Multivariate Distributions from Mixtures of Max-Infinitely Divisible Distributions

Transkript

1 ournal of multvarate analyss 57, (1996) artcle no Multvarate Dstrbutons from Mxtures of Max-Infntely Dvsble Dstrbutons Harry Joe Unversty of Brtsh Columba, Vancouver, Canada and Tazhong Hu Unversty of Scence and Technology of Chna, Hefe, Anhu, People's Republc of Chna A class of multvarate dstrbutons that are mxtures of the postve powers of a max-nfntely dvsble dstrbuton are studed. A subclass has the property that all weghted mnma or maxma belong to a gven locaton or scale famly. By choosng approprate parametrc famles for the mxng dstrbuton and the dstrbuton beng mxed, famles of multvarate copulas wth a flexble dependence structure and wth closed form cumulatve dstrbuton functons are obtaned. Some dependence propertes of the class, as well as some characterzatons, are gven. Condtons for max-nfnte dvsblty of multvarate dstrbutons are obtaned Academc Press, Inc. 1. Introducton. We consder a class of multvarate dstrbutons that are mxtures of the postve powers of a max-nfntely dvsble dstrbuton. (A max-nfntely dvsble multvarate dstrbuton s one for whch all postve powers of t are also proper dstrbuton functons.) Wth approprate parametrzatons of the mxng dstrbuton and the dstrbuton beng mxed, many parametrc famles of multvarate copulas (dstrbutons wth unvarate unform margns) wth flexble dependence structure can be obtaned. The class s a generalzaton of the famles n Marshall and Olkn [15] and Joe [8]. The new dstrbutons have dependence and other propertes that should make them sutable for multvarate models. Receved December 8, 1994; revsed October AMS 1990 subect classfcatons: 62H99, 62H05. Key words and phrases: Max-stable, max-nfntely dvsble, multvarate extreme value dstrbuton, copula, postve dependence, Laplace transform X Copyrght 1996 by Academc Press, Inc. All rghts of reproducton n any form reserved. 240

2 MULTIVARIATE DISTRIBUTIONS 241 Specal cases of multvarate max-nfntely dvsble dstrbutons are multvarate extreme value dstrbutons for maxma, snce these are maxstable (F(x 1,..., x m ) s max-stable, f for each t>0, F t (x 1,..., x m )= F(a 1t +b 1t x 1,..., a mt +b mt x m ) for some vectors (a 1t,..., a mt ), (b 1t,..., b mt ).) If all of the unvarate margns of the multvarate extreme value dstrbuton are the same, then the dstrbuton F resultng from a mxture has the property of closure under weghted mnma or maxma. If (X 1,..., X m )tf, ths means that (a) for all (c 1,..., c m )#R m +, max[c 1X 1,..., c m X m ] (or mn[c 1 X 1,..., c m X m ]) leads to a random varable n the same scale famly as the X 's, f the support of the X 's s n [0, ); or (b) for all (c 1,..., c m )#R m, max[x 1 &c 1,..., X m &c m ] (or mn[x 1 &c 1,..., X m &c m ]) leads to a random varable n the same locaton famly as the X 's, f the support of the X 's s n (&, ). Ths class of multvarate dstrbutons ncludes the class of max-geometrc stable dstrbutons of Rachev and Resnck [18] and some multvarate logstc dstrbutons of Arnold [1]. These results are gven n Secton 2. Secton 3 conssts of some dependence propertes and characterzatons of the class of mxtures of powers of a max-nfntely dvsble dstrbuton. Secton 4 conssts of results and condtons for max-nfntely dvsble dstrbutons. In Secton 5, some new nterestng parametrc famles of copulas are lsted together wth some propertes. Included are parametrc famles of multvarate copulas wth flexble dependence structure and closed form cumulatve dstrbuton functons (cdfs); prevously no famles wth both of these propertes had been obtaned. The key results n ths artcle are: (a) consderaton of multvarate dstrbutons of the form (2.7) to get a new class of copulas, wth the specal case of (2.8) or (2.10); (b) dependence propertes and characterzatons for (2.7); (c) showng that there are parametrc examples of (2.7) that have good propertes (ths s mportant because nonparametrc nference s dffcult n hgher dmensons wthout a lot of data); and (d) condtons for a multvarate dstrbuton to be max-nfntely dvsble. 2. Mxtures of Mn-Stable or Max-Stable Multvarate Extreme Value Dstrbutons and Extensons. In ths secton, we start wth mxtures of powers of a multvarate extreme value dstrbuton and then go to the larger class of mxtures of powers of a max-nfntely dvsble (max-d) dstrbuton. The former class has some closure propertes and the latter class covers a wde range of multvarate dependence structures. For statstcal modellng. the choce of a multvarate model mght be based on closure propertes or dependence propertes.

