P1-0448 Stochastic Methods with Applications

P1-0448 Stochastic Methods with Applications is sponsored by Slovenian Research Agency. This research program supports various topics in modern probability, including stochastic processes and their applications in financial and actuarial mathematics.

MARSHALL-OLKIN COPULAS REVISITED 1 University of Ljubljana, Faculty of Mathematics and Physics, Ljubljana, Slovenia University of Ljubljana, Faculty of Mathematics and Physics, Ljubljana, Slovenia, and University of Zagreb, Faculty of Science, Department of Mathematics, Zagreb, Croatia University of Ljubljana, Faculty of Mathematics and Physics, and Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia Abstract.

Almost seventy years old Marshall-Olkin copulas, then wider Marshall copulas, and finally even wider shock model (SM) copulas constitute a substantial part of nowadays copula theory due to numerous applications. Recently, Christian Genest with some coauthors introduced a new stochastic model for a special subclass of SM copulas which gives not only a new angle on these copulas but also widens the range of applications.

In this paper we extend this type of stochastic model to all known subclasses of SM copulas. We also introduce a novel class of SM copulas and extend the new stochastic model to this class as well. E-mail addresses : tomaz.

kosir@fmf. uni-lj. si, Petra.

Lazic@fmf. uni-lj. si, matjaz@omladic.

net . 2020 Mathematics Subject Classification. Primary: 60E05; Secondary: 62G32, 62H05.

Key words and phrases. Copula; dependence concepts; shock models; extreme points. All authors acknowledge the support of ARIS (Slovenian Research and Innova-tion Agency) research core funding No. P1-0448.

The work of the second author was also supported by Croatian Science Foundation grant no. 2277. > arXiv:2512. 12265v1 [math.

ST] 13 Dec 2025 2T. KO ˇSIR, P. LAZI ´C, AND M.

OMLADI ˇC Copula is simply a joint distribution with uniform margins, but when we insert arbitrary univariate distributions as margins into it, we can get any bivariate distribution. This seminal 1959 result of Sklar [24] has made them the most important tool of dependence modeling [9, 20, 6].

Most of what we do in this paper has an extension to its multivariate version, but we will restrict ourselves to the bivariate case to make intuitively clearer the stochastic model that we are introducing. If one wants to build a stochastic model for describing the depen-dence among two (or more) lifetimes, i.e. positive random variables, one comes to shock models.

In engineering applications, joint models of lifetimes may serve to estimate the expected lifetime of a system composed by several components. In a related situation like portfolio credit risk, instead, the lifetimes may have the interpretation of time-to-default of firms, or generally financial entities, while a stochastic model may estimate the price/risk of a related derivative contract (e.g. CDO).

In both cases, it is of interest to estimate the probability of the occurrence of a joint default. This approach is a modern practitioners’ view on what was started analytically in the 1967 key paper of Marshall and Olkin [18] and was extended later by Marshall in [17]. (Perhaps we should be aware that in the original paper [18] the word copula was not even mentioned.)

An important portion of copula theory today (here is some recent publications [2, 3, 4, 8, 16, 19]) studies or is at least based on these two papers. The first one allows only exponential shocks, i.e. lifetimes, while the second one allows for arbitrary distributions. In 2016 Omladiˇ c and Ruˇ zi´ c [21] propose to study Marshall type of mod-els which result in slightly different kind of copulas they call maxmin copulas.

That paper started a series of papers that introduced some classes of copulas obtained in similar ways, including RMM (reflected maxmin) copulas, proposed by Koˇ sir and Omladiˇ c in 2020 [13], cf. also [5, 10, 11, 12, 14].

Many authors are calling copulas of all these types Marshall-Olkin copulas , but most of them keep this term only for the type introduced in [18], i.e. the ones modelled by exponential shocks, and call the general class Marshall copulas . We will call the whole class shock model copulas and add the term exponential respectively general if our shocks will be distributed exponentially resp. according to more general distributions.

