Diagonal sections of copulas, multivariate conditional hazard rates and distributions of order statistics for minimally stable lifetimes

As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables $T_{1},...,T_{r}$. In any case, we assume that $T_{1},...,T_{r}$ are identically distributed, with a common survival function $\overline{G}$ and their survival copula is denoted by $K$. The diagonal's and subdiagonals' sections of $K$, along with $\overline{G}$, are possible tools to describe the information needed to recover the laws of order statistics. When attention is restricted to the absolutely continuous case, such a joint distribution can be described in terms of the associated multivariate conditional hazard rate (m.c.h.r.) functions. We then study the distributions of the order statistics of $T_{1},...,T_{r}$ also in terms of the system of the m.c.h.r. functions. We compare and, in a sense, we combine the two different approaches in order to obtain different detailed formulas and to analyze some probabilistic aspects for the distributions of interest. This study also leads us to compare the two cases of exchangeable and minimally stable variables both in terms of copulas and of m.c.h.r. functions. The paper concludes with the analysis of two remarkable special cases of stochastic dependence, namely Archimedean copulas and load sharing models. This analysis will allow us to provide some illustrative examples, and some discussion about peculiar aspects of our results.


Introduction
Concerning the basic role of the concept of copula and of the Sklar's theorem in the analysis of stochastic dependence, a main issue is the study of the distributions of the order statistics X 1:r , ..., X r:r for a set of interdependent random variables X 1 , ..., X r . On one hand, the condition of exchangeability is specially relevant (see in particular Galambos (1982) [11]) in such a study. On the other hand, the marginal distributions of X 1:r , ..., X r:r are strictly related to the diagonal, and sub-diagonal, sections of copulas (see, e.g., Jaworski (2009) [12], Durante and Sempi (2016) [9]). For these reasons, in the theory of order statistics, the study of diagonal and sub-diagonal sections of copulas has been mainly concentrated on the case of exchangeable random variables.
Really, in such a study, the assumption of exchangeability can at any rate be replaced by the more general condition that, for d = 2, ..., r − 1, all the d-dimensional diagonal sections of copulas do coincide. Such a condition has been attracting more and more interest in the recent literature, where it has been however designated by means of different terminologies. In fact, such a condition can actually manifest under different mathematical forms, as we will discuss in details. For our purposes it is specially convenient to look at it as the condition that X 1 , ..., X r are minimally stable (see Definition 2 below).
In this note we concentrate attention on the case of non-negative, minimally stable, random variables which we denote by T 1 , ..., T r .
Generally, concerning with non-negative random variables, stochastic dependence can also be conveniently described in terms of stochastic intensities of related counting processes. See in particular Arjas (1981) [1], Bremaud (1981) [3], Arjas and Norros [2]. Such a description, in particular, can be based on the knowledge of the so-called multivariate conditional hazard rates (m.c.h.r.) functions, when attention is restricted to the absolutely continuous case (see in particular the papers by Shaked and Shanthikumar [25,26,27]). In such a case the family of those functions gives rise to a method to describe a joint distribution, which is alternative to the one based on copulas and marginal distributions or on the joint density function.
From an analytical view-point the two methods are actually equivalent in that a formula is clearly available to obtain in terms of the joint density the family of the m.c.h.r. functions, just in view of the very definition of them. On the other hand, an "inverse" formula permits to recover the joint density function when the m.c.h.r. functions are known. As a matter of fact, however, these formulas are not easily handleable in general cases. The two methods, furthermore, are respectively apt to explain completely different aspects of stochastic dependence.
In this paper we aim to establish a bridge between the two different approaches. Maintaining the attention focused on the minimally stable case, then, we are primarily interested in the relations tying the system of the diagonal and subdiagonal sections with the system formed by the m.c.h.r. functions. Such relations will allow us to detect, both in terms of copulas and in terms of the m.c.h.r. functions, which are the minimal sets of functions able to convey sufficient information to recover the family of the marginal distributions of the order statistics T 1:r , ..., T r:r .
In such a framework, interesting questions also concern with understanding the real difference between the cases when T 1 , ..., T r are exchangeable and when they are minimally stable. On this purpose, the differences between the two properties will be detailed both using the language of copulas and the language of the m.c.h.r. functions. Still by using and combining the two approaches, we will also face the problem of constructing examples of random variables T 1 , ..., T r which are minimally stable but not exchangeable.
