Tail Dependence Comparison of Survival Marshall-Olkin Copulas

Transcription

1 Tal Dependence Comparson of Survval Marshall-Olkn Copulas Hajun L Department of Mathematcs and Department of Statstcs Washngton State Unversty Pullman, WA 99164, U.S.A. lh@math.wsu.edu January 2006 Abstract The multvarate tal dependence descrbes the amount of dependence n the upperorthant tal or lower-orthant tal of a multvarate dstrbuton and can be used n the study of dependence among extreme values. We derve an explct expresson of tal dependence of multvarate survval Marshall-Olkn copulas, and obtan a suffcent condton under whch tal dependences of two survval Marshall-Olkn copulas can be compared. Some examples are also presented to llustrate our results. Key words and phrases: Marshall-Olkn dstrbuton, copula, tal dependence, parwse IE transform. 1 Introducton The random extremal events occurred n engneerng and fnancal systems are usually dependent, and such an extremal dependence fundamentally affects the system dynamcs. The multvarate tal dependence s a tool to descrbe the amount of dependence n the upperorthant tal or lower-orthant tal of a multvarate dstrbuton and characterze the dependence among extremal values. Ths paper focuses on the tal dependence of multvarate 1

2 Marshall-Olkn copulas, and studes how a change n the dstrbutonal dependence structure would affect ts copula s tal dependence. The copula s a useful tool for handlng multvarate dstrbutons wth gven unvarate margnals. Formally, a copula C s a dstrbuton functon, defned on the unt cube [0, 1] n, wth unform one-dmensonal margnals. Gven a copula C, f one defnes F (t 1,..., t n = C(F 1 (t 1,..., F n (t n, (t 1,..., t n R n, (1.1 then F s a multvarate dstrbuton wth unvarate margnal dstrbutons F 1,..., F n. Gven a dstrbuton F wth margnals F 1,..., F n, there exsts a copula C such that (1.1 holds. If F 1,..., F n are all contnuous, then the correspondng copula C s unque, and can be wrtten as C(u 1,..., u n = F (F 1 1 (u 1,..., F 1 n (u n, (u 1,..., u n [0, 1] n. Thus, for multvarate dstrbutons wth contnuous margnals, the unvarate margnals and multvarate dependence structure can be separated, and the dependence structure can be represented by a copula. The copula was frst developed n Sklar (1959, and the copula theory and ts applcatons can be found, for example, n Nelsen (1999. In the study of extremal dependence, the survval copula s more effectve. Consder a random vector (X 1,..., X n wth contnuous margnals F 1,..., F n and copula C. Observe that F (X = 1 F (X, 1 n, s also unformly dstrbuted over [0, 1], and thus Ĉ(u 1,..., u n Pr{ F 1 (X 1 u 1,..., F n (X n u n } (1.2 s a copula, and called the survval copula of (X 1,..., X n. The survval functon of random vector (X 1,..., X n can be expressed as F (t 1,..., t n = Pr{X 1 > t 1,..., X n > t n } = Ĉ( F 1 (t 1,..., F n (t n, (t 1,..., t n R n. It also follows that for any (u 1,..., u n [0, 1] n, C(u 1,..., u n Pr{F 1 (X 1 > u 1,..., F n (X n > u n } = Ĉ(1 u 1,..., 1 u n, where C s the jont survval functon of copula C. The tal dependence of a bvarate dstrbuton has been dscussed extensvely n statstcs lterature (Joe 1997, but the tal dependence of the general case has not adequately addressed. Schmdt (2002 proposed one possble extenson, and the followng modfcaton offers more flexblty. Defnton 1.1. Let X = (X 1,..., X n be a random vector wth contnuous margnals F 1,..., F n and copula C. 2