The main goal of this paper is twofold. First, we follow a new sto-chastic model that gives rise to the usual shock model copulas and has some advantages over the original one. This kind of model was first introduced by Genest et al.

in 2018 [7] and later by Bentoumi et al. MARSHALL-OLKIN COPULAS REVISITED 1 3 [1] and by El Ktaibi et al. [15].

They are developing the model only for very special classes of exponential models. We show that this ap-proach extends to all known cases of shock model copulas. Moreover, we introduce a novel class we call the survival RMM copula (SMM for short) .

The proposal of this class is the second goal of this paper. Of course, we extend the new stochastic model to this class as well. The paper is organized as follows: Section 2 provides preliminaries on the Marshall, Maxmin and RMM copulas.

Section 3 exhibits the new stochastic model for the Marshall case. Section 4 presents the new model for the RMM case and Section 5 introduces the new class of SMM copulas. Finally, Section 6 gives the new stochastic model for the SMM copulas.

2. Preliminaries on Marshall, maxmin, and RMM copulas A function C : r0, 1s ˆ r 0, 1s Ñ r 0, 1s is called a copula if conditions (a) and (b) below are fulfilled. (a) C is grounded and 1 is the neutral element of C;(b) C-volume of every rectangle is positive.

Here, condition (a) means that Cpx, 0q “ Cp0, x q “ 0 for all x P r 0, 1s and Cpx, 1q “ Cp1, x q “ x for all x P r 0, 1s. Condition (b) means that (P) VC pr x1, x 2sˆr y1, y 2sq “ Cpx2, y 2q´ Cpx1, y 2q´ Cpx2, y 1q` Cpx1, y 1q ě 0 for all x1, x 2, y 1, y 2 P r 0, 1s such that x1 ď x2 and y1 ď y2.

A set of the form rx1, x 2s ˆ r y1, y 2s Ď r 0, 1s ˆ r 0, 1s, x1 ď x2, y 1 ď y2, is called a rectangle and condition (P) may be seen as the definition of its C-volume denoted by VC , together with the requirement that it is positive. The copulas presented here are sometimes called bivariate copulas as opposed to more general multivariate copulas, where the bivariate domain px, y q P r 0, 1s2 is replaced by px1, x 2, . .

. , x nq P r 0, 1sn for any integer n ě 2. However, we will not study copulas in this generality.

It turns out that for any copula C we have W px, y q ď Cpx, y q ď M px, y q, W px, y q “ max t0, x ` y ´ 1u respectively M px, y q “ min tx, y u are copulas called respectively Fr´ echet-Hoeffding lower bound and up-per bound of all bivariate copulas. For a bivariate distribution F px, y q of a random vector pX, Y q, it is easy to find the distributions of its components. For simplicity we 4 T.

KO ˇSIR, P. LAZI ´C, AND M. OMLADI ˇC assume that the domains of X and Y are contained in the interval FX pxq “ F px, 1q and FY pyq “ F p1, y q are the respective distributions of the components.

We call them the marginal distributions . (For instance, copulas may be seen as bivari-ate distributions whose marginal distributions are both uniform on the interval r0, 1s.) In his seminal 1959 paper [24] A.

Sklar defined copu-las and proved the following theorem. (Actually, he proved it in the multivariate setting.) Sklar Theorem.

For any bivariate distribution F px, y q there exists a copula Cpx, y q such that F px, y q “ CpFX pxq, F Y pyqq where FX , F Y are the corresponding marginal distributions. Conversely, if Cpx, y q is any copula and FX , F Y are any univariate distributions, then Hpx, y q “ CpFX pxq, F Y pyqq is a bivariate distribution whose copula is Cpx, y q. We refer to [6, 20] for further details on general theory on copulas.

Marshall [17] showed that shock models governed by U “ max tX, Z u, V “ max tY, Z u, where X, Y , and Z are independent random variables, are described by copulas of the form Cpu, v q “ min tug pvq, f puqvu, where generating functions f and g satisfy the following conditions (here and in what follows we are using notation I for the closed interval r0, 1s)(M1) f p0q “ 0, f p1q “ 1, (M2) f is nondecreasing on I,(M3) function f ˚puq “ f puq u is nonincreasing on p0, 1s.