We notice that the assumption of absolute continuity allows us to explain, in a simpler way, the main ideas concerning the bridge between the two approaches. However the same ideas remain substantially valid also in more general cases, which may be treated in terms of stochastic intensities of the counting process {N A t , t ≥ 0}, where A ⊂ {1, ..., r} and N A t := i∈A 1 {T i ≤t} , and in terms of concepts related to stochastic filtering. More in details, the plan of this paper goes as follows. In Section 2 we introduce some needed notation and then we review basic facts about distributions of order statistics, about diagonal sections of copulas, and about the relations tying these two families of objects. We also show in details the equivalence among different forms under which one can represent the condition that T 1 , ..., T r are minimally stable. Some relevant remarks are given and an example is presented concerning the construction of random variables which are minimally stable but not exchangeable.
In Section 3 we will first recall, in general, the definition and some basic aspects of the family of the multivariate conditional hazard rate functions. We will then show the special features of the cases where the lifetimes T 1 , ..., T r are exchangeable or minimally stable. In this frame, the relations pointed out in Section 2 will emerge as a natural tool to obtain, in Section 4, the relations existing among diagonal sections of copulas, the distributions of order statistics, and a special subclass of multivariate conditional hazard rates. See in particular Propositions 16 and 17. In order to demonstrate some special aspects of the results presented in the Sections 3 and 4, Section 5 will be devoted to a detailed discussion of the remarkable cases of exchangeable models based on Archimedean copulas and on minimally stable time homogeneous load-sharing models. Some more general examples will be presented in the Appendix.
Often, along the paper, the term lifetime will be used as a short-hand for "non-negative random variable".
For any subset J ⊆ [n], we denote by |J| the cardinality of J, and as usual we denote by J c the complementary set of J, i.e., the set of indices [n] \ J. Furthermore, for any k ≤ |J| we denote by the set of k-permutations of J. When k = |J| we drop the index k and write simply Π(J). The symbol (n) k := n(n − 1) · · · (n − (k − 1))) = |Π k ([n])| denotes the number of k-permutations in Π k ([n]).
For any subset A = {j 1 , ..., j ℓ } ⊂ [n] we denote by e A the vector whose i-th component equal to 1 if i ∈ A, and is equal to 0, otherwise.
We assume that T 1 , ..., T r are identically distributed, and denote by G their common one-dimensional marginal survival function, so that G(t) := P (T j > t) , for j = 1, ..., r, and for t > 0.
Assume also for the moment that their joint survival function F (t 1 , ..., t r ) is exchangeable, The symbol δ r denotes the diagonal section of K, namely the function δ r : [0, 1] → [0, 1] defined as the trace of K on the diagonal: More in general, being K permutation-invariant, the functions δ ℓ : [0, 1] → [0, 1] are defined as the traces of K on the sub-diagonals of the different dimensions: for ℓ = 2, ..., r − 1, An almost obvious, but important, remark is that the sub-diagonal δ ℓ (u) coincides with the diagonal of the exchangeable copula which is obtained as the ℓ-dimensional margin of K. Therefore we will refer to the functions δ ℓ (u) also as diagonal sections. It is clear that δ ℓ (u) is an increasing function and that δ 2 (u) ≥ δ 3 (u) ≥ ... ≥ δ r (u). Conditions, for a function δ : [0, 1] → [0, 1] to be the diagonal section of a copula, are given, in particular, in Jaworski (2009) [12], and Durante and Sempi [9].
As well-known, a direct relationship can be established between δ r and the probability law of the minimal order statistics T 1:r , in fact one immediately obtains, for t > 0, G 1:r (t) = P (T 1 > t, ..., T r > t) = δ r (G(t)).
We emphasize at this point that formula (6) for G ℓ:r (t) is still valid when the joint distribution of T 1 , ..., T r satisfy the specific symmetry conditions recalled in Definitions 1 and 2, below. Such conditions are actually weaker than exchangeability, and turn out to be equivalent each other (see Proposition 4 below). Definition 1. We will say that the random variables T 1 , ..., T r are t-Exchangeable if for every t ≥ 0, the binary random variables X i (t) = 1 {T i >t} , i = 1, ..., r, are exchangeable, or equivalently the events {T i > t}, i = 1, ..., r, are exchangeable.
