Relations between ageing and dependence for exchangeable lifetimes with an extension for the IFRA/DFRA property

We rst review an approach that had been developed in the past years to introduce concepts of “bivariate ageing" for exchangeable lifetimes and to analyzemutual relations among stochastic dependence, univariate ageing, and bivariate ageing. A speci c feature of such an approach dwells on the concept of semi-copula and in the extension, from copulas to semi-copulas, of properties of stochastic dependence. In this perspective, we aim to discuss some intricate aspects of conceptual character and to provide the readers with pertinent remarks from a Bayesian Statistics standpoint. In particular we will discuss the role of extensions of dependence properties. “Archimedean” models have an important role in the present framework. In the second part of the paper, the de nitions of Kendall distribution and of Kendall equivalence classes will be extended to semi-copulas and related properties will be analyzed. On such a basis, we will consider the notion of “Pseudo-Archimedean”models and extend to them the analysis of the relations between the ageing notions of IFRA/DFRA-type and the dependence concepts of PKD/NKD.

All along this note X , ..., Xn are considered to be exchangeable and such that P(X i = ) = , i = , ..., n. By G we denote the one dimensional marginal survival function: G (x) := P X j > x , for j = , ..., n and x ≥ .
As it is well known, for random variables, any property of dependence is generally compatible with any arbitrary choice of a one-dimensional probability distribution. This is guaranteed by the Sklar Theorem (see, e.g., [23,44]). However, compatibility may fail if we impose some extra condition on the joint probability distribution.
When X , ..., Xn are interpreted as lifetimes of di erent individuals, attention is often concentrated on the two di erent phenomena of stochastic dependence and of stochastic ageing. A very rich literature in applied probability has been devoted to this eld and, from a technical viewpoint, we remind that several di erent notions of dependence and of ageing have been considered. Typically, such properties are described in terms of inequalities involving comparisons between probabilities of di erent events or conditional probabilities of a xed event, given di erent information-states. Furthermore dependence and ageing are strictly related one another and, at a time, they are heavily a ected by the actual state of information about (X , ..., Xn). See, e.g., references [1-3, 7, 31, 34, 36, 44, 49, 51, 53]. Concerning the relations among such notions, in this paper we rst review an approach that had been developed in the past years, for the case of exchangeable lifetimes. This approach is based on the role of the family of the level curves of joint survival functions in the description of ageing for inter-dependent lifetimes. See [8,10,12,51]. The earliest motivation for such an approach had been the e ort to extend the property of lack of memory of exponential distributions to the case of a vector of dependent variables. Some of the results and other motivations are brie y recalled in Sections 2, 3, and 5. See also [41] and [18] for further developments. In such a context, the speci c notion of semi-copula emerges as a rather natural extension of the notion of copula and several derivations hinge on the extension to semi-copulas of concepts of stochastic dependence. In [12] a conceptual method was introduced to single out the appropriate notions of dependence and bivariate ageing, "corresponding" to a xed property of onedimensional ageing. This method arises in a Bayesian framework. Actually the relations among dependence, univariate ageing, and bivariate ageing provide us with necessary conditions for reciprocal compatibility among these notions. At a same time, those relations are relevant to understand the e ects of those changes of information which do not destroy exchangeability. One aim of our work amounts to illustrating this method and clarifying some intriguing aspects of it. Furthermore we propose a natural meaning of the term "corresponding" for the special class of dependence properties de ned in terms of Positive Dependence Orderings (see Section 6, step g)).
Some issues, concerning with properties of copulas and semi-copulas, still require further analysis and developments. Speci cally the analysis of the dependence concept of PKD/NKD and the related ageing concepts of IFRA/DFRA had been carried out only for Archimedean models (i.e., models with Archimedean survival copulas). In this respect the further aim of this paper amounts to extending the results in [12] from the Archimedean models to a larger class of models. On this purpose, we will introduce new concepts such as generalized Kendall distributions for semi-copulas and related Kendall equivalence classes of semi-copulas. Of special relevance for our aims are the Pseudo-Archimedean semi-copulas (i.e., Kendall equivalent to Archimedean semi-copulas). Such semi-copulas emerge when looking for necessary conditions for compatibility between PKD/NKD and IFRA/DFRA. Actually, our extension (see Theorem 7.19) concerns the subclass of Pseudo-Archimedean models to be de ned by suitable regularity conditions (see in particular properties (P1), (P2) in Subsection 7.2).
Our arguments and results will be demonstrated by means of some examples and remarks. For our discussion an important class of models is the Archimax one (i.e., models with Archimax survival copulas). As a relevant property, we exhibit a large class of Pseudo-Archimedean Archimax copulas (and semi-copulas) (see Example 7.11) and give necessary conditions for compatibility between PKD/NKD and IFRA/DFRA for Archimax models (see Example 7.20).
In the literature about stochastic dependence, positive dependence properties are often supposed to imply PQD, whereas it has been pointed out that PKD does not generally imply PQD, though such an implication does hold in the Archimedean case. This circumstance suggested us to introduce a stronger concept of positive dependence, de ned by requiring both PKD and PQD and denoted by PKQD. The family of PKQD copulas strictly contains PKD Archimedean copulas. Indeed we give su cient conditions under which Archimax copulas are PKQD. Also for this dependence concept some necessary conditions for compatibility with IFRA/DFRA are found.
The paper has a structure as follows. In Section 2, we concentrate attention on some simple aspects of the basic notions of IFR/DFR for one-dimensional probability distributions. In particular we focus on the circumstance that the IFR property for a family of conditional distributions can go lost under the operation of unconditioning. In Section 3 we recall the speci c bivariate IFR/DFR notion of our interest and introduce the issue of the relations between ageing and dependence notions. Section 4 is divided into three subsections, and is devoted to reviewing necessary notions about bivariate copulas, semi-copulas, dependence properties, Kendall distributions and bivariate ageing functions. The relevant case of Archimedean copulas and semicopulas is treated in details in Subsection 4.3. In Section 5 we concentrate attention on the univariate and bivariate ageing notions of IFR/DFR and review some speci c results concerning with relations between them and the corresponding concept of positive dependence, namely Super/Submigrativity. In Section 6 we explain in details the conceptual method introduced in [12] and leading to appropriate extensions of the arguments reviewed in the previous Section 5. Such extensions concern other notions of dependence and univariate/bivariate ageing. Furthermore we pave the way to the extension, to be developed in Section 7, of the results about the relations among stochastic dependence and IFRA/DFRA-type concepts of ageing. On this purpose, we introduce the concept of generalized Kendall distributions for semi-copulas. Furthermore we present de nitions and results about equivalence classes of semi-copulas, which are de ned in terms of generalized Kendall distributions. Subsection 7.3 is devoted to PKQD dependence property. In Section 8, we present a short discussion containing some comments, concluding remarks, and open problems. In order to let the paper be self-contained, in the Appendix we present arguments and proofs for some already known results, adapting them to our context and notation.

A brief review about one-dimensional IFR and DFR properties
Let T be a non-negative random variable and let the symbol G to be again used to denote the survival function of T. As a basic concept and a paradigmatic notion of univariate ageing we recall that T is Increasing Failure Rate (IFR) when, for any xed s ≥ , the function turns out to be a non-increasing function. Similarly, T is Decreasing Failure Rate (DFR) when it is a nondecreasing function of t. According to a common language, the notion of IFR de nes a concept of positive ageing, whereas DFR de nes a concept of negative ageing. If T is exponentially distributed then it is IFR and DFR, at a time, and the lack of memory property of univariate exponential distributions is also seen as a property of no-ageing.
The arguments in this section will be concentrated on the notions of IFR and DFR. This will constitute a basis for the brief discussion concerning bivariate ageing, that will be presented in the next Sections 3 and 5.
In the following result, attention is preliminarily focused on di erent characterizations of the concept of IFR distribution. The proof is almost immediate and will be omitted. Details can be found, e.g., in [51]. Before stating it, we recall the notion of Schur-Concavity (Schur-Convexity): A function W : It is crucial in our discussion that Schur-Concavity (Schur-Convexity) for a function W(x, y) is a property of the level curves {(x, y) s.t. W(x, y) = c}. For this and other properties of Schur-concave functions, see, e.g., [39].
Proposition 2.1. The following conditions are equivalent (i) T is IFR.
(iv) For two i.i.d. random variables T , T , distributed according to G (·) one has, for given s > and for given ≤ t < t , (2)

Remark 2.2.
In the case when a non-negative random variable T has a probability density function denoted by g(t), we can consider the failure rate function By item (ii) of Proposition 2.1 the latter is, of course, non-decreasing when T is IFR (whence the present terminology just arises).