Observe that (M3) implies that f puq ě u for all u P I. The joint distribution function H of pU, V q is then given by Hpx, y q “ CpFU pxq, F V pyqq , where FU and FV are the marginal distribution functions of pU, V q, i.e., distributions of U and V respectively given by FU pxq “ FX pxqFZ pxq and FV pxq “ FY pxqFZ pxq.

MARSHALL-OLKIN COPULAS REVISITED 1 5 Omladiˇ c and Ruˇ zi´ c [21] introduced a new class of copulas modeling shocks given by U “ max tX, Z u, V “ min tY, Z u. where X, Y , and Z are independent random variables.

A copula for pU, V q is then of the form (1) Cpu, v q “ min tu, ϕ puqp v ´ ψpvqq ` uψ pvqu , where generating function ϕ satisfies conditions (M1)-(M3) with ϕ “ f ,while ψ has the following properties (F1) ψp0q “ 0, ψ p1q “ 1, (F2) ψ is nondecreasing on I,(F3) function ψ˚pvq “ 1 ´ ψpvq v ´ ψpvq is nonincreasing on r0, 1q. Here, ψ˚pvq “ 8 if ψpvq “ v. Functions of the form (1) are called maxmin copulas .

Koˇ sir and Omladiˇ c [13] studied reflected version of the maxmin cop-ulas that are called reflected maxmin copulas (RMM for short). Their generators f and g satisfy the following conditions. (Here, we introduce additionally functions f ˚puq “ f puq u , pf puq “ f puq ` u,and pgpuq “ gpuq ` u in order to make these conditions easier to state and understand.)

(G1) f p0q “ gp0q “ 0, f p1q “ gp1q “ 0, and f ˚p0q “ g˚p0q “ 0; (G2) functions pf and pg are nondecreasing on I;(G3) functions f ˚ and g˚ are nonincreasing on I. An RMM copula is then of the form Cpu, v q “ max t0, uv ´ f puqgpvqu , 3. Shock model for Marshall copulas We first consider a slight variation of Marshall model [17, Proposi-tion 3.

2] that still leads to the same family of copulas. We consider four random variables X, Y , Z1, and Z2 representing shocks. The two triples X, Y, Z 1 and X, Y, Z 2 are independent, while Z1 and Z2 form a comonotonic pair, so that their dependence is modeled by the Fr´ echet-Hoeffding upper bound M pu, v q “ min tu, v u.

Denote by FX and FY the distribution functions of X and Y respectively, and by Gi the distribution functions of Zi for i “ 1, 2. We construct two random variables U “ max tX, Z 1u, V “ max tY, Z 2u,6 T. KO ˇSIR, P.

LAZI ´C, AND M. OMLADI ˇC whose distributions are clearly FU pxq “ FX pxqG1pxq and FV pyq “ FY pyqG2pyq. Before giving the copula connecting U and V we introduce some further notation.

Let F be a distribution function. For u P I we have F ´1puq “ inf tx P R : F pxq ě uu. We assume that infimum of an empty set is equal to `8 and infi-mum of R is equal to ´8 .

Note that F ´1 is nondecreasing. We have F pF ´1puqq ě u for all u P I and F ´1pF pxqq ď x for all x P R. Fol-lowing the notation of [21] we denote further by f px´q the left limit of function f if it exists.

(Note that all our functions will be monotone, so that our left limits will always exist at x.) For u R im F Y t 0, 1u we write u “ F pF ´1puqq and u “ F pF ´1puq´q . Then u P im F and either u P im F or u R im F and pu ´ ε, u q X im F ‰ H for every small enough ε ą 0.

Let us introduce generating functions for our model by puqq , if u P im FU zt 0, 1u u ´ u pu ´ uq ` φpu´q otherwise; and pvqq , if v P im FV zt 0, 1u v ´ v pv ´ vq ` ψpv´q otherwise . A careful reader will have noticed that the first three lines of each definition are the obvious natural choice while the fourth one is a linear interpolation on the intervals left undefined.