We will briefly refer to the previous property as t-EX. Definition 2. The random variables T 1 , ..., T r are said minimally stable, when, for any ℓ = 1, ..., r and for any subset namely Finally we recall the strictly related concept of diagonal dependent copulas (see Navarro, Fernandez-Sanchez (2009) [16]). This concept is related to k-diagonal dependence, for k ≤ r, introduced in Okolewski (2017) [20]. To this purpose for a r-dimensional copula C, and for any A ⊂ [r] we set Definition 3. Let C be an r-dimensional copula C. The copula C is said to be a k-diagonal dependent copula, with k ≤ r, if for any ℓ ≤ k and for any subset When k = r, the copula C is said diagonal dependent.
As in Navarro and Fernandez-Sanchez (2020) [16] we briefly refer to the property of diagonal dependence as DD.
The following result can be obtained by taking into account basic and well known properties of exchangeable binary random variables originally obtained by de Finetti (see [6]). See also Navarro et al. (2021) [17].
(iii) The random variable T 1 , ..., T r are minimally stable; (iv) The random variable T 1 , ..., T r are identically distributed and their copula K is diagonal dependent.
Proof. Properties (i) and (ii) are clearly equivalent: indeed Similarly properties (iii) and (iv) are equivalent: indeed if T 1 , ..., T r are minimally stable, then by taking ℓ = 1 in (7), they are identically distributed, and therefore for all and G(t) is invertible, in view of the regularity condition (H). Finally (iv) is equivalent to (ii), in view of the inclusion-exclusion formula. The interest for the properties (i) and (iv) had independently emerged in the two papers Marichal et al. (2011) [15] and Navarro and Fernandez-Sanchez (2020) [16] with reference to the field of systems' reliability. Still in the same framework, furthermore, the study of conditions for the equivalence between (i) and (iv) has been developed in Navarro et al. (2021) [17]. More precisely the above mentioned articles deal with the so-called signature representation for the survival function R S (t) of the lifetime T S of a binary coherent system S. For a coherent system made with r binary components, the signature is a probability distribution s := (s 1 , ..., s r ) over [r] which is a combinatorial invariant associated to the structure function ϕ of S (see in particular Samaniego (2007) [23]). The signature representation means that the equation holds for any t > 0, where G 1:r ,...,G r:r denote the survival functions of the order statistics of the components' lifetimes. See also the Remark 10 below.
We are now in a position to establish the following result. Actually the validity of (6) hinges on the Eq. (5), which only requires the DD property of the survival copula and the identical distribution of T i , i = 1, ..., r.
From now on we assume that the random variables T 1 , ..., T r are minimally stable.
Since the marginal survival functions G 1:r , ..., G r:r are determined by the knowledge of the joint distribution of T 1 , ..., T r then, in principle, they should depend on the survival copula K. As shown by the formula (6), however, G 1:r , ..., G r:r are actually determined by the only knowledge of the diagonal sections δ 2 , ..., δ r and the common survival function G.
On the other hand, when G 1:r , ..., G r:r are known, we can easily recover the common marginal survival function G(t) and the functions δ 2 , ..., δ r . Indeed, the random variables T 1 ,...,T r are identically distributed and therefore as immediately follows by observing that r h=1 1 {T h >t} = r k=1 1 {T k:r >t} . Furthermore the same formula (6) permits to recover, step-by-step, the functions δ 2 , ..., δ r . A detailed formula is given in the Corollary 8 of the following result. Proposition 7. Let T 1 , T 2 , ..., T r be minimally stable, with order statistics distributed according to G 1:r , G 2:r , ..., G r:r . Then, for every d ∈ {1, 2, ..., r}, and J ⊆ {1, 2, .., r}, with |J| = d where by convention G 0:r (t) = 0, for t > 0, and 0 0 = 1. Before giving the proof we get the expression of δ d in terms of the marginal survival functions G k:r (t), k = 1, ..., r. Indeed, this is a consequence of equations (13) and (5) (as already observed the condition appearing in the latter equation also holds for minimally stable variables).
Remark 9. Taking into account also (11), we can thus conclude that the family of functions G 1:r , ..., G r:r and the family G, δ 1 , ..., δ r convey the same piece of information.
Remark 10. From the equivalence between signature representation and properties (i) and (iv) we can conclude that, under our conditions, the same information contained in G 1:r , ..., G r:r is also contained in the family of all the reliability functions R S (t) in (10). We emphasize that such a family is indexed by all the coherent structures ϕ for binary systems made of r components.