Remark 2.3. Item (iii) of Proposition 2.1 is a property of the level curves of the survival function G(t )G(t ).
We notice furthermore that the characterizations given in both items (iii) and (iv) allow us to express the univariate positive ageing notion of IFR in terms of conditions for the bivariate joint distribution of two i.i.d. random variables. These observations will turn out to be relevant in our discussion.
By suitably modifying the statement of Proposition 2.1, and also the above two Remarks, one can directly obtain corresponding statements that are valid for the univariate, negative-ageing, concept of DFR. So far there is in fact a complete symmetry between the two notions of IFR and DFR. On the contrary, the following result points out an aspect of lack of symmetry between the two notions. Let Ξ be a set of indexes and, for the sake of notational simplicity, let us consider a family {G θ ; θ ∈ Ξ} of absolutely continuous survival functions over the half-line [ , +∞) and let G be given by a mixture of the G θ 's.

Proposition 2.4 ([7]
, see also [37] and [5] for a Bayesian interpretation). If G θ is DFR, ∀θ ∈ Ξ, then also any mixture G is DFR. Remark 2.5. Generally a mixture G will not be IFR, under the condition that G θ is IFR, ∀θ ∈ Ξ. Remark 2.6. It is easy to prove Proposition 2.4. As an immediate consequence, since the exponential distributions are DFR, one obtains that a mixture of exponential distributions is DFR. Since the exponential distributions are IFR as well, this property is a counterexample related with the above Remark 2.5.
In the Appendix we present an argument that allows one to understand both the Proposition 2.4 and the above Remarks from the point of view of Bayesian Statistics. The circumstance, that the IFR property can go lost under the operation of making mixtures, can also be understood as a Simpson-type paradox. (See Remark 3.1 below).

A bivariate concept of IFR for exchangeable variables
In the items (iii) and (iv) of Proposition 2.1, the case of two i.i.d. random times was considered. Focusing attention on the vector X ≡ (X , ..., Xn) of non-negative exchangeable (non-independent) random variables, we notice that it still makes sense to consider for X , ..., Xn a condition such as in (2): namely, for i ≠ j, for ≤ x < y, and t ≥ , Actually, in a subjective-probability or Bayesian framework, the latter can be interpreted as a bivariate condition of positive ageing (see [6,8,9,34,50,51]). As remarked above, (see (2) in Proposition 2.1), such a condition is equivalent to the IFR property of the marginal survival function G, when X , ..., Xn are i.i.d. variables. The opposite inequality is, on the contrary, equivalent to the DFR property of G and can therefore be interpreted as a bivariate condition of negative ageing.
The above inequalities (3) and (4) are conditions on the joint bivariate survival function of any pair X i , X j , with ≤ i ≠ j ≤ n, and we will denote the latter by In order to highlight that (3) and (4) are properties of positive/negative of bivariate ageing, it will be convenient to say that F ( ) (x, y) is Bayesian bivariate Increasing/Decreasing Failure Rate, abbreviated to Bayesian biv-IFR/Bayesian biv-DFR.
Let us now consider the case of conditionally independent, identically distributed, variables X , ..., Xn. What can be said for this case?
Let Θ be a Ξ-valued random parameter (with Ξ ⊆ R d , say), with probability distribution Π Θ and let X , ..., Xn be conditionally independent given Θ, with a conditional one-dimensional survival function G(·|θ) for θ ∈ Ξ, namely In such a case, the condition that G(·|θ) is IFR for all θ ∈ Ξ implies that the bivariate condition of positive ageing (3) holds true. However (see Remarks 2.5, 2.6) such a condition does not imply the IFR property of the marginal survival function G(x) = Ξ G(x|θ)dΠ Θ (θ), which would result, for t > and ≤ x < y, in the inequality The condition that G(·|θ) is DFR for all θ ∈ Ξ implies that both the condition that G is DFR and the condition of bivariate negative ageing (4) hold true. (2) is established between the conditional probabilities of two di erent events given a same conditioning event. In (6), on the contrary, we compare two conditional probabilities containing two di erent conditioning events. The inequality (6) is not implied then by the assumption P (X i > x + t|X i > x; θ) ≥ P X j > y + t|X j > y; θ as it happens when, e.g., Ξ = R+ and Θ| (X > x) is stochastically larger than Θ| (X > y), for ≤ x < y (see the Appendix). This conclusion can be looked at as a Simpson-type Paradox (see [48]).

Remark 3.1. Notice that the comparison in
As it is well known, dating back to the original work by de Finetti (see, e.g., [17]), the other cases of exchangeability di erent from those of conditional independent and identical distribution, are those of nite exchangeability.
What about the relation between the univariate and bivariate properties of ageing in the case when, alternatively to conditional independence, we assume X , ..., Xn to be nitely exchangeable?
Such relations are generally in uenced by the type of stochastic dependence among X , ..., Xn. The property of conditional independence does, in any case, imply some sort of positive dependence among X , ..., Xn. At least, positive correlation between X i , X j for ≤ i ≠ j ≤ n, as is very well-known.
In the case when X , ..., Xn are nitely exchangeable, the relations between the ageing properties of G and the conditions (3), (4) may be a bit more involved. Actually, in such a case, one can meet di erent types of (positive or negative) stochastic dependence (see, e.g., [51]) and the marginal survival function G can still be a mixture of given survival functions, but some of the coe cients of the mixture may be negative. (See, e.g., [30,32,35,38]). For our purposes, however, it is not really relevant to distinguish between nite or in nite exchangeability. Rather, we look at the actual properties of stochastic dependence of bivariate distributions, i.e., of the joint survival function F ( ) , de ned in (5). The rest of the paper is devoted to showing some results relating stochastic dependence and ageing properties. Preliminarily, on this purpose it is convenient to present a brief review of technical de nitions and related properties concerning bivariate survival models as in (5). This will be done in the next section, before continuing our analysis in the subsequent sections.

A review about bivariate copulas and ageing functions
For our convenience, and to x the notation, we start this section by just recalling well-known facts about bivariate copulas. We recall that a bivariate copula is a function C : In other words a bivariate copula is the restriction to [ , ] of a joint distribution function for a pair of random variables U, V, uniformly distributed in [ , ]. From now on the term bivariate will be generally dropped. Three special copulas are the following ones: We remind that any bivariate distribution function F(x, y), with marginal distribution functions F X (x), F Y (y) can be written as where C is a copula. When F X , F Y and F are continuous, then the copula is unique and we refer to it as the corresponding connecting copula. The term distributional copula is also used in the literature. Moreover, under the same conditions, the bivariate survival function F(x, y), with survival marginal distribution functions F X (x), F Y (y) can be written as where C is the corresponding two-dimensional survival copula. Furthermore looking at C and C as joint distribution functions of the pairs (U, V) and ( U, V), respectively, one can write The survival copula C is linked to the connecting copula C by means of the following relation The concept of copula is relevant in the description of dependence properties among random variables (see, e.g., [23,31,44]). Often a dependence property for a joint bivariate distribution, is equivalent to the same dependence property of the connecting copula. Obviously, properties of dependence may be assessed for the survival copula as well. It may happen that assessing a dependence property on the survival copula or on the connecting copula gives rise to di erent conditions for the joint distribution (see in particular Remarks 4.1 and 4.7, below).
In the paper [43] it is shown how the conditional distribution of Y, given di erent types of information on X, can be described in terms of appropriate distortions of the marginal distribution of Y. The speci c forms of such distortions are obtained in terms of the connecting copula. On the other hand, several conditions of stochastic dependence between X and Y, such as SI and LTD (to be recalled in the next subsection), can be characterized in terms of corresponding conditions of stochastic monotonicity for Y given X (or viceversa). In particular, the role of the types of information on Y in determining the type of dependence is analyzed in [26]. Therefore, in view of [43], the e ect of copulas in the analysis of these dependence properties may also be described by means of di erent forms of distortions of a marginal distribution.