Let us give the proof of the following proposition for the sake of completeness. Proposition 1. The copula of the pair pU, V q is a Marshall copula Cpu, v q “ min tuψ pvq, vφ puqu .

MARSHALL-OLKIN COPULAS REVISITED 1 7 Proof. Denote by H the joint distribution function of the random vector Hpx, y q “ PrU ď x, V ď ys “ PrX ď x, Z 1 ď x, Y ď y, Z 2 ď ys“ FX pxqFY pyqPrZ1 ď x, Z 2 ď ys “ FX pxqFY pyq min tG1pxq, G 2pyqu “ min tFU pxqψpFV pyqq , F V pyqφpFU pxqqu “ CpFU pxq, F V pyqq , where C is the Marshall copula with generating functions φ and ψ.

˝ We now state and prove two results that exhibit a modified version of Marshall’s Theorem [17, Proposition 3. 2]. Our results are based on Lemma 5.

6 and Theorem 5. 9 of [22]. Lemma 2.

Suppose that U and V are two random variables with dis-tribution functions FU and FV , and with the joint distribution function H. Then the following two statements sre equivalent. (a) There are two triples of independent random variables X, Y, Z 1 and X, Y, Z 2, where Z1 and Z2 are comonotonic, such that U “ max tX, Z 1u and V “ max tY, Z 2u.

(b) There are distribution functions FX , F Y , G 1, and G2 such that Hpx, y q “ FX pxqFY pyq min tG1pxq, G 2pyqu “ FX pxqFY pyqM pG1pxq, G 2pyqq If any of (a) or (b) holds, then we also have FU pxq “ FX pxqG1pxq, FV pyq “ FY pyqG2pyq,FU ď min tFX , G 1u, and FV ď min tFY , G 2u. Proof. Suppose (a) holds.

Then, we see as in the proof of Proposition 1 that Hpx, y q “ FX pxqFY pyq min tG1pxq, G 2pyqu . Now, assume (b) . Then, Hpx, y q is the joint distribution function of random variables U “ max tX, Z 1u and V “ max tY, Z 2u as above.

Since the random vector is uniquely determined by its joint distribution function, (a) follows. ˝ Remark 3. Lemma 2 remains correct if we replace the word “comono-tonic” in (a) with the word “countermonotonic” and the formula in (b) with Hpx, y q “ FX pxqFY pyqW pG1pxq, G 2pyqq The proof follows in a similar way as above.

8 T. KO ˇSIR, P. LAZI ´C, AND M.

OMLADI ˇC Theorem 4. Let Cφ,ψ be a Marshall copula, and let FU and FV be two distribution functions that satisfy the following assumptions: (a) There is an increasing function χ : R Ñ R such that if x P R is such that FU pχpxqq ą 0 and FV pxq ą 0, then φ˚pFU pχpxqqq “ ψ˚pFV pxqq . (b) Function φ is either continuous at 0 or xU “ inf tx P R, F U pxq ą 0u ą ´8 and FU is not continuous at xU .

Function ψ is either continuous at 0 or xV “ inf tx P R, F V pxq ą 0u ą ´8 and FV is not continuous at xV . (c) Function φ is such that either φ˚p0`q “ 8 or there is x P R such that FU pxq “ 0. Function ψ is such that either ψ˚p0`q “ 8 or there is x P R such that FV pxq “ 0.

Then there are two triples of independent random variables X, Y, Z 1,and X, Y, Z 2, where Z1 and Z2 are comonotonic, such that Hpx, y q “ Cφ. ψ pFU pxq, F V pyqq is the joint distribution function of U “ max tX, Z 1u and V “ max tY, Z 2u. Proof.

We define FX pxq “ φpFU pxqq and FY pyq “ ψpFV pyqq for all x, y P R. Assumptions (b) and (c) imply that FX and FY are distribu-tion functions. Next we define 0, if FU pχpxqq “ FV pxq “ 0,FV pxq ψpFV pxqq , if FU pχpxqq “ 0 and FV pxq ą 0,FU pχpxqq φpFU pχpxqqq , otherwise .

whenever both FU pχpxqq ą 0 and F V pxq ą 0 by assumption (a) . We also take G1pxq “ G2pχ´1pxqq for all x P R. Here, χ´1 is the inverse of χ which is an increasing function since χ is increasing.

