Characterizations of Extreme Value Extended Marshall-Olkin Models with Exponential Marginals

We construct and characterize bivariate extreme value distributions with exponential marginals generated by the stochastic representation (X1, X2) = (min(T1,T3),min(T2,T3)) where the random variable T3 is independent of random variables T1 and T2 which are assumed to be dependent. A building procedure is suggested when the joint distribution of (T1,T2) is absolutely continuous and Ti’s are not necessarily exponentially distributed, i = 1, 2, 3. The Pickands representation of the vector (X1, X2) is computed. We illustrate the general relations by examples.


Introduction
Let us consider the fatal shock model defined by the stochastic representation (X 1 , X 2 ) = (min(T 1 , T 3 ), min(T 2 , T 3 )) , where non-negative continuous random variables T 1 and T 2 identify the occurrence of independent "individual shocks" affecting two devices and T 3 is their "common shock".The random vector (X 1 , X 2 ) presents the joint distribution of both lifetimes.
To give a probability interpretation of model (1), consider a system composed by two items, to be denoted by 1 and 2. We associate with each item j, j = 1, 2, a Bernoulli random variable Z j , indicating whether the item is operational (Z j = 1) or failed (Z j = 0).The bivariate Bernoulli random vector (Z 1 , Z 2 ) represents the state of the system.It is specified in terms of MO construction (1).The vector (X 1 , X 2 ) exhibits the latent state of the system, since the MO model ( 2) is defined in terms of vector (T 1 , T 2 , T 3 ) of latent variables that identify independent exponential shock times.Each shock takes down a given subset of items ({1}, {2} or {both 1 and 2}) and occurs at an exponential time with constant rates λ 1 , λ 2 and λ 3 , respectively.
The stochastic relation (1) is widely used in literature.For example, Li and Pellerey (2011) launched the Generalized MO (GMO) model considering non exponential independent random variables T i in (1), i = 1, 2, 3.The corresponding joint distributions do not possess bivariate lack of memory property, i.e., are "aging".As a further step, Pinto and Kolev (2015) introduced the Extended MO (EMO) model assuming dependence between latent variables T 1 and T 2 , but keeping T 3 independent of them.The motivation is that the individual shocks might be dependent if the items share a common environment.In this case however, the EMO distributions can be "aging" or "non-aging" depending on the parameters of joint distribution of (T 1 , T 2 ) and the distribution of T 3 .
We will assume further that T 1 and T 2 are no more independent, but defined by their joint survival function ) and the random variable T 3 is independent of T 1 and T 2 .Let S T i (x) = P(T i > x) be the survival functions of T i , i = 1, 2, 3 for x ≥ 0. Thus, the joint survival function of EMO model generated by (1) can be written as In Section 2 we establish the extreme value representation and characterizations of a subclass of EMO distributions defined by (3) whose marginals X 1 and X 2 are exponentially distributed, see Theorem 1 and the most general Theorem 3. We suggest a procedure to construct EMO distributions with exponential marginals even if T i 's in (1) are not exponentially distributed under the corresponding additional restriction, e.g., Theorem 2. We obtain in Section 3 the Pickands dependence function and Pickands measure corresponding to (3) if the joint distribution of (T 1 , T 2 ) is absolutely continuous.We illustrate the general relationships with two examples.As a particular case one can find the corresponding representations associated to the MO's bivariate exponential distribution (2) obtained by Mai and Scherer (2010).We finish the article with a short discussion.