3 1. X s sad to be upper-orthant tal dependent f for some subset J {1,..., n}, the followng lmt exsts and s postve. γ C J = lm u 1 Pr{F j (X j > u, j / J F (X > u, J} > 0. (1.3 If for all J {1,..., n}, γ C J = 0, then we say X s upper-orthant tal ndependent. 2. X s sad to be lower-orthant tal dependent f for some subset J {1,..., n}, the followng lmt exsts and s postve. β C J = lm u 0 Pr{F j(x j u, j / J F (X u, J} > 0. (1.4 If for all J {1,..., n}, β C J = 0, then we say X s lower-orthant tal ndependent. The lmts γj C s (βc J s are called the upper (lower tal dependent coeffcents. Obvously, the tal dependence s a copula property, and does not depend on the margnal dstrbutons. Snce Pr{F j (X j > u, j / J F (X > u, J} = Pr{ F j (X j 1 u, j / J F (X 1 u, J}, we obtan a dualty property for multvarate dstrbutons, γ C J = βĉj, for all J {1,..., n}. (1.5 Smlarly, βj C = γĉj. That s, the copula C s upper-orthant (lower-orthant tal dependent f and only f the survval copula Ĉ s lower-orthant (upper-orthant tal dependent. It s well-known that the bvarate normal dstrbuton s asymptotcally tal ndependent f ts correlaton coeffcent ρ < 1. Schmdt (2002 showed that bvarate ellptcal dstrbutons possess the tal dependence property f the tal of ther generatng random varable s regularly varyng. Ellptcal copulas do not have closed form expressons and are restrcted to have radal symmetry (C = Ĉ. In engneerng and fnancal applcatons, there s often a stronger dependence among bg losses than among bg gans (Embrechts, Lndskog and McNel Such asymmetres cannot be modeled wth ellptcal copulas. In ths paper, we dscuss the survval Marshall-Olkn copulas that are asymmetrc and have closed form expressons. We obtan the explct expressons of the tal dependence coeffcents of the Marshall-Olkn copulas and show that the survval Marshall-Olkn copula possesses varous upper tal dependence coeffcents. A more ntrgung ssue s how the dependence structure of a copula would affect ts tal dependence. That s, f we ncrease the dependence strength of the copula n some sense, 3

4 does ths ncrease the upper tal dependence? We show, va the parwse IE transforms (Xu and L 2000, that ths s not always the case, and n fact, an ncrease n the dependence strength can n some cases decrease the tal dependence. The paper s organzed as follows. In Secton 2, we obtan the explct expressons of the Marshall-Olkn copulas. In Secton 3, we derve the explct expressons of the tal dependence for the Marshall-Olkn copulas, and dscuss a suffcent condton under whch the tal dependences of Marshall-Olkn copulas can be compared. Fnally, some comments n Secton 4 conclude the paper. 2 Multvarate Marshall-Olkn Copulas Let {E S, S {1,..., n}} be a sequence of ndependent, exponentally dstrbuted random varables, wth E S havng mean 1/λ S. Let T j = mn{e S : S j}, j = 1,..., n. (2.1 The jont dstrbuton of T = (T 1,..., T n s called the Marshall-Olkn exponental dstrbuton wth parameters {λ S, S {1,..., n}} (Marshall and Olkn In the relablty context, T 1,..., T n can be vewed as the lfetmes of n components operatng n a random shock envronment where a fatal shock governed by Posson process {N S (t, t 0} wth rate λ S destroys all the components wth ndexes n S {1,..., n} smultaneously. In credt-rsk modelng, T 1,..., T n can be vewed as the tmes to default of varous dfferent counterpartes or types of counterparty, for whch the Posson shocks mght be a varety of underlyng economc events. It follows from (2.1 that the survval functon of T can be wrtten as F (t 1,..., t n = Pr{T 1 > t 1,..., T n > t n } (2.2 [ n = exp λ t ] λ j max{t, t j } λ 12...n max{t 1,..., t n }. =1 <j To express ths survval functon n terms of ts margnal survval functons, we ntroduce the followng notatons, for any S {1,..., n}, S t = mn{t, S}, S t = max{t, S}. It s easy to verfy va nducton that for any I {1,..., n}, k I t k = t k + + ( 1 1 k S t k + + ( 1 I 1 k I t k. k I S I,=s 4