Functions G1 and G2 are well defined because φpxq ě 0 and ψpyq ě 0 for all x, y P I. Assumptions (a)–(c) imply that G1 and G2 are distribution functions. If FU pxq “ 0 then FX pxqG1pxq “ φpFU pxqq G1pxq “ 0 “ FU pxq.

MARSHALL-OLKIN COPULAS REVISITED 1 9 FX pxqG1pxq “ φpFU pxqq G2pχ´1pxqq “ φpFU pxqq FU pxq Therefore we have FU pxq “ FX pxqG1pxq for all x P R. If FV pyq “ 0then FY pyqG2pyq “ 0 “ FV pyq. Otherwise, if FV pyq ą 0 then FY pyqG2pyq “ ψpFV pyqq FV pyq Hence we have FV pyq “ FY pyqG2py for all y P R.

Finally, the definitions of H, C φ,ψ , F X , F Y , G 1, and G2 imply Hpx, y q “ Cφ,ψ pFU pxq, F V pyqq “ min tFU pxqψpFV pyqq , φ pFU pxqq FV pyqu “ min tFU pxqFY pyq, F V pyqFX pxqu “ FX pxqFY pyq min tG1pxq, G 2pyqu . The implication (a) ñ(b) of Lemma 2 then completes the proof. ˝ 4.

Shock model for RMM copulas Next we present a shock model setup first proposed in Genest et al. [7] (cf. also [1, 15]).

We consider two triples pX, Y, Z 1q and pX, Y, Z 2q of independent random variables. This time we assume that Z1 and Z2 are a countermonotonic pair, so that their dependence is measured by the Fr´ echet-Hoeffding lower bound W pu, v q “ max t0, u ` v ´ 1u. Again we construct two random variables U “ max tX, Z 1u, V “ max tY, Z 2u, whose distributions are clearly FU pxq “ FX pxqG1pxq and FV pyq “ FY pyqG2pyq as above.

Let us introduce auxiliary generating functions for our model by puqq , if u P im FU zt 0, 1u u ´ u pu ´ uq ` pf pu´q otherwise; 10 T. KO ˇSIR, P. LAZI ´C, AND M.

OMLADI ˇC pvqq , if v P im FV zt 0, 1u v ´ v pv ´ vq ` pgpv´q otherwise . p0qq “ 0 independently of the point 0 being in im FU or not, and similarly for pg. In the same way, if 1 is not in im FU , we get that FX pF ´1 p1qq “ FX p`8q “ 1, and if 1 is in im FU , then also FX pF ´1 p1qq “ 1.

We write also f puq “ pf puq ´ u Theorem 5. The copula of the pair pU, V q is an RMM copula with generating functions f and gCpu, v q “ max t0, uv ´ f puqgpvqu . Proof.

Since X and Z1, respectively Y and Z2, are independent, we have (2) FU puq “ FX puqG1puq, respectively FV puq “ FY puqG2puq.

The joint distribution function H of the random vector pU, V q is then Hpu, v q “ PrU ď u, V ď vs“ PrX ď u, Y ď v, G ´11 pZq ď u, G ´12 p1 ´ Zq ď vs. Since X, Y , and Z are independent, we obtain Hpu, v q “ FX puqFY pvqPr1 ´ G2pvq ď Z ď G1puqs “ FX puqFY pvq max tG1puq ´ p 1 ´ G2pvqq , 0u“ FX puqFY pvqW pG1puq, G 2pvqq , where in the second line we have used the fact that Z is distributed uniformly on I and then the definition of the Fr´ echet-Hoeffding lower bound W .

The copula of pU, V q is then given by We use (2) and the definition of the generating functions f and g to conclude Cpu, v q “ max t0, uv ´ f puqf pvqu . It remains to show that functions f and g satisfy conditions (G1)-(G3) for generators of RMM copulas. (G1): By definition, pf p0q “ 0 and pf p1q “ 1.