3 242 JOE AND HU The three types of unvarate extreme value margns are Gumbel, Webull, and Fre chet. Snce maxma can be transformed nto mnma and vce versa, we wll consder Webull survval margns wth mnma, and Fre chet and Gumbel margns wth maxma so that we can work on ether [0, ) or(&, ) for each unvarate margn. Wthout loss of generalty, we assume that the unvarate margns are dentcal and standardzed. A key property of a multvarate extreme value (MEV) dstrbuton G that s used here s that all postve powers of G are also dstrbutons. More generally, a multvarate cdf (survval functon) that has the property that all postve powers are cdfs (survval functons) s sad to be max-nfntely (mn-nfntely) dvsble. Let G be a mn-stable m-varate exponental survval functon wth unt exponental margns. Let A=&log G be a possble exponent of a mnstable m-varate exponental survval functon. Then A s homogeneous of order 1, A(x 1,..., x m )=x f all arguments are zero except x, and G # (x 1,..., x m )=exp[&#a(x 1,..., x m )]=exp[&a(#x 1,..., #x m )] s a survval functon for all #>0. See Galambos [4] and Joe [6, 9] for some dscusson and examples of mn-stable multvarate exponental dstrbutons. By makng the transformatons x x :, wth :>0, the resultng mnstable m-varate Webull survval functon s G 1 (x 1,..., x m ; :)=exp[&a(x :,..., 1 x: m )]. (2.1) If (X 1,..., X m ) have the dstrbuton n (2.1), then Pr(mn[X 1 c 1,..., X m c m ]>t)=exp[&a((tc 1 ) :,..., (tc m ) : )]=exp[&t : A(c :,..., 1 c: )], t>0, m for all (c 1,..., c m )#R m. That s, mn[x + 1c 1,..., X m c m ] has a scaled Webull dstrbuton for all (c 1,..., c m )#R m. + Smlar results hold for transforms to other extreme value margns. By makng the transformatons x x &;, wth ;>0, the resultng max-stable m-varate Fre chet dstrbuton functon s G 2 (x 1,..., x m ; ;)=exp[&a(x &; 1,..., x &; m )]. (2.2) If (X 1,..., X m ) have the dstrbuton n (2.2), then Pr(max[X 1 c 1,..., X m c m ] t) = exp[&a((tc 1 ) &;,..., (tc m ) &; )] =exp[&t &; A(c &; 1,..., c &; )], m t>0, for all (c 1,..., c m )#R m. That s, max[x + 1c 1,..., X m c m ] has a scaled Fre chet dstrbuton for all (c 1,..., c m )#R m. + By makng the transformatons x e &x, the resultng max-stable m-varate Gumbel dstrbuton functon s G 3 (x 1,..., x m )=exp[&a(e &x 1,..., e &x m )]. (2.3)