5 Applyng these equaltes to (2.2, we obtan that F (t 1,..., t n ( n = exp λ S t + + ( 1 ( k λ S I t + + ( 1 n λ 12...n n =1 t = = =1 =1 S S I =k I =k S:I S [ ( ] n exp λ S t [ ( ] exp ( 1 k λ S I t... exp [( 1 n λ 12...n n =1 t ] [ ( ] n exp ( 1 k λ S I t. I =k S:I S S:I S Defne I I t = { I t I t f I s odd f I s even. (2.3 Snce exp [ ( 1 ( k S:I S λ ] S I t s non-decreasng (non-ncreasng n t when k s even (odd, we have, for any I wth I = k, [ ( ] [ ( ] exp ( 1 k λ S I t = k I exp ( 1 k λ S t. S:I S S:I S Thus, [ ( ] n F (t 1,..., t n = k I exp ( 1 k λ S t. (2.4 I =k S:I S For each and I {1, 2,..., n}, defne, α I = I S λ S S λ S. (2.5 Note that α I 1 for any I. It follows from (2.2 that the survval functon of the -component of Marshall-Olkn vector T s gven by [ ( ] F (t = exp λ S t, 1 n. (2.6 Thus, for any 1 n, I, [ ( ] exp ( 1 k λ S t = ( F (t ( 1 k+1 α I, S:I S S 5

6 whch, together wth (2.4, mply the followng. n ( F (t 1,..., t n = k I F (t ( 1 k+1 α I. (2.7 I =k That s, the jont survval functon of T s determned by the margnal survval functons and relatve rates α I s. Let u = F (t, 1 n, (2.7 yelds the explct expresson of the survval copula of the Marshall-Olkn dstrbuton. Proposton 2.1. Let T = (T 1,..., T n be a random vector wth the Marshall-Olkn dstrbuton wth parameters {λ S, S {1,..., n}}. Its survval copula s gven by n Ĉ(u 1,..., u n = k I u ( 1k+1 α I, (2.8 I =k where k and α I are gven by (2.3 and (2.5, respectvely. The survval functon C of the correspondng Marshall-Olkn copula C can be also easly expressed as C(u 1,..., u n = n I =k k I (1 u ( 1k+1 α I. (2.9 To obtan an expresson of the Marshall-Olkn copula, we defne, for any S {1, 2,..., n}, and u = (u 1, u 2,..., u n, C(u S = I S, I =k k I (1 u ( 1k+1 α I. Usng the ncluson-excluson formula, the Marshall-Olkn copula C s gven by n n C(u 1, u 2,..., u n = u n ( 1 k C(u S. (2.10 =1 k=2 =k Note, however, that t s easer to work wth the survval functon C of C. Example 2.2. Consder the bvarate case. The survval copula of a Marshall-Olkn vector T = (T 1, T 2 s gven by Ĉ(u 1, u 2 = u 1 u 2 2 =1 u α12 = u 1 u 2 mn{u α12 1 1, u α }. For the three dmensonal case, the survval Marshall-Olkn copula s gven by Ĉ(u 1, u 2, u 3 = u 1 u 2 u 3 mn{u α12 1 1, u α } mn{u α13 1 1, u α13 3 mn{u α23 2 2, u α } max{u α , u α , u α }. 3 } 6