MARSHALL-OLKIN COPULAS REVISITED 1 11 (G2): Consider function pf puq “ f puq ` u “ FX pF ´1 puqq ; since dis-tribution functions FX and FU are nondecreasing, so that F ´1 is also nondecreasing, and since composite of two nondecreasing functions is nondecreasing, the claim follows for pf . The conclusion for pg is shown in a similar way. (G3): We want to show that function f ˚puq “ f puq u is nonincreasing on p0, 1s.

It suffices to show that f ˚puq ` 1 “ pf puq p0, 1s. Choose u P im FU to get (using (2)) This function has a nondecreasing function in its denominator, so that it is nonincreasing on im FU . Since we defined pf outside this region by linear interpolation, the claim follows.

Using similar arguments we prove that g˚ is also nonincreasing. ˝ Theorem 6. Let U and V be two random variables with respective distribution functions FU and FV such that there exists an x0 P R with FU px0q, F V px0q P p 0, 1q.

Furthermore, let the copula of the pair pFU , F V q be an RMM copula with generating functions f and g, i.e., Cpu, v q “ max t0, uv ´ f puqgpvqu . Then there exist random variables X, Y, Z 1, Z 2 such that X, Y, Z i are independent for i “ 1, 2, Z2 is countermonotonic to Z1, and function Hpx, y q “ CpFU pxq, F V pyqq is the joint distribution function of the variables max tX, Z 1u and max tY, Z 2u. Proof.

For all x, y P R define FX pxq “ ˆf pFU pxqq and FY pyq “ ˆgpFV pyqq ,where ˆf and ˆ g are defined just before Theorem 5. Moreover, let 1 ` g˚p1 ´ FV pxqq , if FU pxq “ 0 ˆf pFU pxqq , otherwise , 1 ` f ˚p1 ´ FU pyqq , if FV pyq “ 0 ˆgpFV pyqq , otherwise . 12 T.

KO ˇSIR, P. LAZI ´C, AND M. OMLADI ˇC Since f and g are nondecreasing, the same holds for ˆf and ˆ g, so that FX and FY are nondecreasing and c` adl` ag.

Continuity of ˆf and ˆ g implies that lim and similarly for FY . We conclude that FX and FY are distribution functions of some random variables we denote by X and Y . The con-dition given in the theorem tells us that if F pxq “ 0 for some x, then x ă x0, so that FV pxq ď FV px0q ă 1.

Similarly, FV pxq “ 1 implies x0 ă x, so that FU pxq ě FU px0q ą 0. Now, choose x P R with FU pxq ą 0 and FV pxq ă 1 to get 1 G1pxq “ 1 ` g˚p1 ´ FV pxqq g˚p1 ´ FV pxqq “ ˆf pFU pxqq and similarly for G2. When x goes to ´8 , we use the fact that g˚ is continuous at 0 to get that G1 approaches to 0.

When x goes to `8 we use the fact that ˆf is continuous at 1 to get that G1 approaches to 1. Next we prove that G1 is nondecreasing. If FU pxq ą 0 then 1 FU pxq ` 1which is nonincreasing by (G3).

If FV pxq ă 1 then 1 g˚p1 ´ FV pxqq ` 1which is nonincreasing by (G3). So, it remains to consider the case that FU pxq “ 0 and FV pyq “ 1 for some x and y. Clearly, we have x ă y and the point x0 given in the theorem belongs to the interval 1 ` g˚p1 ´ FV pxqq ď g˚p1 ´ FV px0qq 1 ` g˚p1 ´ FV px0qq “ ˆf pFU px0qq The first one of the inequalities above holds because FV pxq, F V px0q ă 1and the second one because FU px0q, F U pyq ą 0.

Furthermore, since ˆf and g˚ are continuous and there exists some x0 with FV pxq ă 1 and FU px0q ą 0, it follows that G1 is c` adl` ag. Similar considerations and conclusions apply to G2. Finally, we have shown existence of random variables Z1 and Z2 whose distribution functions are respectively G1 and G2.