As far as the proof of Proposition 7 is concerned, one could apply well-known and simple results (see, e.g., de Finetti (1937) [6]) about exchangeable events.
The above formulas (12)-(15) in Proposition 7 and Corollary 8, may also be obtained in terms of the Möbius transform starting from formula (6).
For the ease of the reader we give however a self-contained, and detailed, proof. One important ingredient of the proof is the observation that, for any subset J ⊂ [r] one has Proof of Proposition 7. We start by observing that so that, thanks to Eq. (9) On the other hand, we observe that Then the thesis follows immediately: indeed, for every J ⊂ {1, 2, ..., r} with |J| = d, Eq.s (18) and (19), together with (16), imply and formula (12) follows by observing that Finally, from (12), taking into account the convention that G 0:r (t) = 0, one obtains Therefore, by observing that Then formula (13) follows by setting h = r − k in the last sum.
Remark 11. For non-exchangeable, but minimally stable lifetimes T 1 , ..., T r there still exist exchangeable lifetimes T 1 , ..., T r , such that P( T j > t) = P(T j > t) = G(t), and the sub-diagonal sections δ ℓ (u) of their survival copula K coincide with the sub-diagonal δ ℓ (u) of the survival copula K. Indeed K may be contructed by symmetrizing K: The above construction can be of help in obtaining the explicit form of P(T j > t, ∀ j ∈ A) in some special cases (see in particular Section 5.2).  [17]. See also Example 31 in the Appendix.
Example 13. First of all we notice that, when r = 2, then any pair of identical distributed random variables T 1 and T 2 are minimally stable, but in general is not exchangeable. Similarly, and trivially, any 2-dimensional copula C is minimally stable. Indeed δ C 2 (u) = C(u, u), and δ C 1 (u) = C(u, 1) = C(1, u) = u. Starting from the copula C one may define two 3-dimensional copulas as follows respectively obtained as the symmetric mixture over the cyclic permutations of (1, 2, 3) and the cyclic permutations of (3, 2, 1). Notice that when C is non-exchangeable, then C (1,2,3) and C (3,2,1) are non-exchangeable: indeed if u, v ∈ (0, 1) are such that C(u, v) = C(v, u) then, for example, which is clearly different from By iterating the above construction, one can also obtain a DD, non-exchangeable, copula which is n-dimensional. Details can be found in the Appendix (see Example 30).

Multivariate conditional hazard rates
In this section we restrict attention to vectors of lifetimes with absolute continuous joint probability distributions. In such a case, the probabilistic properties of the latter distributions can be alternatively described in terms of the multivariate conditional hazard rate functions.
Before concentrating attention on minimally stable or exchangeable lifetimes T 1 ,...,T r of our concern, we briefly recall some definitions and basic properties of multivariate conditional hazard rate functions, more in general. On this purpose, consider n nonnegative random variables V 1 , ..., V n with an absolutely continuous joint distribution whose joint density function is denoted by f V .
For the given In the sequel we will use the convention The above limits make sense in view of the assumption of absolute continuity of the joint distribution of V 1 , ..., V n and the m.c.h.r. functions can be seen as direct extensions of the common concept of hazard rate function of a univariate non-negative random variable.
For the random vector V ≡ (V 1 , ..., V n ), the system of the m.c.h.r. functions in (22) and (23) can be computed in terms of the joint density f V . It is remarkable the circumstance that the function f V can be obtained from the knowledge of the set of all the m.c.h.r. functions. In fact, the following formula holds: for (x 1 , ..., x n ), let j = (j 1 , ..., j n ) a permutation in Π([n]) such that then the joint density may be written as Then, setting, for k > 1, using the convention that, when k = 1, λ j|∅ (u), (27) and the further convention that x j 0 = 0, we can write shortly For proofs, details, and for general aspects of the m.c.h.r. functions see Shaked, Shantikumar (1990), (2007) [25,26].
See also the reviews contained within the more recent papers Shaked, Shantikumar (2015) [27], Spizzichino (2019) [28] For our purposes it is relevant to highlight that, in particular, the functions λ j|∅ (v), for j ∈ [n], are strictly related to the marginal law of the minimal order statistics V 1:n ≡ V 1:[n] = min j=1,...,n V j . In this respect the following identity holds: (See, e.g., De Santis et al. (2021) [7], where a more detailed description of the probabilistic behavior of V i:n in terms of λ j|∅ (v), for j ∈ [n], is pointed out).