7 The bvarate copulas of Marshall-Olkn dstrbutons are obtaned n Mulere and Scarsn (1987, and also dscussed n Embrechts, Lndskog and McNel ( Tal Dependence of Marshall-Olkn Copulas To obtan the upper tal dependence coeffcents of a Marshall-Olkn copula, we utlze the dualty property (1.5. Consder a Marshall-Olkn vector T = (T 1,..., T n wth parameters {λ S, S {1,..., n}}. It follows from (2.8 and (1.5 that ts copula has the followng upper tal dependence coeffcents, for any J {1,..., n}, n γj C I =k k I u ( 1k+1 α I = lm u 0 J I J, I =k k I u ( 1k+1 α I n = lm u 0 = lm u 0 I α I I =k u( 1k+1 J I J, I =k u( 1k+1 I α I P n P I =k ( 1k+1 I α I u u P J = lm u Pn u 0 P I J, I =k ( 1k+1 I α I P I =k ( 1k+1 I α I P J P I J, I =k ( 1k+1 I α I, where the second equalty follows from the fact that k I u ( 1k+1 α I = u ( 1k+1 I α I for 0 u 1. Notce that θ J = J n ( 1 k+1 I α I I =k for any J {1,..., n}. Thus, we have I J, I =k ( 1 k+1 I α I 0 (3.1 Proposton 3.1. Let T = (T 1,..., T n be a random vector wth the Marshall-Olkn dstrbuton wth parameters {λ S, S {1,..., n}}. The upper-orthant tal dependence coeffcents of ts copula are gven by, for any J {1,..., n}, γ C J where θ J s gven by (3.1. = lm u 1 Pr{F j(t j > u, j / J F (T > u, J} = { 0 f θ J > 0 1 f θ J = 0, Thus, asymptotcally, a Marshall-Olkn copula s ether tal ndependent or perfectly dependent, dependng on the parameter θ J. Consder the bvarate case where the parameters θ J s become θ 1 = θ 2 = 1 α 12 1 α

8 A bvarate Marshall-Olkn copula s asymptotcally ndependent f λ 1 > 0 or λ 2 > 0. If λ 12 > 0, and both λ 1 and λ 2 are zero, then the Marshall-Olkn copula s perfectly tal dependent. The survval Marshall-Olkn copulas offer more flexblty n descrbng the upper-orthant tal dependence. Theorem 3.2. Let T = (T 1,..., T n be a random vector wth the Marshall-Olkn dstrbuton wth parameters {λ S, S {1,..., n}}. The upper-orthant tal dependence coeffcents of ts survval copula are gven by, for any L {1,..., n}, γĉl = lm Pr{ F j (T j > u, j / L F (T > u, L} = n =1α 12...n, (3.2 u 1 L α L where α 12...n and α L are defned as n (2.5. Proof. Wthout loss of generalty, assume that L = {1, 2,..., j} for some 1 j < n. Consder We have Pr{ F j (T j > u, j / L F (T > u, L} = Pr{ F (T > u, {1, 2,..., n}} Pr{ F (T > u, {1, 2,..., j}}. Pr{ F (T > u, {1, 2,..., n}} = 1 Pr{ F (T u, {1, 2,..., n}} = 1 ( 1 +1 Pr{ F (T u, S}. S {1,2,...,n} Snce k I u ( 1k+1 α I = u ( 1k+1 I α I for any 0 u 1 and I = k, we have, Thus Pr{ F (T u, S} = Pr{ F (T > u, {1, 2,..., n}} = 1 Smlarly Pr{ F (T > u, {1, 2,..., j}} = 1 I =k,i S = u P S {1,2,...,n} S {1,2,...,j} 8 k I u ( 1k+1 α I P I =k,i S ( 1k+1 I α I. ( 1 +1 u P P I =k,i S ( 1k+1 I α I. ( 1 +1 u P P I =k,i S ( 1k+1 I α I.

