alternative but mathematically equivalent to the one expressed in terms of f X ,...,Xn , the joint density function of (X , . . . , Xn). See, e.g., the recent review paper [21] and references cited therein.
In a sense, the MCHR functions arise as direct extensions of the univariate concept of hazard rate function for a single non-negative random variable X. For our purposes, we can limit ourselves to formulating the de nition of MCHR functions for the case of n = non-negative variables X , X . In this case the family of the MCHR and, for < x < t, The assumption of absolute continuity of the joint distribution is essential since it simultaneously guarantees the meaningfulness of the conditional probabilities appearing above and the needed condition of no-tie: The functions λ ( ) (t), λ ( ) (t), λ ( ) (t, x) and λ ( ) (t, x) belonging to L can be obtained in terms of the joint density function f X ,X of (X , X ). One can also check that, viceversa, f X ,X can be recovered from the knowledge of the family L. The description of a joint distribution in terms of L turns out, furthermore, to be very e cient in the analysis of some problems of applied probability and in de ning appropriate models of dependence. In particular, special models arise from the condition that λ ( ) (t, x) and λ ( ) (t, x) do not depend on x. The corresponding models can be called models of Load-Sharing. See, e.g., [24] or the monograph [23] and references cited therein for more details and further remarks. In particular we can consider the time-homogeneous case de ned by the constants In this case we have, for all t, s ≥ , and Thus, the stochastic order X ≥ st X , which is satis ed when λ ( ) ≥ λ ( ) and λ ( ) ≥ λ ( ) , is maintained under the conditioning {X > t, X > t}, so that X ≥ st X ⇔ X ≥ hr:wj X . Moreover, observe that conditions give raise to a positive dependence, while the conditions give raise to a negative dependence. Note that if the inequalities λ ( ) ≤ λ ( ) ≤ λ ( ) ≤ λ ( ) hold, recalling that X ≥ hr X ⇒ X ≥ st X , then one has a case where X ≥ hr X ⇒ X ≥ hr:wj X holds true for negatively dependent random variables.

Relationships with multivariate aging notions
The notion of supermigrative copula had originated from the analysis of some questions concerning the concept of aging for a pair of exchangeable lifetimes. In the previous section we saw that the extension of supermigrativity to non-exchangeable copulas can be relevant in the analysis of the ≥ hr:wj property. In this vein, we point out in this section some relations between ≥ hr:wj and the Bivariate Increasing Hazard Rate property that will be recalled below.
First, we recall the de nition of IHR property, which is a well-known notion used in the description of the reliability of engineering systems (see, e.g., [1] or, equivalently, if their joint survival function F is Schur-concave.
The following statement provides a relation between B-IHR and joint weak hazard rate order. Proof. Fix any t , t such that ≤ t ≤ t . By letting δ = t − t it holds: where the inequality follows by assumption X + s ≥ hr:wj X for all s ≥ .
Relations between univariate aging, bivariate aging and dependence properties have been extensively studied in [2]. Among other results, the following relationship between IHR, B-IHR notions and supermigrativity property follows from Theorem 5.2(1) in [2]. Concerning the language and notation used in that paper, we observe that supermigrativity is denoted as P + in there and that P-positive 2-aging reduces to Bivariate IHR when one refers to the dependence property of supermigrativity.
Theorem 3.2. Let (X , X ) be a couple of exchangeable lifetimes, and let C denote its survival copula. If the margins X and X satisfy the IHR univariate aging notion and C is supermigrative, then (X , X ) satis es the bivariate ageing notion B-IHR.
As a consequence of the Theorem 3.1, we immediately obtain that Theorem 3.2 above follows as a corollary of Theorem 2.3. In fact, assume that (X , X ) has identically distributed and IHR margins, and that the corresponding survival copula C is supermigrative. Then, from the IHR property of X and Theorem 1.B.38(iii) in [20] one has X + s ≥ hr X ∀ s ≥ , which in turn implies X + s ≥ hr X ∀ s ≥ , by exchangeability of X and X . Now, by supermigrativity (thus also generalized supermigrativity) of C, and applying Theorem 2.3, it follows X + s ≥ hr:wj X ∀ s ≥ , i.e., that (X , X ) satisfy the B-IHR property by Theorem 3.1.
A statement describing relations between the B-IHR property and the joint weak hazard rate order, in the opposite direction with respect to what stated in Theorem 2.3, is now given. Observe that the assumption on the function ϕ considered here is satis ed, for example, by any di erentiable increasing concave function ϕ such that ϕ ( ) ≤ .

Theorem 3.3.
Given the couple (X , X ) of exchangeable lifetimes, if it satis es the B-IHR property then X ≥ hr:wj ϕ(X ) for any non negative increasing function ϕ which is subadditive and such that ϕ(t) ≤ t for all t ≥ .
Proof. Let H denotes the survival function of ϕ(X ), i.e., let H(t) = G(ϕ − (t)), where G is the survival function of the margins of X i . Since ϕ(t) ≤ t and X = st X , it clearly holds X ≥ st ϕ(X ). From subadditivity of ϕ follows the superadditivity of ϕ − (t) = G − (H(t)), which, in turns, implies, for all t, w ≥ , where the rst inequality follows from w ≤ G − (H(w)) for all w (i.e., X ≥ st ϕ(X )), while the second one follows from superadditivity of ϕ − (t).
As an immediate example of application of Theorem 3.3, consider an exchangeable vector (X , X ) whose marginal distributions are IHR, and assume its survival copula is of Archimedean type, having log-convex generator. As shown in [2], in this case (X , X ) satis es the B-IHR bivariate aging property, thus, by previous statement, one has X ≥ hr:wj ϕ(X ) for any di erentiable increasing concave function ϕ such that ϕ ( ) ≤ .