Extreme Value EMO Distributions
There is a number of mathematical results characterizing multivariate extreme value distributions and extreme value copulas, see Chapter 6 in Joe (1997) for example.The marginals of any multivariate extreme value distribution must be members of the class of univariate generalized extreme value distributions, i.e., location-scale families of distributions based on Weibull, Fréchet and Gumbel distributions, see Theorem 2.4.1 in Galambos (1978).
Without loss of generality, we will assume hereafter marginal exponential distributions in (3), i.e., X i ∼ Exp(λ X i ), i = 1, 2. Such a choice does not have influence on the corresponding survival copula to be obtained, being invariant on strict monotone transformations.It is also well known that these transformations can be used to move from one member to the other in the class of univariate generalized extreme value distributions, see Galambos (1978) and Beirlant et al. (2005) for a related discussion.
The joint survival function of the EMO distributions specified by (3) can be equivalently represented by Note that the right hand side in ( 4) is a product of two bivariate distributions.The first one is defined by S T 1 ,T 2 (x 1 , x 2 ), and the second one refers to a bivariate random vector with comonotonic components sharing the same marginal distribution as T 3 .Such a product construction technique in terms of copula has been discussed by Genest et al. (1998), see their Proposition 2. Consult Liebscher (2008) for a general power function based approach as well.
Starting from the latent random vector (T 1 , T 2 ) we "modify" it via (4) to get (X 1 , X 2 ), which can be interpreted as follows: the "modified" joint distribution (X 1 , X 2 ) can be used to model a complementary amount of bivariate asymmetry induced by (T 1 , T 2 ).Note that, in general, this additional asymmetry does not necessarily imply an increase of upper tail dependence (if exists) governed by (T 1 , T 2 ), see supporting comments in Joe (2015), page 184.
The simplest way to ensure exponentially distributed marginals X 1 and X 2 in (4) is to advocate that T i ∼ Exp(λ i ), i = 1, 2, 3.This distributional choice has nice properties and will be justified in Theorem 1.
Other distributional possibilities for T i , i = 1, 2, 3, do exist in order to construct EMO model with exponentially distributed marginals X j with parameter λ X j , j = 1, 2. To believe, denote by r T i (x) = d dx [− ln S T i (x)] the failure rates of T i , i = 1, 2, 3. Note that the marginal survival functions in (4) are given by S X i (x i ) = S T i (x i )S T 3 (x i ) and the only condition in terms of failure rate functions to get exponential marginals in (4) is Denote by BEV E the set of bivariate extreme value distributions with exponential marginals and let BEV E − EMO be a subclass of BEV E satisfying (4), where (T 1 , T 2 ) ⊂ BEV E and T 3 ∼ Exp(λ 3 ).This means that the following functional equation is fulfilled (5) Joe (1997) observed that when multivariate copulas take on univariate generalized extreme value distributions one obtains multivariate extreme value distributions for maxima, but when the copulas take on generalized extreme value survival margins for minima, multivariate extreme value survival functions for minima results, see page 176.A general related discussion in terms of functional equations involving joint survival functions can be found in Marshall and Olkin (1991) as well.
Relation (5) implies that the survival copula C T 1 ,T 2 (u, v) associated to S T 1 ,T 2 (x 1 , x 2 ) is also an extreme value copula for all t > 0, i.e., It follows our first characterization statement.
Proof.Let (X 1 , X 2 ) ⊂ BEV E − EMO.Taking into account that X i and T 3 are exponentially distributed and relations But T 3 is exponentially distributed, so that [S T 3 (max{x 1 , x 2 })] t = S T 3 (max{tx 1 , tx 2 }) and we arrive to (5).Now assume that (T 1 , T 2 ) ⊂ BEV E and T 3 ∼ Exp(λ 3 ) in (4).Since the comonotonic copula M(u, v) = min(u, v) is an extreme value copula, applying Sklar's theorem in (4) and taking into account (6) we obtain Thus, the functional equation ( 5) for the vector (X 1 , X 2 ) is satisfied and (X 1 , X 2 ) belongs to the class BEV E − EMO.
To proceed, we will need to recall that any bivariate survival function S T 1 ,T 2 (x 1 , x 2 ) can be represented as where ] is the dependence function introduced by Sibuya (1960), to be referred as Sibuya's dependence function.It exhibits interesting relationships with dependence phenomena, see Kolev (2016) or Pinto and Kolev (2015a) for a deep discussion.
With an additional assumption of absolute continuity of the joint distribution of (T 1 , T 2 ), we justify in the next Lemma 1 the choice of exponential T i 's in order to get a member of BEV E − EMO via (4), i = 1, 2, 3. Let us denote by BEV E − EMO AC this subclass.
Applying the exponential representation for bivariate survival functions (7) in (3) we obtain Taking logarithms in both sides of former equation and calculating the mixed partial derivatives, we get Integrating we have Thus, we arrive to the functional equation Since S X 1 ,X 2 (x 1 , x 2 ) is an extreme value survival function with exponential marginals, we deduce that Applying the exponential representation ( 7) in ( 9) and taking into account (8), we obtain the homogeneous of order 1 functional equation H T 3 (tx 2 ) = tH T 3 (x 2 ), for all x 2 , t > 0, where H T 3 (x 2 ) is the cumulative hazard of T 3 .Substitute λ 3 = H T 3 (1) to get the general solution H T 3 (x 2 ) = λ 3 x 2 with λ 3 > 0, since H T 3 (x 2 ) is nonnegative for all x 2 ≥ 0, see Aczel (1966) as well.
The conclusion is the same when x 2 ≥ x 1 ≥ 0. Thus, T 3 ∼ Exp(λ 3 ).Applying the "only if" branch of Theorem 1 we finish the proof.
Linking Theorem 1 and Lemma 1 we obtain the following characterization.
The absolute continuity assumption for (T 1 , T 2 ) in Theorem 2 is an important condition.In the next we suggest a building procedure showing that it is possible to construct a member of BEV E − EMO class where the random variables T i in (3) are not necessarily exponentially distributed, i = 1, 2, 3, but as a compensation we have to relax the assumption of absolutely continuity of the survival function S T 1 ,T 2 (x 1 , x 2 ).
Example 1 (Building procedure).Let us consider three nonnegative absolutely continuous and independent random variables defined by The following two-step procedure generates a vector (X 1 , X 2 ) ⊂ BEV E − EMO with non-exponentially distributed T i 's: 2. Select a random variable T 3 , independent of (T 1 , T 2 ), with a survival function S T 3 (x) = exp{−bx − f (x) + a}.Apply (3) to get the corresponding EMO survival function S X 1 ,X 2 (x 1 , x 2 ) = exp{−x 1 − x 2 − 2b max(x 1 , x 2 )} with exponentially distributed marginals.
Notice that S T 1 ,T 2 (x 1 , x 2 ) obtained in the first procedure step is neither an extreme value survival function nor absolutely continuous, since has a singular component with support on the set In addition, T 3 defined in the second step is not exponentially distributed.Finally, inequalities a − bx < f (x) < a + bx and f (0) = a guarantee that S Y 3 (x) and S T 3 (x) are proper survival functions.Now observe that the functional equation ( 8) involving Sibuya's dependence function D T 1 ,T 2 (x 1 , x 2 ) is homogeneous of order 1 for all x 1 , x 2 ≥ 0 and t > 0. As a consequence of Theorem 6.2 in Joe (1997), equation ( 8) can serve as a characterization of bivariate extreme value survival functions S T 1 ,T 2 (x 1 , x 2 ) with exponential marginals even when absolute continuity for (T 1 , T 2 ) does not hold true.Hence, we deduce the following characterization.