9 Hence lm Pr{ F j (T j > u, j / L F (T > u, L} u 1 1 S {1,2,...,n} ( 1+1 u P = lm u 1 Observe that 1 lm u 1 S {1,2,...,n} 1 S {1,2,...,j} ( 1+1 u P ( 1 +1 u P P I =k,i S ( 1k+1 I α I P I =k,i S ( 1k+1 I α I P I =k,i S ( 1k+1 I α I = 1 = k=0 S {1,2,...,n}. (3.3 ( 1 +1 ( n n 1 n k ( 1 k k = (1 1 n = 0. (3.4 Then the lmt n (3.3 s of 0 0 form. Applyng L hosptal s rule to (3.3 yelds lm Pr{ F j (T j > u, j / L F (T > u, L} u 1 = S {1,2,...,n} ( 1+1 I =k,i S ( 1k+1 I α I S {1,2,...,j} ( 1+1. (3.5 I =k,i S ( 1k+1 I α I We need to show that for any n 1, S {1,2,...,n} ( 1 +1 I =k,i S For ths, consder, for any j 1, and J = {j + 2,..., n}, ( 1 k+1 I α I = n =1 α12...n. (3.6 = S {1,2,...,j,j+1} S {1,2,...,j} ( 1 +1 ( 1 +1 S=S {j+1},s {1,2,...,j} I =k,i S I =k,i S ( 1 S +2 S +1 ( 1 k+1 ( I α ( 1 k+1 ( I α I =k,i S {j+1} J α J α + (3.7 ( 1 k+1 ( I α J α. 9

10 The second term of the rght hand sde can be wrtten as follows. S=S {j+1},s {1,2,...,j} = {j+1} J α {j+1} J + ( 1 S +2 S +1 ( 1 S +2 S +1 I =k,i S {j+1} I =k,i S {j+1} ( 1 k+1 ( I α ( 1 k+1 ( I α J α J α = {j+1} J α {j+1} J + + ( 1 S +2 = {j+1} J α {j+1} J + ( 1 S +2 S +1 S ( 1 S +2 I=I {j+1}, I =k 1,I S ( 1 k+1 S k=0 ( 1 S +1 I =k,i S ( 1 k+2 S ( ( 1 k+1 I α I =k,i S ( I α I =k,i S ( 1 k+1 ( I α I {j+1} J J α ( I α J α J α {j+1} J α I {j+1} J. Notcng that the second term n the last expresson above and the frst term n (3.7 are canceled out, t follows from (3.7 that S {1,2,...,j,j+1} = {j+1} J α {j+1} J + = {j+1} J α {j+1} J = ( 1 +1 S ( 1 S +1 k=0 S ( 1 S +1 I =k,i S ( 1 k+1 ( I α J α ( ( 1 k+1 I α I {j+1} J I =k,i S ( 1 S +1 {j+1} J α {j+1} J S ( 1 S +1 I =k,i S ( 1 k+1 I =k,i S ( 1 k+1 + ( I α I {j+1} J ( I α I {j+1} J {j+1} J α I {j+1} J {j+1} J α I {j+1} J {j+1} J α I {j+1} J, 10

11 where the second equalty follows from the fact that α {j+1} J = 1, and the thrd equalty follows from (3.4. Thus, we have, for any j 1, and J = {j + 2,..., n}, S {1,2,...,j,j+1} S {1,2,...,j} ( 1 +1 ( 1 +1 I =k,i S I =k,i S ( 1 k+1 ( I α ( 1 k+1 ( I α I {j+1} J J α = {j+1} J α I {j+1} J.(3.8 We apply (3.8 on the left hand sde of (3.6, startng at j + 1 = n and J =. By applyng (3.8 recursvely n 1 tmes, (3.6 follows. Therefore, for L = {1, 2,..., j}, lm Pr{ F j (T j > u, j / L F (T > u, L} = n =1 α12...n. u 1 The general (3.2 can be obtaned by the ndex rearrangement. j =1 α12...j For example, for the three dmensonal case that n = 3, the varous tal coeffcents of the survval Marshall-Olkn copula are gven below. γĉ1 = γĉ2 = γĉ3 = α1231 α2 123 α3 123, γĉ12 = α123 1 α2 123 α3 123 α1 12, γĉ23 = α123 1 α2 123 α3 123 α12 2 α2 23, γĉ13 = α123 1 α2 123 α3 123 α23 3 α1 13. α13 3 The two dmensonal case has been dscussed n Embrechts, Lndskog and McNel (2003. Note, however, that they stated ther survval copula result n terms of the copula. Usng the same dea, we can obtan a general verson of Theorem 3.2. Theorem 3.3. Let T = (T 1,..., T n be a random vector wth the Marshall-Olkn dstrbuton wth parameters {λ S, S {1,..., n}}. For any L I {1,..., n}, lm Pr{ F j (T j > u, j I\L F (T > u, L} = Iα I u 1 L α L where α I and α L are defned as n (2.5., (3.9 We are now n the poston to dscuss how the dependence of Marshall-Olkn random varables would affect ther tal dependence. That s, we fx the margnal dstrbutons of a Marshall-Olkn vector, and ncrease the dependence n some sense, and see how ths would change the tal dependence. Let S λ S be fxed for each, and t follows from (2.6 that all the margnals are fxed. Let for any J {1, 2,..., n}, m J = n =1 P 1 S λ S 1 J P S λ S 11.