. A brief review of dependence properties
We now move to recalling a few relevant de nitions of dependence properties. We start with the Positive Quadrant Dependence (PQD) property, i.e., the events {Y ≤ y} and {X ≤ x}, are positively correlated for all x and y: Another dependence property to be recalled is the Stochastic Increasingness (SI) in X: which is equivalent to the following properties of the two pairs of random variables U, V and U, V, i.e. of their respective copulas C(u, v) and C(u, v): A further dependence property is the condition: Y is Left Tail Decreasing (LTD) in X, i.e., the function x → P(Y ≤ y|X ≤ x) is decreasing: for all x ≤ x , and for all y, i.e., V is LTD in U, where U and V have been de ned in (12).
is instead equivalent to the condition that Y is Right Tail Increasing (RTI) in X, i.e., for all y ≥ , the function We are now going to recall further dependence properties that are specially relevant for what follows. We start with the so called Supermigrativity property: for an arbitrary copula D, i.e., D (us, v) ≥ D (u, sv) , for all ≤ v ≤ u ≤ and s ∈ ( , ).
This condition, applied to the survival copula C, has emerged to describe the Schur-concavity of F (see [12], see also Proposition 5.2, below). The term Supermigrativity has been coined in [19]. In a sense, Supermigrativity can be seen as a property of positive dependence: in particular it implies the PQD property.
In such a case the survival copula coincides with the maximal copula: and is obviously supermigrative.
For each of these positive dependence notions (PQD, SI, LTD, Supermigrativity), one can consider the corresponding notions of negative dependence (NQD, SD, LTI, Submigrativity). The latter ones are de ned in an obvious way, in the present bivariate case.
For what concerns Super/Submigrativity properties, in order to highlight that they are notions of positive/negative dependence, it will be convenient to use also the terms Positive/Negative Migrativity Dependence, abbreviated to PMD/NMD.

Remark 4.3.
As pointed out in [12] (see Proposition 6.1 therein), Supermigrativity (or equivalently PMD property) does coincide with the LTD property in the case of an Archimedean copula (a brief review of Archimedean copulas will be given next). Other features related with Supermigrativity have been analyzed in [19]).
The last dependence property to be recalled here is the Positive Kendall Dependence (PKD) property (see De niton 4.6, below). This property will have a relevant role for our results in Section 7 and is connected with the Kendall distribution function associated to F (see, e.g., [45]) We point out that K(t) depends only on the connecting copula C, and we will also use the notation K C (t). We also recall that K C (t) ≥ t in [ , ], as is easily checked.
In particular for the independence copula Π(u, v) = uv, one has In analogy with the Kendall distribution one can consider (see [41]) the upper-orthant Kendall distribution associated to F, i.e., In other words K(t) = K C (t).

Remark 4.4. Note that
where µ C is the probability measure on [ , ] such that The above remark suggests an alternative way to compute K C (t). This will be done in the following Lemma 4.5, the proof of which is almost immediate and will be omitted.
be a nite partition of ( , ], and denote by P the class of all nite partitions of ( , ]. Set furthermore

Lemma 4.5. Let C(u, v) be a copula, then
We now remind the de nition of Positive (Negative) Kendall Dependence (see in particular [14]) while D is Negative Kendall Dependent (NKD) when the reverse inequality holds. Two random variables X, Y are Positive (Negative) Kendall Dependent (PKD/NKD) when their connecting copula C is PKD/NKD. Two random variables X, Y are Positive (Negative) upper-orthant Kendall Dependent (PuoKD/NuoKD) when their survival copula C is PKD/NKD.

Remark 4.7.
We notice that the PKD property for the connecting copula C and for the survival copula C give rise to di erent dependence properties for the random variables X, Y.

. Bivariate ageing functions, semi-copulas, and related dependence properties
In the case of exchangeability, a survival function F(x, y) takes the form where C is an exchangeable copula and G denotes the common marginal survival function. Restricting our attention to this case we also associate to F(x, y) a function that describes the family of its level curves, according to the following de nition (see [10]).
Notice that in the above de nition we have used the convention − log( ) = +∞, F(x, +∞) = F(+∞, y) = F(+∞, +∞) = , for all x and y. From (18) one immediately obtains Any other survival function M(x, y) sharing with F(x, y) the same family of level curves, must also share the same ageing function B. Therefore M(x, y) must be of the following form However the function B is not always a copula. A bivariate copula, in fact, must have the properties of a bivariate probability distribution function. Whereas, on the contrary, examples of ageing functions B can be given such that Nevertheless, B turns out to be a copula in some special cases. (see [51]). In such a case the function B does coincides with the product copula Π(uv) = uv. It is immediate to

is a survival function if and only if G(x) is convex.
In the limiting case of "comonotone" dependence, (see Example 4.2) one has The term semi-copula has been proposed to designate functions which, like the ageing function B above, are increasing in each variable, satisfy the margin conditions (20), and are such that the inequality in (21) may hold for some values < u < u < , < v < v < . For a more formal de nition of semi-copula, basic properties, extensions and technical details about semi-copulas and ageing functions, also in a multivariate context, see in particular, the papers [10-12, 18, 21, 22, 24, 41], and the book [23] with references cited therein. We point out that the class of bivariate ageing functions is strictly contained in the one of semi-copulas (see, e.g., [23]).
Taking into account the expression (17) for the survival function we can also rewrite (18) as By setting whence we can also write The above expression suggests to introduce the following family of operators on the semi-copulas: for any increasing homeomorphism of [ , ], i.e., for any continuous strictly increasing function η : such that η( ) = and η( ) = , one may de ne the operator Φη which transforms a semi-copula S into the semi-copula Then, by using the above notation and recalling that G( ) = and x → G(x) is continuous strictly decreasing, we can actually write B = Φγ( C).
Viceversa, given the ageing function B and the marginal survival function G then the copula C is given by For our purposes it is useful to extend to semi-copulas de nitions of stochastic dependence that have been formulated for copula functions. In particular, it is immediate to de ne the property of Supermigrativity (Submigrativity) for any semi-copula S: We say that S is supermigrative (submigrative) i Similarly we point out that all the de nitions of dependence properties considered so far can be extended in a direct way, but the PKD/NKD property. The extension to semi-copulas of the latter property requires an appropriate de nition of the concept of Kendall distribution for semi-copulas. Such a de nition will be given in Subsection 7.1.

. Archimedean copulas and semi-copulas
One important class of copulas is the one of Archimedean copulas: Let ϕ : ( , ] → [ , +∞) be a continuous, convex, and decreasing function and denote by C ϕ the bivariate Archimedean copula with additive generator ϕ. Namely, for ≤ u, v ≤ , set It is convenient, though not strictly necessary, to assume ϕ strictly decreasing and such that Similarly, we say that a semi-copula S is Archimedean when it has the form where φ : ( , ] → [ , +∞) is a continuous decreasing function, not necessarily convex. We maintain the assumptions lim u→ + φ (u) = +∞, φ ( ) = . Since C ϕ is a copula, and Sφ is a semi-copula it makes sense to consider their dependence properties. Since ϕ − and φ − are, technically, one-dimensional survival functions, it makes sense to consider their ageing properties. On the other hand, the inverse of any one-dimensional survival function G, if it exists, can be seen as the generator of the Archimedean semi-copula In this case the ageing function B(u, v) is Archimedean as well. More precisely, recalling (22), (26), and (23) one has where Relating Dependence to Ageing Properties: the IFR/DFR case As announced at the end of Section 3, we continue to analyze the case of exchangeable variables X , ..., Xn, with two-dimensional marginal survival function F ( ) given in (5). We denote by G and C(u, v) the marginal survival function and the survival copula of F ( ) . We are now in a position to discuss the interrelations among dependence properties of PMD/NMD, the univariate positive/negative ageing properties of IFR/DFR, and the bivariate positive/negative ageing conditions of Bayesian biv-IFR/biv-DFR. In this respect we remind the following simple result, which can be obtained as a consequence of the general results given in Section 5 of [12]. We prefer to state this speci c result for di erent reasons. First of all it admits a self-contained proof, di erent from the proof in [12] (see the Appendix). Furthermore it provides the reader with a paradigmatic scheme of those results, and will help us to explain in the next section the general idea behind them.