4 MULTIVARIATE DISTRIBUTIONS 243 If (X 1,..., X m ) have the dstrbuton n (2.3), then Pr(max[X 1 &c 1,..., X m & c m ] t) = exp[&e &t A(e &c 1,..., e &c m )], &<t<, for all (c 1,..., c m )#R m. That s, max[x 1 &c 1,..., X m &c m ], a maxmum of shfted random varables has a locaton shfted Gumbel dstrbuton for all (c 1,..., c m )#R m. The weghtng s done wth addtve constants rather than multplcatve constants n ths case. Note that a postve power of (2.1), (2.2), or (2.3) s a survval or dstrbuton functon snce ether a scale or locaton shft occurs. By takng mxtures of powers of a multvarate extreme value dstrbuton, we get dstrbutons wth other unvarate margns whch satsfy the closure property of weghted mnma or maxma n the same scale or locaton famly. Let M be the dstrbuton functon of a postve random varable and let ts Laplace transform (LT) be. The mxtures of (2.1), (2.2), (2.3) lead to 0 0 exp[&#a(x :,..., 1 x: m )]dm(#)=(a(x:,..., 1 x: m )), (2.4) exp[&#a(x &; 1,..., x &; m )]dm(#)=(a(x&; 1,..., x &; m )), (2.5) exp[&#a(e &x 1,..., e &x m )]dm(#)=(a(e &x 1,..., e &x m )). (2.6) 0 The unvarate survval margns n (2.4) are (x : ) and the unvarate dstrbuton functons n (2.5), (2.6) are respectvely (x &; ) and (e &x ). If (X 1,..., X m ) has the dstrbuton n (2.4), then mn 1m [X c ] has the survval functon ((t_) : ), for t>0, where _=[A(c :,..., 1 c: m )]&1:. If (X 1,..., X m ) has the dstrbuton n (2.5), then max 1m [X c ] has the dstrbuton ((t') &; ), or t>0, where t>0, where '= [A(c &; 1,..., c &; m )]1;, and f (Y 1,..., Y m ) has the dstrbuton n (2.6), then max 1m [Y &c ] has the dstrbuton (e &[t&+(c 1,..., c m )] ), &<t<, where +(c 1,..., c m )=log A(e &c 1,..., e &c m ). Remarks. 1. The above results overlap wth those n Robertson and Strauss [19] and n Secton 5 of Strauss [20]. 2. More generally, one can have F= H # dm(#)=(&log H), (2.7) where H s a max-stable m-varate dstrbuton wth arbtrary extreme value unvarate margns or a non-mev dstrbuton that s max-d (H # s a dstrbuton for all #>0). Ths s then a generalzaton of constructons n Marshall and Olkn [15] and Joe [8]. If H s not a MEV dstrbuton,

5 244 JOE AND HU then the closure property of weghted maxmamnma does not hold. Subfamles of (2.7) are presented below, and condtons for an m-varate dstrbuton to be max-d are gven n Secton A specal case of (2.7) arses when (s)=(1+s) &1. For H beng a general max-stable dstrbuton, F becomes a max-geometrc stable dstrbuton (see [18]). For H beng the Gumbel dstrbuton, the unvarate margns of (2.6) or (2.7) become the logstc dstrbuton (1+e &x ) &1 and F s a max-geometrc stable multvarate logstc dstrbuton (see [1]). 4. Marley [13] obtans a class of dstrbuton of the form (2.4) wth satsfyng some condtons (the boundary and complete monotoncty condtons of a Laplace transform) and A satsfyng some condtons for dervatves. The class consdered here generalzes Marley's class. In the remander of ths secton, we look at specal cases, wth the form of (2.7), that can lead to parametrc famles of multvarate dstrbutons or copulas wth closed form cdfs, flexble dependence structure and partal closure under takng of margns. The number of parameters s at most ( m 2 )+m+1 for m dmensons. Let K,1<m, be bvarate copulas that are max-d. Let H 1,..., H m be unvarate cdfs. Let M be the dstrbuton of a postve random varable, and let ts LT be. Consder the mxture 0 ` 1<m = \ & : m K : (H, H ) ` 1< m =1 H & : dm(:) m log K (H, H )& : =1 & log H +, (2.8) where usually & 's are nonnegatve, although they can be negatve f some of the copulas correspond to ndependence. The unvarate margns of (2.8) are F =(&(& +m&1) log H ). (2.9) Hence (2.8) s a copula wth unform (0, 1) margns, f H (u ) s chosen to be exp[&p &1 (u )] wth p =(& +m&1) &1, =1,..., m. Wth these substtutons, the copula s C(u 1,..., u m ) (2.10) = \& : 1<m m log K (e &p &1 (u ), e &p &1 (u ) )+ : =1 & p &1 (u ) +. A rough nterpretaton s that the LT leads to a mnmal level of (parwse) dependence, the copulas K add some ndvdual parwse