Lemma 2 A bivariate distribution is BEV E if an only if its Sibuya's dependence function is homogeneous of order 1.
To characterize distributions belonging to the subclass BEV E − EMO when the joint distribution of (T 1 , T 2 ) is not absolutely continuous, we will need an additional assumption since is satisfied for all x 1 , x 2 ≥ 0 and t > 0.
Proof.Under conditions in the theorem, the Sibuya's dependence function for the EMO model ( 3) is given by To finalize the proof, just apply Lemma 2.
Example 2 (BEV E −EMO distributions when (T 1 , T 2 ) is not absolutely continuous).The Marshall-Olkin survival function is of EMO-type, being an example of extreme value survival function as well.We use relation ( 4) and the two step procedure from Example 1: first selecting random variables (T 1 , T 2 ) with survival function , where H Y 3 (x) = x − cos(x) + 1, and second, choosing T 3 independent of (T 1 , T 2 ) with cumulative failure rate H T 3 (x) = x + cos(x) − 1.Notice that S T 1 ,T 2 (x 1 , x 2 ) is neither an extreme value survival function nor absolutely continuous, as well as T 3 is not exponentially distributed.In this case we have D T 1 ,T 2 (x 1 , x 2 ) = H Y 3 (min(x 1 , x 2 )), so that the functional equation given in Theorem 3 is satisfied.

Pickands Representation and Examples
Given the characterization established in Theorem 2, in the sequel we assume absolutely continuous distributions for (T 1 , T 2 ) to ensure uniqueness in our construction of extreme value EMO distributions.In the next theorem we obtain the general form of the corresponding survival function and related copula.We will need to remind basic facts related to Pickands measure involved first.Pickands (1981) proves that each multivariate extreme value distribution is uniquely characterized by a finite measure satisfying boundary conditions and related dependence function.In the bivariate case, the bivariate extreme value (survival) copulas can be completely characterized by the relation where , for a positive finite measure H on [0, 1], denominated Pickands measure, see Joe (1997).
The so-called Pickands dependence function A(w) : [0, 1] → [ 1 2 , 1] must be convex and should satisfy max(w, 1 − w) ≤ A(w) ≤ 1.The lower bound of A(w) corresponds to the comonotonic copula and related Pickands measure puts mass 2 at w = 1 2 ; the upper bound of A(w) is associated to the independence copula with Pickands measure assigning mass 1 to both w = 0 and w = 1.
The Pickands dependence function can be recovered from the copula C by setting and is uniquely related to the measure H via equation where d dw A(w) is the right-hand derivative of A(w), see Theorem 3.1 in Pickands (1981) and section 8.5.3 in Beirlant et al. (2005).Moreover, the point masses of H at 0 and 1 are given by , where A T 1 ,T 2 (w) is the Pickands dependence function corresponding to C T 1 ,T 2 (u, v) and λ i > 0, i = 1, 2, 3.
Whenever the derivative of A T 1 ,T 2 (w) is continuous from ( 15) we obtain We will apply relations established in Theorem 4 in the next two examples.