For an arbitrary subset of indices A ⊂ [n], we can also consider the survival function of V 1:A , the minimal order statistics among the variables V j , with j ∈ A.
It is important to stress that formulas similar to (29) can be obtained for the survival function of V 1:A , provided a different set of m.c.h.r. functions is appropriately considered. On this purpose we can notice that, for A ⊂ [n], the |A|-dimensional joint distribution of {V j , j ∈ A} is absolutely continuous as well. Thus, for any k ≤ |A| and for any k-permutation (j 1 , ..., j k ) ∈ Π k (A), j ∈ A\{j 1 , ..., j k }, 0 < v 1 < · · · < v k ≤ v, we can consider the m.c.h.r. functions defined as follows and, for j ∈ A, v > 0 With the above notation, one can write where In view of the equivalence of condition (9) and minimal stability (i.e., conditions (ii) and (iii) of Proposition 4), for our purposes it is also relevant to express in terms of the m.c.h.r. functions. To this end we set for any d-permutation j = (j 1 , ..., j d ) Then, for any subset A ⊂ [n], we can write In some applications we will use also the following alternative expression for Ψ(t; [n], j) Concerning the above arguments and coming back to our lifetimes T 1 , ..., T r , we start with the case of exchangeable lifetimes T 1 ,...,T r by noticing that the very condition of exchangeability leads to a remarkable simplification of notation, technical results, and conceptual aspects.
First notice that the symmetry conditions among the different random variables, as requested by exchangeability, imply a specially simple form for the m.c.h.r. functions. More precisely, the function λ j|i 1 ,...,i k (v|v i 1 , . . . , v i k ) cannot depend on the index j / ∈ {i 1 , ..., i k }. Furthermore all the k-permutations (i 1 , ..., i k ) are to be considered as similar one another and thus the dependence of a m.c.h.r. function w.r.t. to (i 1 , ..., i k ) is encoded in the number k.
where we have used the notation (33).

4.
Relations among diagonal sections of DD copulas, distributions of order statistics, and multivariate conditional hazard rates Concerning with the joint distribution of minimally stable lifetimes T 1 , ..., T r , it has been pointed out in Remark 9 that the two systems of functions (a) {G; δ 2 , ..., δ r }, .., G r:r } convey the same information about the joint distribution of T 1 ,...,T r and that they can be then recovered one from the other as shown by the formula (6) and Proposition 7.
We now restrict attention on the absolutely continuous case, where the joint distribution of T 1 ,...,T r can be described in terms of the corresponding m.c.h.r. functions. In this respect, in terms of those functions, we aim to single out characteristics of the joint distribution, whose knowledge may be equivalent to that of the systems of functions in (a) and (b).
It is convenient on this purpose to fix attention on the marginal distributions, of the different dimensions, for the random vector (T 1 , ..., T r ). The following simple remark has a central role for our task.
As already observed (see Eq. (29)) the m.c.h.r. functions λ 1|∅ (t), ..., λ r|∅ (t) are strictly related to the marginal laws of the minimal order statistics, indeed where Λ ∅ (t) = r j=1 λ j|∅ (t). Similarly (see Eq.s (31) and (32)) the m.c.h.r. functions λ A j|∅ (t), j ∈ A, are related to the law of the minimum on an arbitrary set A ⊂ [r], indeed where Λ A ∅ (t) = j∈A λ A j|∅ (t) is the one-dimensional failure rate of T 1:A . Concerning with this notation, observe that the failure rate Λ By the assumption of minimal stability we may restrict attention on the subset of variables T 1 , ..., T d , which are minimally stable, as well. Indeed, since P(T 1:A > t) = P(T 1:B > t) for any A, B ⊆ [r] such that |A| = |B|, then for any d = 1, ..., r, and therefore It will emerge that the information contained in the systems (a) and (b) is equivalent to the knowledge embedded in the systems of functions defined by and furthermore, for d = 2, ..., r and for any u ∈ [0, 1], (ii): for ℓ = 1, ..., r and for any t > 0, Proof of (i). Due to minimal stability the random variables T i share the marginal survival function with T 1 , i.e., ∅ (t), for any t > 0, and (43) follows.
As already observed, we may concentrate attention on the random variables T 1 , ..., T d . Therefore to prove (44), on the one hand one has On the other hand, taking into account (5) we can also write Thus Eq. (44) is immediately achieved by comparing the preceding two formulas.