6 MULTIVARIATE DISTRIBUTIONS 245 dependence on top of the global dependence, and the parameters & 's can be used for bvarate and multvarate asymmetry (the asymmetres are represented through & (& +& ), {). Also the parameters & are ncluded n order that the famly (2.10) s closed under margns. For example, f H m 1 n (2.8), then (2.8) becomes 0 ` 1<m&1 = \ & : m&1 K : (H, H ) ` 1<m&1 =1 and the resultng margnal copula s C 1}}}m&1 (u 1,..., u m&1 ) = \& : H (& +1) : dm(:) m&1 log K (H, H )& : 1<m&1 m&1 + : =1 =1 (& +1) log H +, (2.8$) log K (e &p &1 (u ), e &p &1 (u ) ) (& +1) p &1 (u ) +. (2.10$) (Hence the ``parameters'' K reman the same but the parameters & 's change wth takng margns; ths can be shown notatonally by & (m&1) =& (m) +1 and & (2) =& (m) +m&2.) The (, ) bvarate margnal copula of (2.10) s C (u, u )=(&log K (e &p &1 (u ), e &p &1 (u ) ) (2.10") +(& +m&2) p &1 (u )+(& +m&2) p &1 (u )). The copula (2.10") s more concordant (or more postve quadrant dependent) than C (u, u )=( &1 (u )+ &1 (u )), (2.11) whch explans the above nterpretaton for. The proof of ths s gven n Theorem 3.5 and the defnton of concordance s gven at the begnnng of Secton 3. Specal cases of (2.8) are 1. m=3, K 13 (x, y)=xy, & 1 =& 3 =&1, & 2 =0, K 12 (u, v)=k 23 (v, u)= K(u, v). Then (2.8) becomes 0 K : (H 1, H 2 ) K : (H 3, H 2 ) dm(:), (2.12)

7 246 JOE AND HU wth copula C(u 1, u 2, u 3 ) (2.13) =(&log K(e &&1 (u 1 ), e &0.5&1 (u 2 ) )&log K(e &&1 (u 3 ), e &0.5&1 (u 2 ) )); f H 1, H 2, H 3 are chosen approprately so that the unvarate margns of (2.12) are all the same, then the (1, 2) and (2, 3) bvarate margns of (2.12) are the same and are more concordant than the (1, 3) margn. Hence ths model would be approprate for generatng a second-order statonary Markov chan. The proof of concordance s smlar to above snce the (1, 2) and (2, 3) bvarate margns of (2.13) have copula (&log K(e &&1 (u ), e &0.5&1 (u 2 ) )+0.5 &1 (u 2 )) (2.14) and the (1, 3) bvarate margn s C (u 1, u 3 ), gven n (2.11). 2. & 1 =& 2 =&1, & 3 =}}}=& m =0, K 12 (x, y)=xy, p 1 =p 2 =(m&2) &1, p 3 =}}}=p m =(m&1) &1. Then (2.8) becomes wth copula 0 ` 1<m,(,){(1, 2) C(u 1,..., u m )= \& : 1< m, (,){(1, 2) K : (H, H ) dm(:) log K (e &p &1 (u ), e &p &1 (u ) ) +. If s a one-parameter famly of LTs and each K s a one-parameter famly of copulas, then ths becomes a famly wth m(m&1)2 parameters. The labellng s such that the ndces 1, 2 are assgned to the par of varables wth the least amount of dependence. The (1, 2) bvarate margn has the copula n (2.11). 3. Dependence Propertes and Characterzatons In ths secton, we obtan some characterzatons of (2.4)(2.6) and some dependence propertes of (2.4)(2.6) and (2.8)(2.10). In general, for a gven possble unvarate margn, the dstrbutons have postve dependence (from the mxng). We show that the case of ndependence can occur only f corresponds to a Webull dstrbuton for (2.4), a Fre chet dstrbuton for (2.5), or a Gumbel dstrbuton for (2.6). Propertes that are obtaned are the postve dependence condton of assocaton for (2.4)(2.6), and concordance and tal dependence propertes for (2.10) and (2.10").