Proposition 5.1. With the notations introduced above
In Proposition 5.1, for a joint exchangeable survival function F X attention has been xed on the twodimensional marginal survival function F ( ) and on the pair ( C, G) of the two-dimensional survival copula and the marginal survival function, so that F ( ) is given as in (17).
It is convenient however to rephrase that result in terms of the triple ( C, G, B), where B is the bivariate ageing function of F ( ) (see (18)). On this purpose we recall a result connecting the Bayesian biv-IFR/DFR properties (3) and (4), with the PMD/NMD properties for B (see [12]).
Lemma 5.2. The following three conditions are equivalent: Analogously also the following dual conditions are equivalent For the readers' convenience, in the Appendix we provide an autonomous proof adapted to our language and notation.
It is important to stress that the previous result shows that the Super/Submigrativity property for the ageing function B is a bivariate ageing property.
In view of the previous equivalences, we are now in a position to rephrase the result in Proposition 5.1 in terms of the survival copula C, the marginal survival function G and the bivariate ageing function B, as follows.
Remark 5.4. In [4] it was shown that an Archimedean copula, with a generator ϕ is LTI/LTD if and only if ϕ − is a IFR/DFR survival function, respectively. As pointed out in the previous Section 4, it makes sense however, to extend dependence properties to semi-copulas. It was observed in [12] that also for an Archimedean semicopula the LTI/LTD property is equivalent to the IFR/DFR property for the inverse of its generator. In particular G is IFR/DFR if and only if the semi-copula S G − (see (28)) is LTI/LTD. Moreover, again for Archimedean semicopulas, the LTI/LTD property is equivalent to the NMD/PMD property (see Remark 4.3). We can conclude therefore that the above result in Proposition 5.3 can be formulated in terms of the survival copula C, and the semi-copulas B and S G − . For instance item (i) can be reformulated as and similarly for items (ii)-(vi).
Let us concentrate again our attention on the three semi-copulas C, B, and S G − : Summarizing the above arguments, we can claim that PMD/NMD conditions imposed over two semi-copulas imply a condition -of type either PMD or NMD -for the third semi-copula.