Proposition 17. Let the joint distribution of the minimally stable T 1 , ..., T r be absolutely continuous.
(ii): If the marginal survival functions G 1:r , ..., G r:r for the order statistics are given, respectively denoting by g 1:r , ..., g r:r the probability density of the order statistics T 1:r , ..., T r:r , then for ℓ = 1, ..., r and any t > 0.

Special cases
The arguments developed in the previous sections will now be illustrated by considering the two remarkable classes of models respectively defined by Archimedean copulas and by multivariate hazard rate functions satisfying the load-sharing condition. These choices in a sense correspond to the simplest possible forms admitted in the two types of descriptions of a joint distribution for lifetimes, respectively.
In particular, the analysis of these classes will allow us to obtain some examples of application for some of the results derived above, by showing the special form taken by related formulas. The arguments in Section 5.2 also permit to pave the way for a better understanding of the differences between the cases when lifetimes are exchangeable or minimally stable and between non-order dependent and strictly order dependent load sharing models. Furthermore we will be in a position to present some heuristic ideas at the basis of the construction of minimally stable, but non-exchangeable, multivariate models.
When the common marginal survival function G is given, then T 1 , ..., T r is an exchangeable model, and by (6) we get Furthermore, by Eq. (52) one can get the m.c.h.r. functions Conversely, when the survival distribution functions G k:r are given then by Corollary 8 we immediately get the expression of ψ −1 (dψ(u)) for any d = 1, ..., r. In particular, when d = r then ψ −1 (rψ(u)) = δ r (u) = G 1:r G −1 (u) and d = r − 1 Concerning the problem of identifying the generator ψ starting from the knowledge of the survival functions G k:r , the following remark can be of help.
Remark 18. It is interesting to point out (see Jaworski (2009) [12]) that in general, when r > 2, there exist infinite generators with the same diagonal section δ r , wheras the generator ψ is uniquely determined (up to a multiplicative constant) by the pair δ r and δ r−1 . Unfortunately, the proof of the latter property is not constructive. However, when the diagonal δ r satisfy the condition δ ′ r (1 − ) = r, then Erderly et al. (2013) [10] show that ψ is uniquely determined (up to a multiplicative constant) by δ r : where δ −m r is the composition of δ −1 r with itself m times.

Example 19. Let us consider the Archimedean model with
) α is a completely monotonic function, so that C ψ is a copula fon any r ≥ 2. Then, for any A ⊂ [r] with |A| = ℓ, one has P(T 1:A > t) = ψ −1 (ℓψ(G(t))) = 1 ℓ(e tα − 1) Therefore by (54) and taking into account that In the Schur-constant case with the same generator, i.e., with by Eq. (56) we get When the functions v → λ j|{j 1 ,...,j k } (v) = λ j|{j 1 ,...,j k } are constant w.r.t. time v, then the model T 1 , ..., T r is said a time homogeneous load sharing model (THLS) Load Sharing models are well known and recurrently studied in the reliability literature (see, e.g., Spizzichino (2019) [28], Rychlik and Spizzichino (2021) [22], and the references cited therein). In particular the joint and marginal distributions of the order statistics has been studied in some details. For what concerns the special case of exchangeability see also Kamps (1995) [14].
Concerning the above definition, notice that the m.c.h.r. functions -which generally depends on the set of the failure units -are required to be independent of their failure times order. It is interesting here to extend such a definition to a generalized class of models in which instead also the order of failure times may influence the m.c.h.r. functions.
Actually some of the existing results on LS models can be easily extended to this class.
When the functions v → λ j|j 1 ,...,j k (v) = λ j|j 1 ,...,j k are constant w.r.t. time v, then the model T 1 , ..., T r is said an order dependent time homogeneous load sharing model (odTHLS) A Load Sharing model is also an order dependent Load Sharing model. We will say that a model is a strictly order dependent Load Sharing model when the m.c.h.r. functions do depend on the order.
From an engineering-oriented viewpoint, strictly order-dependent load-sharing models do not seem very significant for applications in the field of reliability. However models with this property can come out in a natural way, from a mathematical standpoint. Furthermore, they may emerge in different ways when showing the general interest of load-sharing models within the family of all the joint absolutely continuous probability distributions for lifetimes, as shown in De Santis,   [8] in the analysis of aggregation paradoxes. See also Example 27, where the condition of load sharing must be limited to strict order dependent THLS models on the purpose of finding minimally stable models which satisfy specific symmetry properties without falling in the exchangeable case.