8 MULTIVARIATE DISTRIBUTIONS 247 We next gve defntons of concordance, tal dependence, postve quadrant dependence, left-tal ncreasngdecreasng, assocated random varables, and TP 2. These are used n the remanng results n ths artcle. References for these defntons are Barlow and Proschan [2] and [7, 8]. Defntons. Two m-varate dstrbutons F 1, F 2 are sad to be ordered by concordance, denoted as F 1 O C F 2,fF 1 (x)f 2 (x) and F 1(x)F 2x) for all x # R m, where F 1, F 2 are the survval functons. (For two bvarate copulas, one set of nequaltes mples the others.) A bvarate copula C has upper tal dependence f C (u, u)(1&u) converges to a constant c 1 n (0, 1] as u 1. Here C s the survval functon defned by C (u, v)=1&u&v+c(u, v). Smlarly, lower tal dependence exsts f C(u, u)u converges to a constant c 2 n (0, 1] as u 0. The constants c 1 and c 2 are referred to as tal-dependence parameters. A bvarate copula K(x, y) satsfes the postve quadrant dependence (PQD) property f K(x, y)xy for all 0x, y1. K satsfes the left-tal decreasng property of the frst varable, denoted LTD1, f K(x, y)x s decreasng n x for all y. (K(x,})x s the condtonal dstrbuton of the second varable gven that the frst varable s less than or equal x). K satsfes the left-tal decreasng properly of the second varable, denoted LTD2, f K(x, y)y s decreasng n y for all x. A vector of random varables X=(X 1,..., X m ) s assocated f the nequalty E[a(X) b(x)]e[a(x)] E[b(X)] holds for all real-valued functons a, b whch are ncreasng or nondecreasng (n each component) and are such that the expectatons exst. A nonnegatve functon h(x, y) stotally postve of order 2, denoted TP 2, n x, y (on ts doman) f for all x 1 <x 2, y 1 <y 2, h(x 1, y 1 ) h(x 2, y 2 ) h(x 1, y 2 ) h(x 2, y 1 ). The frst result s on assocaton. The mxture famles n (2.4), (2.5), (2.6) consst of assocated random varables and hence the dstrbutons satsfy several postve dependence nequaltes. Theorem 3.1. Let F= 0 H # dm(#) be the survval functon or dstrbuton functon gven n (2.4), (2.5), or (2.6). Then F s the dstrbuton of assocated random varables. Proof. Let X=(X 1,..., X m ) have the dstrbuton F. For (2.4), H # s a multvarate mn-stable survval functon for all #>0, and for (2.5) and (2.6), t s a multvarate max-stable dstrbuton functon for all #>0. Hence H # s stochastcally decreasng n # n the former case and stochastcally ncreasng n the latter case. Let 1 be a random varable wth dstrbuton M. To show the assocaton, t s requred to show that Cov(a(X), b(x))0 for all ncreasng functons a, b.