MARSHALL-OLKIN COPULAS REVISITED 1 13 We now want to study the connection of the joint distribution func-tion H of U and V to the copula Cpu, v q “ max t0, uv ´ f puqgpvqu “ uv max t1 ´ f ˚puqg˚pvqu . Indeed, after introducing marginal distributions FU and FV into this formula, we get CpFU pxq, F V pyqq “ “ FU pxqFV pyq max “ ˆf pFU pxqq ˆgpFV pyqq max “ FX pxqFY pyqW pG1pxq, G 2pyqq .

Choose independent random varables with respective distributions FX , F Y ,and G1. Define Z2 “ G´12 p1 ´ G1pZ1qq to get a countermonotonic ran-dom variable whose distribution equals G2. The proof that Hpx, y q is the joint distribution function of random variables max tX, Z 1u and max tY, Z 2u follows by Remark 3.

˝ Example 7. The so-called Eyraud-Farlie-Gumbel-Morgenstern (EFGM) distributions have been considered by many authors (cf. [23] ).

We will follow the approach of Durante and Sempi [6, Section 6. 3] . We want to present shocks X, Y, Z 1 and Z2 such that the corresponding RMM copula is of the form Capu, v q “ uv ´ a2uv p1 ´ uqp 1 ´ vq, where 0 ď a ď 1.

So, Ca is of the EFGM type. (We assume actually 0 ă a ď 1.) Proof.

Observe that pfaptq “ p a`1qt´at 2 and that in the model pfaptq “ x, if 0 ď x ď 11, otherwise , i.e., X „ U p0, 1q. To obtain FU and then also G1, we have to solve pa ` 1qt ´ at 2 “ x14 T. KO ˇSIR, P.

LAZI ´C, AND M. OMLADI ˇC , if 0 ď x ď 11, otherwise . It is easy to verify that FU is a distribution function of a continuous random variable whose density function is equal to apa ` 1q2 ´ 4ax , if 0 ă x ă 10, otherwise .

To obtain the distribution function of G1 we use the relation FU pxqq “ FX pxqG1pxq. Therefore, we get after a simplification that pa ` 1q ` apa ` 1q2 ´ 4ax , if x ď 11, otherwise . With not much more effort we can also compute the density of this continuous random variable pp a ` 1q2 ´ 2ax q apa ` 1q2 ´ 4ax ` p a ` 1q3 ´ 4pa ` 1qax , if x ă 10, otherwise .

Since Capu, v q is symmetric we take FX pxq “ FY pxq and G1pxq “ G2pxq, while Z1 and Z2 are related via copula W pu, v q. ˝ Example 8 (Case of exponential shocks) . If X „ exp pλ1q, Y „ exp pλ2q,Z1 „ exp pμ1q, Z2 „ exp pμ1q, then we have a copula for any α, β P p 0, 1s by letting Cα,β pu, v q “ max t0, uv ´p uα´uqp vβ ´vqu , where α “ λ1 Proof.

This can easily be determined by verifying conditions pG1q´p G3q for the generating functions of the form f ptq “ tα ´ t, for α P p 0, 1s. More generally, a function f ptq “ tαp1 ´ tβ q,MARSHALL-OLKIN COPULAS REVISITED 1 15 Figure 1. Up – Example 7: Copula for the value of the parameter a “ 0.

95. Down – Example 8: Copulas for α “ β “ 0. 1 (left) and α “ 0.

4, β “ 0. 9 (right). is a generating function f of an RMM copula if β P p 0, 1s for α “ 1and β ě 1 ´ α for α P p 0, 1q.

˝ 5. Survival copulas of RMM copulas Cpu, v q “ max t0, uv ´ f puqgpvqu we have its survival copula pCpu, v q “ u`v´1`Cp1´u, 1´vq “ max tu`v´1, uv ´f p1´uqgp1´vqu . We will now acquire the properties of functions h, k : I Ñ I defined by hpuq “ f p1 ´ uq and kpvq “ gp1 ´ vq; this time we need auxiliary functions 1 ´ v , ph:puq