Exchangeable THLS models. We first analyze the special case of exchangeability. An exchangeable load sharing model clearly cannot be strictly order dependent, in that its m.c.h.r. functions are such that µ(t|k; t 1 , . . . , t k ) = µ(t|k). Furthermore it is time homogeneous if and only if for any k = 0, 1, .., r − 1 there exists a constant L(r − k) such that In such a case it is easily seen that (1), i = 0, 1, 2..., r − 1, are indipendent random variables (see in particular Spizzichino (2019) [28], Kamps (1995) [14], Cramer and Kamps (2003) [4], and references therein). In other words, for any k = 1, .., r, the distribution of T k:r coincides with the distribution of the sum of k independent exponential distributions of parameters γ 1 = L(r), ...., γ k = L(r − (k − 1)). In the literature such a distribution is known as Hyperexponential distribution (see Cramer and Kamps (2003) [4] and references therein): for a fixed vector γ = (γ 1 , ..., γ r ) ∈ R r + , for Y 1 , ..., Y r , independent and standard exponential random variables. When γ = (γ 1 , ..., γ r ) is such that γ i = γ j for all i = j, the survival function and the probability density are respectively given by Therefore, denoting by L the vector L := L(r), L(r − 1), ..., L(1) , we can write G k:r (t) = G L k (t). Furthermore, on the one hand formula (11) takes the special form On the other hand, taking into account (13),, for any A ⊂ [r], with |A| = d, one has P T 1: and consequently, recalling the notation in (48), (53) becomes In particular, assuming that L(i) = L(j) for i = j, and setting one has Note that, when d = r then we obviously get that the function t → µ [r] (t|0) is constant and µ [r] (t|0) = 1 r L(r), whereas for d < r the function t → µ [d] (t|0) is not constant. This fact is related to the circumstance that the d-dimensional marginal distributions of a load sharing model is generally not load sharing.
Before passing to the non-exchangeable case, we observe that in the present THLS exchangeable case the functions Ψ(t; [r], j) defined in (33) and (35) can be explicitly computed: Expression (61) turns out to be useful also in the analysis of minimally stable conditions. Indeed for any odTHLS model and therefore, for Λ = Λ ∅ , Λ j 1 , ..., Λ j 1 ,...,j d , by (61), one gets Before continuing we exhibit an example of non-exchangeable random lifetimes which are minimally stable.
However the random variables T 1 , T 2 , T 3 are minimally stable. Indeed by the previous observation (ii), and by Eq. (62), one has: for any j 1 ∈ {1, 2, 3}, for any (j 1 , Taking into account that, for any (j 1 , j 2 ) we may apply Proposition 15 to conclude that T 1 , T 2 , T 3 are minimally stable, and that For the two models it turns out that the family of the marginal survival functions of the order statistics coincide. We can see however that also their joint distributions coincide. Actually the latter circumstance is a consequence of the condition that the functions (j 1 , ..., j k ) → Λ j 1 ,...,j k are constant, only depending on k (see condition (69) in Example 28 and Remark 29 below).
For the minimally stable random variables T 1 , T 2 , T 3 , we now compute explicitly the following survival functions Furthermore, taking into account that G 1:r (t) = e − t 0 Λ ∅ (s) ds = P(N(t) = 0), and we get Minimally stable odTHLS models. We now pass to considering general properties of minimally stable odTHLS models. We start with a simple necessary condition for minimal stability.
Lemma 24. Let T 1 , ..., T r be an odTHLS model. If T 1 , ..., T r are minimally stable then necessarily and Proof. By Proposition 4, we know that for T 1 , ..., T r minimally stable, the probabilities P(T i ≤ t, T j > t, ∀j = i) necessarily assume the same value for any i ∈ [r] and any time t. Taking into account that If there exist an i 0 ∈ [r] such that Λ i 0 = Λ ∅ then necessarily Λ i = Λ ∅ , for any i ∈ [r]. Consequently, also λ i|∅ = λ i 0 |∅ . Viceversa if there exist an i 0 ∈ [r] such that Λ i 0 = Λ ∅ , then necessarily Λ i = Λ ∅ , for any i ∈ [r]. Furthermore one necessarily has Then the thesis follows by the linear independence of the functions t → e at for different values of a ∈ R.