9 248 JOE AND HU The covarance can be wrtten as Cov(a(X), b(x)) =E[Cov(a(X), b(x) 1)]+Cov(E(a(X) 1), E(b(X) 1)). For the frst term, Cov(a(X), b(x) 1=#)0 for all # because H # s assocated from a theorem n Marshall and Olkn [14]. For the second term, E(a(X) 1=#), E(b(X) 1=#)) are decreasng n # for (2.4) and ncreasng n # for (2.5), (2.6) because of the stochastc monotoncty result referred to n the precedng paragraph. Hence the covarance of the two condtonal expectatons s nonnegatve because a sngle random varable 1 s assocated. K In Secton 2, we menton that (2.6) could result n multvarate dstrbutons wth unvarate logstc margnals wth the approprate choce of the LT (s)=(1+s) &1. However, for logstc margns only strctly postvely dependent multvarate dstrbutons can result; t s easly checked that the multvarate dstrbuton wth ndependent unvarate logstc margns does not satsfy the property of closure under weghted maxma. Ths dvson of ndependence versus postve dependence s true n general. We show below that multvarate dstrbutons wth ndependent margns can arse from (2.4) only f the margns are Webull (wth exponental as a specal case). Smlarly, margns must be Fre chet for (2.5) and Gumbel for (2.6). The result for the Gumbel margn s also gven n Theorem 2 of Robertson and Strauss [19]. Theorem 3.2. Suppose (A(x : 1,..., x: m )) n (2.4) s equal to >m =1 (x: ) for all (x 1,..., x m )#R m +. Then all possble solutons are covered by takng (s)=exp[&*s 1_ ] for some postve constants * and _. Proof. Let X 1,..., X m be..d. wth survval functon F (x)=(x : ). Then F 2 (t)=pr(x 1 >t, X 2 >t)=((ta) : )=F (ta) for all t>0, where a : =A(1, 1, 0,..., 0) s a constant (exceedng 1). Let r(t)=&log F (t). Then r(0)=0, r()=, r s ncreasng and 2r(t)=r(at) for all t>0. Let b be a constant satsfyng a=2 b so that 2 b r b (t)=ar b (t)=r b (at). Next, let '(t)=r b (t) so that a'(t)='(at) for all t>0. Snce the LT s dfferentable, ' s dfferentable and a'$(t)=a'$(at) for all t>0. The condtons on r and ' then mply that '$ s a constant and ' s lnearly ncreasng. Snce '(0)=0, '(t)=d b t for a postve constant d. Hence r(t)=dt 1b and b=log alog 2, or F (t)=exp[&dt 1b ] for some constants b, d. K Theorem 3.3. Suppose (A(x &; 1,..., x &; m )) n (2.5) s equal to > m =1 (x &; ) for all (x 1,..., x m )#R m +. Then all possble solutons are covered by takng (s)=exp[&*s 1_ ] for some postve constants * and _.

10 MULTIVARIATE DISTRIBUTIONS 249 Proof. The proof s accomplshed by a smlar argument to that of the above theorem. We omt the detals. K Theorem 3.4. Suppose (A(e &x 1,..., e &x m )) n (2.6) s equal to > m =1 (e&x ) for all (x 1,..., x m )#R m.then all possble solutons are covered by takng (s)=exp[&*s 1_ ] for some postve constants * and _. Proof. Let X 1,..., X m have the dstrbuton (A(e &x 1,..., e &x m )). Set Y =exp(&x :), =1,..., m, for :>0. Then (Y 1,..., Y m ) have the survval functon Pr(Y 1 >y 1,..., Y m >y m )=Pr(X 1 <&:log y 1,..., X m <&: log y m ) =(A( y :,..., 1 y: )) m for all ( y 1,..., y m )#R m. If X + 1, X 2,..., X m are ndependent wth survval functon (e &x ), then Y 1, Y 2,..., Y m are also ndependent wth survval functon ( y : ). From the proof of Theorem 3.2, we get that ( y : )=exp(&*y1b ) for some postve constants *, b, and hence (e &x )=exp(&*e &x[:b] ). Ths completes the proof. K Next, we obtan some general results on bvarate tal dependence and concordance for (2.10"); these are appled to specfc examples n Secton 5. Theorem 3.5. The bvarate copula gven n (2.10") s more concordant than that gven n (2.11). Proof. K(e &p &1 (u ), e &p&1 (u ) ) } e &(&+m&2) p &1 (u ) e &(&+m&2) p&1 (u ) e &&1 (u )& &1 (u ) f K(e &p &1 (u ), e &p &1 (u ) )e &p &1 (u )&p &1 (u ) for all 0<u, u <1, or f K(x, y)xy for all 0<x, y<1. The latter postve quadrant dependence nequalty holds snce K max-d mples t s TP 2 (see Secton 4) whch n turns mples the postve quadrant dependence. (If K s TP 2, that s, K(x, y) K(x 2, y 2 )&K(x, y 2 ) K(x 2, y)0 for all 0x<x 2 1, 0y<y 2 1, then lettng x 2, y 2 1 yelds K(x, y)xy for all 0x, y1, that s, K s PQD.) K Remark. Also note that as K ncreases n concordance n (2.10") wth and &, & held fxed, then C ncreases n concordance. Let K be a bvarate copula and be a LT. Wth (, )=(1, 2) and m=2, (2.10") becomes C(u 1, u 2 )=(&log K(e &p 1 &1 (u 1 ), e &p 2 &1 (u 2 ) ) +& 1 p 1 &1 (u 1 )+& 2 p 2 &1 (u 2 )), (3.1) where & 1, & 2 0 are arbtrary and p =(& +1) &1, =1, 2.