A path to a more general analysis
As mentioned in the previous sections, the Supermigrativity property for a copula is looked at as a condition of positive dependence, while the IFR property of a marginal survival function is a condition of positive one-dimensional ageing. The terms Submigrativity and DFR respectively refer to the corresponding dual conditions of negative dependence and of negative one-dimensional ageing.
For the survival function F(x, y) in (17), one can say that Proposition 5.3 concerns with the properties Supermigrativity/Submigrativity and IFR/DFR for C and G, respectively. It concerns furthermore with Supermigrativity/Submigrativity of B, which in view of Lemma 5.2 can be seen as a notion of positive/negative bivariate ageing.
Several other concepts of stochastic dependence and of ageing have been considered in the literature. Obviously we can say that "dependence" is a property of the survival copula C, "one-dimensional ageing" is a property of the marginal survival function G. Furthermore one can consider notions of "bivariate ageing" de ned in terms of the ageing function B. See [12] for both technical details and some heuristic argument, and [21-24, 41, 52, 53] for further related discussions and results. As it will be also discussed below, positive/negative bivariate ageing can be de ned in terms of positive/negative dependence of the ageing function B, in analogy with the IFR/DFR case.
On such a basis, results of the same form as in Proposition 5.3 can be formulated for some other pairs of «dependence, one-dimensional ageing» properties and notions of "bivariate ageing", appropriately corresponding to them. Informally, we can describe as follows the general format of such results: 1. Positive dependence and positive one-dimensional ageing imply positive bivariate ageing 2. Positive bivariate ageing and negative one-dimensional ageing imply positive dependence 3. Positive bivariate ageing and negative dependence imply positive one-dimensional ageing. Furthermore, dual statements -where the terms "positive" and "negative" are interchanged -do hold.
In order to setting the above statements in a more precise form, we need to give appropriate answers to the following natural questions: Q1: What are the appropriate notion of dependence and property of one-dimensional ageing, "corresponding" to each other? Q2: What might be an appropriate notion of bivariate ageing, "corresponding" to a pair «dependence, one-dimensional ageing»?
Q3: What appropriate meaning should be given to the term "corresponding"?
Finally, assuming that one has given satisfying answers to the preceding questions, one wonders furthermore Q4: What about a general and synthetic method for proving statements of the same form as in Proposition 5.3?
In what follows, on the one hand, we explain the path that gave rise to the general answers to the above questions, as given in [12]. On the other hand we slightly modify those answers in a way which allows us to add a further result to those obtained in [12] (see Section 7). The path will be divided into several steps, reaching the answers in the last step g). a) As also pointed out in Remark 2.3, Proposition 2.1 ensures that the IFR/DFR conditions of onedimensional ageing for G can be translated into inequalities for the bivariate survival function F Π (x, y) := G (x) G (y) of two independent variables distributed according to G (see (iv) of Proposition 2.1, see also (iii), which turns out to be an inequality in view of (1) ). For instance, we can consider, in place of IFR/DFR conditions, the NBU/NWU conditions for G, i.e., b) As observed in Remark 2.3 the inequalities in the above step a), concerning the IFR/DFR properties, can be seen as properties of the family of the level curves of the function F Π (x, y) = G (x) G (y). Also the NBU/NWU properties (33) are properties of the family of the level curves for F Π . Any exchangeable bivariate survival function F(x, y), which shares with F Π (x, y) the family of level curves, also shares with F Π qualitative properties of bivariate ageing, as rst discussed in [8,10,51]. On the other hand, we recall that two di erent survival functions share the same family of level curves if and only if they share the same ageing function B. Whence, di erent bivariate ageing properties de ned in terms of the family of the level curves of F(x, y) turn out to coincide with properties of B.
c) It is remarkable that the bivariate ageing properties, mentioned in step b) and de ned in terms of properties of the ageing function B, actually coincide with di erent "dependence" properties of B (see [12]). The special case of PMD/NMD is considered here in Lemma 5.2. A heuristic argument for the above claim goes as follows. When B is a copula, then F B (x, y) := B(e −x , e −y ) is a true bivariate survival function, with standard exponential marginal distributions, as well as Π(e −x , e −y ). The special case when B is the independence copula Π corresponds to a bivariate no-ageing property. Furthermore, in general, since B does clearly coincide with the survival copula of F B (x, y), and simultaneously is also the bivariate ageing of it, the dependence property of such a model coincides with the bivariate ageing property. The latter is also the ageing property of F, since F and F B share the same family of level curves.
Before continuing our conceptual path, we point out that di erent papers in the literature had been devoted to the analysis of connections between dependence properties of C ϕ and ageing properties of ϕ − . Very precise and detailed results had, in particular, been obtained in [4,40]. Some notions of positive dependence for C ϕ have been shown to be equivalent to negative ageing properties of ϕ − . For the reader's ease, we recall the results in [4] concerning the relations between the dependence properties PQD, PKD, LTD, SI for an Archimedean copula C ϕ , and the ageing properties of the survival functions ϕ − . For simplicity sake we assume that the generator ϕ is di erentiable in ( , ) and denote by f ϕ (x) the density of ϕ − .
The DFRA/IFRA properties are recalled below in De nition 7.1. In the literature, property (37) for f ϕ is a well known property, also called Decreasing/Increasing Likelihood Ratio (DLR/ILR), see, e.g., [42]. We recall that an Archimedean copula is LTD if and only if is PMD (see also Remark 5.4). We also notice the following implications between notions of positive univariate negative ageing (see, e.g., [49]) Whence one obtains the following implications among notions of bivariate dependence for Archimedean copulas SI ⇒ LTD ⇒ PKD ⇒ PQD.
We can now resume our conceptual path by means of the following steps d)-g). d) We associate to the Archimedean copula C ϕ the bivariate survival model in i.e., the one with survival copula C = C ϕ and marginal survival function G := ϕ − . In other words this is the Schur-constant model (see Example 4.9 and (29)) The interest of this model is due to the circumstance that the equivalences (34)-(37) become nothing else but equivalences between properties of the survival copula and those of the marginal distribution. We remind that the ageing function B of the above model (38) is the independence copula Π and, in agreement with step c), the condition B = Π actually describes a bivariate no-ageing property. e) We now consider a survival model, with the same survival copula C = C ϕ as in step d), but with marginal survival function G, di erent from ϕ − . In this case the ageing function B is di erent from Π. As pointed out in Section 4, B is not necessarily a copula, however it turns out to be the Archimedean semicopula Sφ in (30) with generator φ given in (31), i.e., φ(u) = ϕ G(− log(u)) .
Actually B = Sφ is a copula if and only if φ is convex. Even when this is not the case, one is still allowed to consider "extended" dependence properties of B = Sφ . We remind that such "extended" dependence properties are looked at as bivariate ageing properties of the survival model. In this perspective, the results obtained in [4,40] can be extended to Archimedean semi-copulas and reformulated as equivalences between dependence properties of B = Sφ and univariate ageing properties of the generator's inverse φ − . f) We remind that one purpose of our discussion is the analysis of the interrelations among stochastic dependence of F = F ( ) , and ageing properties of the marginal survival function G. In the Archimedean case of step e), stochastic dependence of F(x, y) = F ( ) (x, y) = C ϕ (G(x), G(y)) can be characterized by ageing properties of the survival function ϕ − , in view of the equivalences (34)-(37) above. By the same token, bivariate ageing can be characterized in terms of the univariate ageing property of survival function φ − . By imposing one xed condition of ageing on each of the three univariate survival functions, ϕ − , φ − , G, we respectively obtain a property of dependence, of bivariate ageing, and of univariate ageing for the survival model F = F ( ) . These survival functions are linked together by the relation (31), which we rewrite in the form whence, imposing the xed property of ageing on two of these survival functions implies a condition for the third one. We give an example of this procedure in Lemma 7.21 (see also Remark 7.23). By taking again into account the equivalence properties of type (34)-(37), we convert such implications (for ϕ − , φ − , G) into implications (for C ϕ , Sφ, G) of the form appearing in items 1., 2., and 3. at the beginning of this section. Finally, observing that the positive/negative univariate ageing properties of G can be characterized by the negative/positive dependence properties of the semi-copula S G − , the previous implications can be rephrased as implications on the semi-copulas C ϕ , Sφ, S G − (as already pointed out in Remark 5.4 for the IFR/DFR property).
The announced conceptual path concludes with the following step. g) From the arguments in the above steps, and in particular step f), we see that, for the Archimedean case, it is equivalent to state results in terms of stochastic dependence of semi-copulas or in terms of univariate ageing of the inverse of their generators. However the description in term of stochastic dependence allows for a more general analysis. Actually, we can deal with stochastic dependence of semi-copulas, even for non-Archimedean semi-copulas.
In the Introduction of [47] the authors claim that "many of the modern notions of positive dependence are de ned by means of some comparison of the joint distribution of X and Y with their distribution under the theoretical assumption that X and Y are independent". We highlight that such an approach does hold for the dependence properties considered in this paper. Indeed they can be characterized in terms of appropriate comparisons between the two models F(x, y) = C G(x), G(y) and F Π (x, y) = Π(G(x), G(y)), and furthermore these comparisons become suitable comparisons between C and Π. For instance the PQD property means that Similarly, properties of bivariate ageing we considered can be characterized in terms of appropriate comparisons with the Schur-constant case, i.e., between the two functions F(x, y) and G(x + y).
In this respect we observe that the above two functions can be written as F(x, y) = G − log B(e −x , e −y ) and G(x + y) = G − log Π(e −x , e −y ) , and therefore the mentioned comparisons become now suitable comparisons between B and Π. For instance (see [10]) the bivariate NBU property, which is de ned by can also be written as The remarks presented so far provide us with a justi cation of the de nition of bivariate ageing properties of F(x, y) given in terms of dependence properties of B, as done in [12] and as stated at the beginning of this section. On this basis we are now in a position to reformulate the answers to the questions Q1-Q3 as given in [12]: Suppose that we start with a xed positive dependence property and the related negative dependence property for copulas. Then, as usual, one de nes • DEP the model F(x, y) is positive/negative dependent i the survival copula C has the positive/negative dependence property.
As discussed in [47] (see Remark 5.6 therein), many positive dependence notions are de ned by means of a positive dependence ordering in terms of the survival copula, rather than through the joint distribution function, namely: Once the positive/negative properties are extended to semi-copulas, then the bivariate and univariate ageing properties "corresponding" to the xed positive/negative dependence properties are respectively de ned as follows • biv − AG the model F(x, y) has the bivariate positive/negative ageing property i the ageing function associated to F, i.e., B = Φγ( C), has the positive/negative dependence property, where γ is de ned in (23) and, for any give increasing homeomorphism η, the operator Φη is de ned in (26).
• univ − AG the model has the univariate positive/negative ageing property i S G − has the negative/positive dependence property. The de nition of bivariate positive ageing as a positive dependence arises in a natural way as explained in step c), whereas the choice of giving positive univariate ageing the speci c form of negative dependence arises from the analysis of the Archimedean case, as justi ed in step f).
When the considered dependence property is de ned by means of a positive dependence ordering , then the previous de nitions biv − Ag and univ − Ag respectively become As far as question Q4 is concerned, again in [12] it is shown that the implications in items 1., 2., and 3. hold, provided suitable technical conditions are veri ed. Such conditions concern the interrelations between the chosen positive dependence property and the operators Φη. In particular we recall the condition that, for any increasing homeomorphism η As a matter of fact condition (42) does hold true for all the positive dependence properties considered in [12] and in this paper. By taking into account the identity S G − = Φ γ − (Π), when the above condition (42) does hold, then and therefore, we now de ne univariate positive/negative ageing property, in a form di erent, but equivalent to univ − Ag: • univ − Ag the model has the univariate positive/negative ageing property i the ageing function associated to the model F Π (x, y) = G(x)G(y) has the bivariate positive/negative ageing property, i.e., the Archimedean semi-copula Φγ(Π) has the positive/negative dependence property.
We also add that when the considered dependence property is de ned by means of a positive dependence ordering as in (40), then the above de nitions biv − Ag and univ − Ag can be written in terms of the ageing functions of the two models F(x, y) and F Π (x, y), namely Φγ( C) Π, Φγ(Π) Π. (43) As emphasized by (41), the original de nitions biv − Ag and univ − Ag manifest an asymmetric form, which is eliminated in the alternative formulation biv − Ag and univ − Ag .
Remark 6.1. The path a)-g), illustrating the de nitions of dependence, univariate and bivariate ageing and the interrelations among them, has brought us from the analysis of Archimedean models to the analysis of more general (non-Archimedean) models. We notice however that Archimedean semi-copulas play a fundamental role even for the general case. In particular such a role manifests in the de nitions of univariate ageing and of Schur-constant models, and also in the condition (42), which actually deals only with Archimedean semi-copulas.
We now conclude this section with a couple of further remarks concerning stochastic dependence. First of all we point out that in the present setting the dependence properties, to be considered, should not be necessarily de ned on the class of all copulas, but only on convenient subclasses. Consequently also the extension of dependence properties to semi-copulas can be restricted to corresponding subclasses. We notice that, also in contexts di erent from ours, the study of the dependence notions considered in the literature has been often restricted to special subclasses of copulas (see, e.g., [4,16,40]). In the frame of our setting, an instance of such a circumstance, emerges in [12] when dealing with the PKD property, which has been analyzed only for the class Archimedean copulas and semi-copulas. In order to treat the PKD property for a larger class of models, we need to introduce the concept of generalized Kendall distribution and the related equivalence classes of semi-copulas (see De nitions 7.3 and 7.8 below). Such an extension will be obtained in the next Section 7.
Finally we point out that the concepts of positive dependence which have been considered in the literature, often require a set of desirable properties, among which the condition that positive dependence be stronger than PQD. Such a condition, also required in [12], does not appear however to be strictly needed. In particular, the positive dependence property PKD, which implies PQD for Archimedean semi-copulas as recalled above, does not imply PQD on the family of all semi-copulas.