In the next result (see Proposition 25 below) we show that the survival functions G h:r of a minimally stable odTHLS model is a mixture of Hyperexponential distributions.
We will also use the shorter notation Λ j instead of Λ j 1 ,...,jr .
Then we consider the partition of Π[r] generated by the equivalence relation then each element of the partition may be labeled by the vectors L ∈ L: Clearly the set L is always finite. For our purposes it is convenient to label the coordinates of the vectors in L as follows: With this position, in view of Lemma 24, when furthermore T 1 , ..., T r are minimally stable, then L(r) assumes the same value for any L ∈ L, and the same happens for L(r − 1). More precisely one has Proposition 25. Let T 1 , ..., T r be an odTHLS model. Suppose that the lifetimes T 1 , ..., T r are minimally stable. Then, with the notation introduced above, the survival functions G h:r , h = 1, ..., r, can be obtained as the following mixture of Hyperexponential survival functions Note that .., r, is the survival function of the order statistics of an exchangeable THLS model with In Finally the thesis follows by Remark 11 and the previous representation of the survival functions of the order statistics of an exchangeable THLS model.
As a generalization of the arguments presented in Example 22, we characterize the set of all minimally stable odTHLS models T 1 ,T 2 , T 3 in terms of the m.c.h.r. functions.
Conditions (A1) and (A2) are the necessary conditions of Lemma 24 with L(3) := Λ ∅ and L(2) := Λ 1 and guarantee that the following probabilities take the same value for any Proposition 15 guarantees that T 1 , T 2 , T 3 are minimally stable if and only if the following probabilities assume the same value for any t > 0 and for any {j 1 , j 2 } ⊂ {1, 2, 3}: Taking into account the necessary conditions (A1) and (A2), and that when r = 3, then Λ j 1 ,j 2 = λ j 3 |j 1 ,j 2 the previous condition is equivalent to require that the following sums assume the same value, for any t > 0 and any {j 1 , j 2 } ⊂ {1, 2, 3} In its turn the above requirement is equivalent to either condition (A3) or (A3) ′ .
Among load sharing models, an interesting subclass is the class of the so-called uniform frailty models, whose m.c.h.r. functions are such that, for any k = 0, 1, 2, ..., r − 1, In the next example we analyze the condition of minimal stability for the model of the previous Example 26, under the additional condition of uniform frailty.
Remark 29. As we are going to show, the above condition (69), i.e., the condition that L is a singleton, though does not imply minimal stability, it does imply the following condition: for any permutation (j 1 , ..., j r ) ∈ Π([r]), P T k:r > t|T j 1 < T j 2 < · · · < T jr = P T k:r > t .
Such a condition emerges in a natural way even in more general settings beyond loadsharing, as pointed out in Navarro et al. (2008) [18], where it has been referred to as a condition of weak exchangeability (see also Navarro et al. (2021) [17]).
More precisely the joint distribution of T 1:r , ..., T r:r coincides with the joint distribution of , . . . , Y 0 L(r) where Y k , k = 0, 1, ..., r − 1, are i.i.d. standard exponential, i.e., the joint distribution of the order statistics of an exchengeable THLS model. Indeed, one can easily extend Corollary 3 in Rychlik and Spizzichino (2021) [22] for THLS models, to odTHLS ones: for any permutation (j 1 , ..., j r ) ∈ Π([r]), the conditional joint distribution of T 1:r , ..., T r:r given the following event T j 1 < T j 2 < · · · < T jr , coincides with the law of In the frame of load-sharing models, condition (69) also emerges in De Santis and Spizzichino (2021) [8], where it plays an important role for the special type of problems studied therein.
Since we are particularly interested in examples with absolutely continuous joint and marginal distributions, observe that if we start this procedure with a absolutely continuous copula we get absolutely continuous copulas.
Furthermore we recall the class of absolutely continuous examples given in Navarro and Fernandez-Sanchez (2020) [16] (see in particular Proposition 1 therein). The class in [16] may be seen as a particular case of the larger class considered in the next example.
If α is strictly positive and sufficiently small, then K α is an absolutely continuous DD copula, but not exchangeable.
We now proceed with the proof of the previous statement.
Finally it is interesting to note that K α is a negative mixture of copulas, and that negative mixture of i.i.d. random variable are linked to finite exchangeability, and the problem of extendibility.