11 250 JOE AND HU Theorem 3.6. If $(0) s fnte, then the copula C (u, v)= ( &1 (u)+ &1 (v)) n (2.11) does not have upper tal dependence. If C has upper tal dependence, then $(0)=& and the tal dependence parameter s Proof. lm C (u, u)(1&u) u 1 $ U =2&2 lm [$(2s)$(s)]. (3.2) s0 = lm [1&2u+(2 &1 (u))](1&u) u 1 =2&2 lm $(2 &1 (u))$( &1 (u))=2&2 lm [$(2s)$(s)]. u 1 s 0 If $(0) # (&, 0), then the lmt s zero and C does not have upper tal dependence. s strctly decreasng so that $(0) cannot equal 0. The rest of the result follows. K Theorem 3.7. The copula (3.1) has upper tal dependence only f ether $(0)=& or K has upper tal dependence or both. (The detals of the tal dependence parameter are n the proof.) Proof. Suppose that the copula K n (3.1) has upper tal dependence parameter ; #[0, 1] (;=0 mples no tal dependence). We consder frst the case p 1 =p 2 or & 1 =& 2. Subsequently, for the case of p 1 {p 2, bounds wll be obtaned. For x less than and close to 1, K (x, x)t;(1&x) so that K(x, x)t 2x&1+;(1&x)=1&(2&;)(1&x). Let p 1 =p 2 =p=(&+1) &1. Then for u near 1, &log [K(e &p&1 (u), e &p&1 (u) )]+2&p &1 (u) t&log [1&(2&;)(1&e &p&1 (u) )]+2&p &1 (u) t&log [1&(2&;) p &1 (u)]+2&p &1 (u) t(2&;) p &1 (u)+2&p &1 (u)=# &1 (u), where #=(2(&+1)&;) p=2&;(&+1) # [1, 2]. Hence, for u near 1, [1&2u+C(u, u)](1&u)t[1&2u+(# &1 (u))](1&u) t2&#$(# &1 (u))$( &1 (u))