12 Then (3.2 can be rewrtten as lm Pr{ F j (T j > u, j / J F (T > u, J} = m J u 1 λ 12...n J S λ S. (3.10 We ntroduce the parwse IE transforms on the set of the parameters {λ S }, whch s defned on a lattce {S : S {1, 2,..., n}} wth respect to the set ncluson relaton. Let, for any I 1, I 2 {1, 2,..., n}, Ψ I 1,I 2 ({λ S } = {λ S }, (3.11 where > 0 s a real number such that λ S f S / {I 1, I 2, I 1 I 2, I 1 I 2 } λ S = λ S 0 f S = I 1, or S = I 2 λ S + f S = I 1 I 2, or S = I 1 I 2. That s, a parwse IE transform re-dstrbutes some weghts of I 1 and I 2 to ther common unon I 1 I 2 and common ntersecton I 1 I 2. Such parwse IE transforms have been dscussed n Tchen (1980 and n Xu and L (2000 to study the dependence of dstrbutons. Let T = (T 1,..., T n be a Marshall-Olkn random vector wth parameters {λ S }, and T = (T 1,..., T n be a Marshall-Olkn random vector wth parameters Ψ I 1,I 2 ({λ S }. Xu and L (2000 showed that T and T have the same margnal dstrbutons, and T s less upperand lower-orthant dependent than T n the sense that Pr{T 1 > t 1,..., T n > t n } Pr{T 1 > t 1,..., T n > t n} (3.12 Pr{T 1 t 1,..., T n t n } Pr{T 1 t 1,..., T n t n}. (3.13 However, as the next result shows, the change on tal dependence depends on how we apply the IE transform on the parameter lattce. Theorem 3.4. Let γ J (γ J, J {1, 2,..., n}, be the upper-orthant tal dependence coeffcents of the survval copulas of T (T, as descrbed n ( If I 1 I 2 = {1, 2,..., n}, then γ J γ J 2. If I 1 I 2 {1, 2,..., n}, then γ J γ J for all J {1, 2,..., n}. for all J {1, 2,..., n}. Proof. If I 1 I 2 or I 2 I 1, the results are trvally true. Assume now that I 1 I 2 I for = 1, 2. Note that T and T have the same margnals, and t then follows from (3.10 that we λ need only to compare the ratos P 12...n and J S λ P λ 12...n n (3.10. If I S 1 I 2 = {1, 2,..., n}, J S λ S then λ 12...n λ 12...n + = λ 12...n. For the denomnator n the rato of (3.10, consder the followng cases. 12