Relating Dependence to Ageing Properties: the IFRA/DFRA case
In the Reliability literature (see [7,13,34,49]) relevant concepts of positive/negative one-dimensional ageing are those of Increasing/Decreasing Failure Rate in Average (IFRA/DFRA), which, respectively, generalize those of IFR/DFR and are de ned as follows.

De nition 7.1. A one dimensional survival function G is IFRA if and only if
is an increasing function (see e.g. [7]); G is DFRA if and only if it is a non-decreasing function.
This notion of ageing is strictly related with the notion of PKD/NKD. As shown in [4], and recalled in Section 6 (see (35)), an Archimedean copula C = C ϕ (with a di erentiable generator ϕ) is NKD/PKD if and only if ϕ − is a IFRA/DFRA survival function, respectively. Namely, as it is well known (see [27]), and therefore C ϕ is PKD if and only if Let us now come to the exchangeable model F(x, y) in (29) with marginal survival function G and Archimedean survival copula C = C ϕ (with a di erentiable generator ϕ, as above). As we observed in Remark 4.11, the corresponding ageing function is B = Sφ, where we are using the notation given in (30) and (31), i.e., φ(u) = ϕ G(− log(u)) . Inspired by the latter circumstance, in [12] the PKD/NKD property was extended to Archimedean ageing functions (not necessarily copulas) by requiring φ(u)/φ (u) ≥ u log(u), and it was pointed out that the above equivalence between NKD/PKD and IFRA/DFRA holds even for Archimedean ageing functions.
For such an exchangeable model, a result similar to Proposition 5.3 was given in [12] (see Example 7.4 therein), for what concerns relations between PKD/NKD properties for ageing functions and IFRA/DFRA. Such a result can be rephrased here as follows.

Proposition 7.2. For the above model the following implications hold. (i) If G is IFRA and C
The previous Proposition 7.2 deals then with the concept of PKD for an Archimedean ageing function B. Our aim in this section is to show that the previous result admits a natural extension to a larger class of exchangeable models, where the survival copula C is not necessarily Archimedean. On this purpose, in the next subsection we extend the concepts of Kendall distributions and NKD/PDK property to semi-copulas.

. Generalized Kendall distributions and PKD/NKD property for semi-copulas
The next de nition of generalized Kendall distributions for semi-copulas is given in analogy with Lemma 4.5. The generalized Kendall distribution associated to S is the function K S : [ , ] → R + , de ned by where P is the class of all nite partitions of [ , ], of the form {u i , i ∈ I}, such that u = and u i ≤ u i+ .

be an Archimedean semi-copula (see Subsection 4.3)
where φ(t) is strictly decreasing, di erentiable, and such that φ( ) = . Furthermore assume that φ (t) > in ( , ). Then In fact, when φ(t) is also convex, then Sφ is a copula and the above identity coincides with (44). Moreover, still remaining in this case, by Lemma 4.5, K Sφ (t) can be computed by (45), i.e., The general case can be deduced observing that the latter equality also holds when φ is not convex.
We notice that, with this new de nition, the generalized Kendall distribution has the properties for any semi-copula S as it happens for copulas. Indeed, rst of all, w.l.o.g., we may consider only partitions containing t as an element and observe that, when u i+ ≤ t, and the rst property in (46) follows by observing that the increments The other two properties are obvious. Furthermore we point out that, when S is a Lipschitz semi-copula, i.e., Indeed in the latter case one has ≤ S(u i+ , v i ) − S(u i , v i ) ≤ u i+ − u i . In particular K S (t) ≤ when L S = , i.e., when S is a quasi-copula (see, e.g., [23]). Finally we observe that, when S is not a copula, K S (t) may not be a probability distribution function as shown by the following example.
Then, by taking into account Example 7.4, is not a distribution function, since, in particular, K S (t) converge to ∞ as t goes to .
De nition 7.6. A semi-copula S has the PKD property when the generalized Kendall distribution satis es the following inequality and similarly for the NKD property.
We recall that, for two copulas C , C , one says that C PKD C when K C (t) ≤ K C (t). This ordering can be naturally extended to semi-copulas so that the above de nition of Positive Kendall Dependence for S is equivalent to S PKD Π. Remark 7.7. We point out that Proposition 7.2 also holds when giving the concept of PKD/NKD property our extended De nition 7.6. Indeed Example 7.4 shows that, for Archimedean semi-copulas, our extension of the PKD property coincides with the one proposed in [12], which concerns exclusively with Archimedean semi-copulas.
Our aim now is to show that Proposition 7.2 admits a natural extension to a larger class of exchangeable models, where the survival copula C is not necessarily Archimedean. On this purpose we start by extending to semi-copulas the de nition of the equivalence relation based on the Kendall distribution, introduced in [46] for copulas.

De nition 7.8. Two bivariate semi-copulas S (u, v) and S (u, v) are Kendall-equivalent, written as S ≡ K S , if and only if the Kendall distributions K S (t) and K S (t) do coincide. For a xed semi-copula S(u, v) we denote by K S the equivalence class containing S, and by C S the subclass of the exchangeable semi-copulas belonging to K S .
Before continuing we present three examples related with the above de nitions. Example 7.9 shows a case of a non-exchangeable copula belonging to the same Kendall equivalence class of two exchangeable copulas, one of which is Archimedean. Example 7.11 shows that each copula in the Archimax family is Kendallequivalent to an Archimedean copula or to the maximal copula M(u, v). Starting from these examples, in Example 7.12 we exhibit instances of non-Archimedean exchangeable semi-copulas belonging to the same Kendall equivalence class of Archimedean semi-copulas. Example 7.9. Consider the following copulas and the Archimedean copula C ϕ , with generator As observed in Example 1 in [46], ϕ is a convex generator and K C (t) = K C (t) = K C ϕ (t) = min( t, ). Then C , C ∈ K C ϕ , C ∈ C C ϕ , but C ∉ C C ϕ .
For the next example we need to recall the de nition of Archimax copulas, introduced in [15].
As shown in [15], the Kendall-τ coe cient of C ϕ,A can be computed as τ ϕ, and τ ϕ is the Kendall-τ of C ϕ .
is a convex generator and the copula C ϕ,A belongs to the equivalence class K C ϕ . Indeed, on the one hand, as observed in [28], by Proposition 5.1 in [15], On the other hand, we notice that

When A(t) = A( − t), i.e., when A(t) is symmetric with respect tot = / , then C ϕ,A is exchangeable and
therefore belongs to C C ϕ . As an example we can take, for θ ∈ [ , ], .
A di erent situation is met when τ A = , indeed then K C ϕ,A (t) = t and therefore C ϕ,A ∈ C M . For example one can consider the case The above claim could be checked directly or it can be seen as a consequence of Proposition 7.18, below. Note that when β < then G and φ are convex and Sφ is copula, while when β > then G and φ are not convex and Sφ is only a semi-copula. For β = , i.e., when the marginal distribution is exponential of parameter 1, then B = C(= C ), as already observed in Section 6 (step c)).

Let us now consider the function
obtained by replacing the convex generator ϕ with a non-convex generatorφ in De nition 7.10 of an Archimax copula. As it is easy to check, Sφ ,A is a semi-copula, and we will call it an Archimax semi-copula. Then, similarly to Example 7.11, one gets that Sφ ,A is Kendall-equivalent to the semi-copula Sφ, where φ =φ /( −τ A ) , when τ A < . Indeed the generalized Kendall distribution of Sφ ,A can be obtained by following a procedure analogous to the one in Example 7.4 for the Archimedean case.
In view of De nitions 7.6 and 7.8 we can claim that if a semi-copula S is PKD/NKD then all the semi-copulas in the equivalence class C S share the same Kendall dependence property. The latter claim suggests us that the appropriate extension of Proposition 7.2 is obtained by replacing the conditions C = C ϕ and B = Sφ with the conditions C ∈ C C ϕ and B ∈ C Sφ , respectively. A precise result will be given in Theorem 7.19 below. To this end we also introduce the following further de nition.