13 1. If J I for exactly one = 1 or = 2, then J S λ S = J S λ S + = J S λ S. Thus γ J γ J. 2. If J I for = 1 and = 2, then J I 1 I 2. Thus, J S λ S = J S λ S = J S λ S. Hence γ J γ J. 3. Otherwse, I 1 / {S : J S}, I 2 / {S : J S}, I 1 I 2 / {S : J S}. We must have J S λ S = J S λ S +. Snce J S λ S λ 12...n, then λ 12...n+ s non-decreasng PJ S λ S+ n, and thus λ 12...n λ γ J = m J 12...n + J S λ m J S J S λ S + = γ J. If I 1 I 2 {1, 2,..., n}, then λ 12...n = λ 12...n. For the denomnator n the rato of (3.10, consder the followng cases. 1. If J I 1 I 2, then J S λ S J S λ S. Thus, γ J γ J. 2. Otherwse, I 1 I 2 / {S : J S}. Thus J S λ S = J S λ S, and agan, γ J γ J. Thus, f we ncrease the dependence of a Marshall-Olkn random vector va an IE transform that nvolves λ 1...n, the upper tal dependence of the survval copula s also ncreased, whereas f the IE transform does not nvolve λ 1...n, the upper tal dependence s decreased. Asymptotcally, the tal dependence among components 1,..., n depends heavly on the shock arrval rate λ 1...n that affects all these components. Marshall-Olkn random vector T wth the parameters, λ S = 1, for all S {1, 2, 3}. Consder a three dmensonal If another three dmensonal Marshall-Olkn random vector T has the followng parameters, λ 12 = λ = 2, λ 1 = λ 2 = 0, and λ S = 1 otherwse, then T s less tal dependent than T. Note that T s more upper- and lower-orthant dependent than T n the sense of (3.12 and (3.13. If a three dmensonal Marshall-Olkn random vector T has the followng parameters, λ 123 = λ = 2, λ 12 = λ 3 = 0, and λ S = 1 otherwse, then T s more tal dependent than T. 13

14 4 Concludng Remarks As we llustrated n the prevous sectons, the Marshall-Olkn copulas are asymmetrc and have closed form expressons, and ther tal dependence coeffcents can be explctly expressed n terms of ther parameters. Asymptotcally, the Marshall-Olkn copula s ether ndependent or perfectly dependent, whereas the survval Marshall-Olkn copula possesses varous postve upper tal dependence. The tal dependence s a lmtng condtonal probablty that descrbes the dependence of extremal events, and as such, t should depend heavly on the dependence parameter that affects all the components. Usng the parwse IE transforms, we show that asymptotcally, a survval Marshall-Olkn copula becomes more tal dependent as ts underlyng shock arrval rate that affects all the components ncreases. We also llustrate that ncreasng the upperor lower-orthant dependence of a Marshall-Olkn dstrbuton can sometmes decrease ts tal dependence. It s known that upper- or lower-orthant dependence can be characterzed by tree majorzaton ntroduced n Xu and L (2000. The parwse IE transform s a specal case of the general transformaton method dscussed n Xu and L (2000 for upper- or lower-orthant dependence. It s not clear whether or not our comparson result presented here can be extended to more general transformatons. Acknowledgment: The author would lke to thank Moshe Shaked for hs comments on the related references. 14

15 References [1] Embrechts, P., F. Lndskog and A. McNel (2003. Modelng dependence wth copulas and applcatons to rsk management. Handbook of Heavy Taled Dstrbutons n Fnance, ed. S. Rachev, Elsever, Chapter 8, [2] Joe, H. (1997. Multvarate Models and Dependence Concepts. Chapman & Hall, London. [3] Marshall, A. W. and Olkn, I. (1967. A multvarate exponental dstrbuton. J. Amer. Statst. Assoc. 2, [4] Mulere, P. and Scarsn, M. (1987. Characterzaton of a Marshall-Olkn type class of dstrbutons. Annals of the Insttute of Statstcal Mathematcs, 39, [5] Nelson, R. (1999. An Introducton to Copulas. Sprnger, New York. [6] Schmdt, R. (2002. Tal dependence for ellptcally contoured dstrbutons. Mathematcal Methods of Operatons Research 55, [7] Sklar, A. (1959. Fonctons de répartton à n dmensons et leurs marges. Publcatons de l Insttut de statstque de l Unversté de Pars, 8, [8] Tchen, A. H. (1980. Inequaltes for dstrbutons wth gven margnals. Ann. Probab., 8, [9] Xu, S. and H. L (2000. Majorzaton of weghted trees: A new tool to study correlated stochastc systems. Mathematcs of Operatons Research, 35,