De nition 7.13. A semi-copula S is Pseudo-Archimedean whenever there exists an Archimedean semi-copula
Sφ such that S ∈ K Sφ . The generator φ of Sφ will be referred to as the pseudo-generator of S.
Clearly when a semi-copula S is exchangeable and Pseudo-Archimedean with pseudo-generator φ then S ∈ C Sφ .
In view of the above de nition, Examples 7.11 and 7.12 can be rephrased by saying that, when τ A < , then the Archimax copula C ϕ,A and semi-copula S φ,A are Pseudo-Archimedean, with pseudo-generators ϕ = ( ϕ) /( −τ A ) and φ = ( φ) /( −τ A ) , respectively. We end this subsection with an example of a family of PKD copulas. Example 7.14. Assume that the Archimedean copula C ϕ is PKD. Then, any Archimax copula C ϕ,A in Example 7.11 is PKD, whenever τ A ∈ [ , ). Indeed by assumption and Obviously the same holds for C ϕ , with ϕ = ( ϕ) /( −τ A ) , the pseudo-generator of C ϕ,A . In view of Example 7.12 similar considerations hold when considering an Archimax semi-copula. On the contrary, when C ϕ is NKD, no similar implication can be obtained.

. Generalization of Proposition 7.2
In this subsection we show the generalization of Proposition 7.2 to the models F(x, y) = C G(x), G(y) , with C (not necessarily Archimedean) and G satisfying the following conditions: (P1) for any u ∈ ( , ) the function v → Cu(v) := C(u, v) is strictly increasing and continuous (and therefore invertible), (P2) the function x → G(x) is strictly positive in [ , ∞), strictly decreasing, and continuous (and therefore invertible). For a copula C satisfying (P1), Genest and Rivest in [28] have proved a transformation result for Kendall distribution (see Proposition 1 therein), which we reformulate in our notation as follows. Fix an increasing homeomorphism η : ) is a copula, then the Kendall distribution K C * can be obtained in terms of K C as follows: In the perspective of generalizing Proposition 7.2, we will extend the above transformation result to the case when C * is not necessarily a copula. In this way we will obtain the generalized Kendall distribution for the aging function B (see Proposition 7.16 below). Indeed, when G(x) satis es (P2), then the functions γ(u) and γ − (z) are increasing homeomorphism and then, taking η = γ and C = C, the function C * coincides with the ageing function B, namely Assume furthermore that the density g of G is continuous, then Proof. We essentially need to prove (51) for t ∈ ( , ). In fact -Equation (50) is exactly Proposition 1 in [28], -Equation (52) just amounts to a reformulation of (51), by rewriting γ − (t) and γ (γ − (t)) in terms of G.
-Equation (53) easily follows by (50) and (51) by taking into account that and by the change of variable u = γ(u) within the previous integral. In this respect we point out that (51) is obvious for t = and t = .
In view of (47), the proof of (51) for t ∈ ( , ) is obtained by proving that As it is well known, any copula is a Lipschitz function, with Lipschitz constant equal to , therefore C(u, v) being a copula, we have where we have set δ := max(w i+ − w i ; t ≤ w i ≤ w i+ ≤ ).
Without loss of generality we may assume that t + δ < , so that Finally, observing that C(w i , z i ) = γ − (t), we have that Recalling that g(x) denotes the continuous density of G, we rewrite γ γ − (t) = t g(− log(t)) . By assumption the latter function is continuous in ( , ), and the proof of (51) is accomplished by observing that sup{ i∈I w i+ >γ − (t) Remark 7.17. Apparently, the equation (51) is nothing else but an equivalent form to write the thesis of Proposition 20 in [41]. Actually the di erence between the two results relies on the present generalization of Kendall distributions to semi-copulas. In [41] such a generalization was not considered, rather the right hand side of (53) was taken as an operator on ageing functions. Here, on the contrary, K B (t) is de ned directly by (45), or, equivalently by (47), and equation (53) is obtained as a direct consequence.
The present approach allows us to show that B is Pseudo-Archimedean if and only if C is such, and to obtain the pseudo-generators of them. Finally, as a crucial issue, we can thus obtain the relation tying G with such pseudo-generators. The precise statements follow: conditions (P1) and (P2). Fix a t ∈ ( , ) and de ne and Then (i) ϕ is the generator of an Archimedean copula, namely C ϕ , and φ is the generator of an Archimedean semicopula, Sφ, (ii) C and B are Pseudo-Archimedean and, in particular, C ∈ C C ϕ and B ∈ C Sφ . Assume furthermore that the density g of G is continuous, then the following equivalent relations hold Proof. By Proposition 1.2 in [27], and as observed in [46], we know that, for any "true" Kendall distribution function K(t) of a bivariate survival function, the function ϕ K (t) := exp t t s−K(s) ds is the generator of an Archimedean copula, i.e., ϕ K (t) is increasing and convex, with ϕ K ( ) = , if and only if K(t − ) > t for all t ∈ ( , ). Moreover K(t) is the Kendall distribution of the Archimedean copula C ϕ K . Under (P1) and (P2), by (50) in the previous Proposition 7. 16, the Kendall distribution function of C satis es K C (t − ) > t for all t ∈ ( , ), so that ϕ(t) is the generator of C ϕ , and clearly and (i)-(ii) follow as far as ϕ(t) and C are concerned. Similar results hold except for the convexity property (see also [41]), as far as φ(t) and B are concerned, in particular Finally, by using the above expressions of K C (t) and K B (t), together with (51), one immediately gets that φ(t) = ϕ(γ − (t)), i.e., (iii) follows.
We are now in a position to extend Proposition 7.2 by considering the case when the survival copula C is Pseudo-Archimedean. Actually such a generalization is proved by collecting the results in Proposition 7.18, Remark 7.7, and Proposition 7.2 itself.

Theorem 7.19. Let F(x, y) be a bivariate exchangeable model satisfying the conditions (P1) and (P2). Besides the standing assumptions on G, assume that the corresponding density g(x) is continuous. Finally assume that the generator ϕ(t) de ned in (55) is di erentiable. Then (i) If G is IFRA and C is
Example 7.20. When C = C ϕ,A is an Archimax copula, with A(t) = A( − t), then the ageing function B = S φ,A , as de ned in (49), with φ = ϕ G(− log(u)) . Moreover, when τ A < , by Example 7.14 and Proposition 7.18, then B is Kendall-equivalent to Sφ, with φ(u) = ϕ G(− log(u)) , where ϕ = ( ϕ) /( −τ A ) . In particular, when Sφ is PKD and G(x) = e −x β , with β < , then B is PKD and G is DFR and therefore also DFRA. As a consequence of the above Theorem 7.19, the Archimax copula C is PKD. Similarly we can obtain that Sφ PKD and C ϕ NKD necessarily imply that G is IFRA, whence β ≥ .
We have proved the above Theorem 7.19 by using, among other results, Proposition 7.2, presented in [12]. An alternative proof, avoiding Proposition 7.2, can be given by using the following Lemma 7.

. The PKQD dependence property
One of the desirable axioms for a positive dependence notion, as proposed by Kimeldorf and Sampson (see, e.g., [33]), is that it implies PQD. As pointed out at the end of Section 6, however, PKD does not generally imply PQD. This fact is related with the circumstance that PKD does not satisfy the antisymmetric property, namely it is a preorder, rather than an order. Actually the implication "PKD ⇒ PQD" is not needed in the proof of Theorem 7.19, when using Lemma 7.21. Generally when a notion of positive dependence does not satisfy the above mentioned axiom, as it happens for PKD, one may follow two di erent approaches. A rst simple approach amounts to referring to such a notion as a weak positive dependence notion. When instead, for some purposes, the PQD property is needed, one may switch to another notion of positive dependence de ned by adding the requirement that PQD is also satis ed. More speci cally, along the latter approach, for a semicopula S we now consider the positive dependence condition obtained by requiring S to be simultaneously PKD and PQD. We will refer to this condition with the symbol PKQD. When considering PKQD as positive dependence and IFRA as univariate ageing, under the conditions (P1) and (P2), implications of the kind 1., 2., and 3. in Section 6 still hold, for C Pseudo-Archimedean (and exchangeable). The latter claim follows by taking into account the implications of the kind 1., 2., and 3. for the PQD/NQD notions of dependence and the NBU/NWU notions of univariate ageing (see [12]), Theorem 7.19, and the implication "IFRA ⇒ NBU". For example, we obtain that if G is IFRA and C is PKQD then B is PKQD, as well. As already mentioned in Section 6, for the Archimedean semi-copulas the implication "PKD ⇒ PQD" holds and therefore PKQD coincides with PKD. We now present a class of PKQD semi-copulas, larger than the class of Archimedean ones. Example 7.24. As in Example 7.14, we consider a generator ϕ such that C ϕ is PKD, and therefore also PQD. We claim that for any A(t) the Archimax copulas C ϕ,A are simultaneously PKD and PQD. Indeed the PKD property holds, as already observed in Example 7.14. As far as the PQD property is concerned, we start by observing that C ϕ,A is PQD if and only if the following condition is veri ed Furthermore, when A(t) = A (t) = , then C ϕ,A coincides with C ϕ and the PQD property for C ϕ implies the same property for C ϕ,A for any A, since max(t, − t) ≤ A(t) ≤ A (t) = . Same arguments hold for Archimax semi-copulas. When G satis es (P2), g is continuous, C = C ϕ,A satis es (P1), and A(t) = A( − t), the implications of the kind 1.,2.,3. hold for PKQD and IFRA. Note that however we cannot use this result when A(t) = max(t, − t), since the Archimax copula C ϕ,A is not Pseudo-Archimedean, and therefore it does not satis es (P1). Remark 7.25. We observe that Examples 7.14 and 7.24 respectively show that the PKD and PQD conditions for the Archimedean semi-copula S φ are inherited by all the Archimax semi-copulas S φ,A . A further property of this type has been pointed out in [29], where it is shown that the SI dependence condition for the copula C φ is inherited by all the Archimax copulas C φ,A (see Theorem 3.1,therein). Notice however that for the corresponding negative dependence properties similar implications do not hold.

Summary and open problems
For vectors of exchangeable non-negative random variables with the meaning of lifetimes, we have reconsidered the relations between dependence and ageing already studied in [12]. Thus, as to stochastic dependence, we have considered properties of described in terms of the associated bivariate survival copula C. Concerning the common, one-dimensional, marginal distribution for the single variables, ageing properties are considered, such as IFR/DFR, NBU/NWU, IFRA/DFRA. A basic role in this study is played by the bivariate ageing function B, and bivariate ageing properties de ned in term of it. Using the latter properties, in [12] the equivalence relations of [4] have been extended to obtain di erent types of implications between dependence and ageing. Those equivalence relations can be re-obtained by imposing the condition B(u, v) = uv (i.e., by considering Schur-constant models).
As a main motivation of our work, we made an e ort to demonstrate the logic underlying such an approach for exchangeable variables. A further motivation comes from the need to enrich the analysis of the relations between the ageing notions of IFRA/DFRA and the dependence concepts of PKD/NKD. Actually a model is Pseudo-Archimedean (i.e., its survival copula is Kendall equivalent to an Archimedean copula) under suitable regularity conditions and the results obtained in [12] for Archimedean models have been extended here to this subclass of Pseudo-Archimedean models. Even though the latter subclass is rather wide, the problem remains however open whether our extension can be still obtained when dropping the above mentioned regularity conditions and/or the Pseudo-Archimedean property. In this respect one might rely on the result, proved in [46], showing that each Kendall equivalence class of copulas is characterized by a unique associative copula, along with the well-known fact that an associative copula coincides with the ordinal sum of a collection of copulas C i , where C i is either Archimedean or equal to M. One step towards the mentioned extension is the characterization of the PKD/NKD property for an associative copula C. On this purpose one can use the formula giving the Kendall distribution of C (see [46]). A further step might amount to limiting attention to models with survival copula Kendall-equivalent to an ordinal sum of Pseudo-Archimedean copulas.
As pointed out by the arguments in Subsection 7.3, another issue for further analysis is the characterization of exchangeable copulas such that the implication "PKD ⇒ PQD" holds. In any case this implication has not been used to obtain the interrelation results between dependence and ageing in Theorem 7.19. Note that on the one hand PKD does not in general imply PQD, on the other hand it corresponds to a dependence ordering (or preordering) as in (40). This fact may suggest that analogous interrelation results hold for any dependence property de ned in terms of a positive ordering, or preordering, even when not implying PQD. Indeed, as mentioned in Section 6 (see in particular (43) at the end of step g)), for this kind of dependence notions, the ageing properties can be written in terms of the ageing functions of the two models F(x, y) and F Π (x, y). This observation can be exploited for a simpli cation and an extension of the Consider now the failure rate function r of T r(t) = g (t) G (t) = +∞ g θ (t) dΠ Θ (θ) +∞ G θ (t) dΠ Θ (θ) and the conditional probability distribution of Θ given the event {T > t}, which in view of Bayes' theorem is de ned by the equation Thus we can write We can now compare the mixing measure Π Θ (θ) and the mixing measure Π Θ (θ|T > t), appearing in (59) and (60), respectively. The latter measure does obviously depend on t, while the former does not. This aspect is at basis of the argument aiming to justifying Proposition 2.4, Remark 2.5 and Remark 2.6 . In this respect we assume, for simplicity's sake, that Π Θ is absolutely continuous, denote its density function by π Θ (θ), and consider, for ≤ t < t , the ratio between the conditional densities of Θ, given the events {T > t } and {T > t }. By using the Bayes Formula, we obtain As a substantial simpli cation, furthermore, we consider the case when r θ (t) is monotone w.r.t. the variable θ, for any t ≥ : for example, r θ (t) ≤ r θ (t), for any θ ≤ θ .
Namely we consider the case when the ratio is non-increasing as a function of t. Under the assumption (62), we separately consider now the cases when the IFR property holds for all the conditional distributions of T given Θ = θ and when the DFR property holds for all the conditional distributions of T given Θ = θ. Thanks to (62), for ≤ t < t , we have that the ratio G θ (t ) G θ (t ) , and therefore the ratio in (61), is a non-increasing function of θ. This condition implies (see, e.g., [49]; see also Chapter 3 in [51]) that the conditional distribution Π Θ (θ|T > t ) is stochastically greater than Π Θ (θ|T > t ), namely, for any nondecreasing function δ (θ), one has +∞ δ (θ) π Θ (θ|T > t ) dθ ≥ +∞ δ (θ) π Θ (θ|T > t ) dθ.
On the contrary, in the conditionally IFR case, when, for any θ ≥ , the functions t → r θ (t) are non-decreasing, one is not allowed to conclude that also t → r(t) is non-decreasing since, in this case, r θ (t ) ≤ r θ (t ).
It is thus clear that, even assuming the IFR property for all the conditional distributions of T given Θ = θ, we cannot generally conclude that the marginal distribution of T is IFR as well (notice that the latter conclusions would also stand valid under the condition that r θ (t) is decreasing w.r.t. the argument θ). In this respect, see also arguments in [5].

A. Arguments from Section 5
Proof of Proposition 5.1. It is su cient to prove the implication in item (i). The other implications can be proven similarly. Fix ≤ x < y, t > , so that G (x) > G (y). Set, furthermore so that G (y + t) G (x) ≤ s for G IFR.
By reminding (11) and that C is supermigrative we get By using the assumption of G IFR, we have G(y) · s ≥ G(y + t). Furthermore, C being a copula and then nondecreasing in each variable, we can conclude P X i > x + t|X i > x, X j > y ≥ C G(x), G(y) · s F ( ) (x, y) ≥ C G(x), G(y + t) F ( ) (x, y) = P X j > y + t|X i > x, X j > y .
Proof of Lemma 5.2. For ≤ x < y, the inequality (3) just means F ( ) (x + t, y) ≥ F ( ) (x, y + t), for any t ≥ , namely, in view of exchangeability, the condition (1) of Schur-concavity for F ( ) , i.e., condition (a) is equivalent to condition (b). It only remains to prove that condition (b) is equivalent to condition (c). Indeed, taking into account (19), and the fact that the function ξ → G(− log(ξ )) is increasing, the above Schurconcavity condition is equivalent to B(e −(x+t) , e −y ) ≥ B(e −x , e −(y+t) ), for any t ≥ , and ≤ x < y.
Setting u = e −x , v = e −y and s = e −t , the above condition is equivalent to B being PMD, i.e., to the Supermigrativity condition (27) for B. The proof of the equivalence of the other conditions is similar.