12 MULTIVARIATE DISTRIBUTIONS 251 and the upper tal dependence parameter of C n (3.1) s $ U = 2&# lm s 0 $(#s)$(s). If C does not have upper tal dependence, then $ U =2&#=;(&+1) and C has upper tal dependence f and only f K has upper tal dependence (and the tal dependence parameter of K s larger snce &0). If C does have upper tal dependence, then # lm s 0 $(#s)$(s) should be ncreasng and $ U decreasng as # ncreases or as & ncreases (ths follows from Theorem 3.8 below). If ;=0 so that #=2, then $ U = 2&2 lm s 0 $(2s)$(s) s the tal dependence parameter of C.If;=1 and &=0 so that #=1, then $ U =1. Hence the tal dependence parameter of (3.1) s greater than or equal to that of C. For the asymmetrc case wth p 1 p 2 (& 1 & 2 ), K(e &p 2 &1 (u), e &p 2 &1 (u) )K(e &p 1 &1 (u), e &p 2 &1 (u) )K(e &p 1 &1 (u), e &p 1 &1 (u) ) so that from above, the tal dependence parameter $ U s bounded as 2&# 2 lm $(# 2 s)$(s)$ U 2&# 1 lm $(# 1 s)$(s), s 0 s 0 where # =(2&;)(& +1)+& 1 (& 1 +1)+& 2 (& 2 +1), =1, 2. Note that # 1 # 2. K Theorem 3.8. Let C be as gven n (3.1). (a) Then C ncreases n concordance as p 1 ncreases ( from 0 to 1) and & 1 decreases f K satsfes the LTD1 property. (b) Also C ncreases n concordance as p 2 ncreases and & 2 decreases f K satsfes the LTD2 property. (c)if p 1 =p 2 =p and & 1 =& 2 =&, then C ncreases n concordance as p ncreases f K satsfes both LTD1 and LTD2. Hence the upper tal dependence parameter of C, $ U = 2&# lm s 0 $(#s)$(s), ncreases as & decreases, wth #=2&;(&+1), K (x, x)t;(1&x). Remark. K LTD mples K PQD but not the converse (see [2]). If K s TP 2 as n max-d copulas, then t s easy to show that K s LTD. Proof. Detals wll manly be gven for case (a). Let 0<p$ 1 = (&$ 1 +1) &1 <p 1 1 and &$ 1 >& 1 0. Then f (&log K(e &p$ 1 &1 (u 1 ), e &p 2 &1 (u 2 ) )+&$ 1 p$ 1 &1 (u 1 )+& 2 p 2 &1 (u 2 )) (&log K(e &p 1 &1 (u 1 ), e &p 2 &1 (u 2 ) ) +& 1 p 1 &1 (u 1 )+& 2 p 2 &1 (u 2 )) \u 1, u 2 K(e &p$ 1 &1 (u 1 ), y) e &&$ 1p$ 1 &1 (u 1 ) K(e p 1 &1 (u 1 ), y) e && 1p 1 &1 (u 1 ) \u 1, y

13 252 JOE AND HU or f (wth x=e &&1 (u 1 ) and &=& 1, &$=&$ 1 ) K(x 1(&$+1), y) x &$(&$+1) K(x 1(&+1), y) x &(&+1) \x, y #(0,1). Ths s the same as K(x 1(&+1), y) x &(&+1) decreasng n &0 for all x, y or K(x 1&*, y) x * =[K(x 1&*, y)x 1&* ] x decreasng n * # [0, 1]. Fnally, ths s the same as the LTD1 condton of K(z, y)z decreasng n z for all y. For (c), the concordance orderng s equvalent to [K(x 1&*, y 1&* ) [x 1&* y 1&* ]]xydecreasng n * # [0, 1] for all x, y. Ths follows from the LTD1 and LTD2 condtons because f 0*<*$1, then the condtons mply K(x 1&*$, y 1&*$ ) x *$ y *$ K(x 1&*, y 1&*$ ) x * y *$ K(x 1&*, y 1&* ) x * y *. K Analogous results for lower tal dependence are gven next. Theorem 3.9. The copula C (u, v)=( &1 (u)+ &1 (v)) has a lower tal dependence parameter equal to $ L =2 lm [$(2s)$(s)]. (3.3) s Proof. The proof s smlar to that of Theorem 3.6. K Theorem If p 1 =p 2 =p (and & 1 =& 2 =&) n (3.1) and the lower tal dependence parameter of K s ; #(0,1], then the lower tal dependence parameter of C n (3.1) s $ L =# lm $(&log ;+#s)$(s), (3.4) s where #=p(1+2&)1. If the lower tal dependence parameter of K s 0, then the lower tal dependence parameter of C s less than the rght-hand sde of (3.4) for all ;>0 (wth #=p(1+2&)). If the behavor at the lower tal s K(x, x)t;x \ as x 0 wth p>1, then the lower tal dependence parameter of C s gven by (3.4) wth #=p(\+2&)1. Proof. If ;>0, then K(x, x)t;x for x near 0. Hence, C(u, u)t(&log[;e &p&1 (u) ]+2&p &1 (u)) t(&log ;+p(1+2&) &1 (u)). Equaton (3.4) follows. The case K(x, x)t; 2 x \ s proved smlarly. K Examples that llustrate these tal dependence results are gven n Secton 5.

14 MULTIVARIATE DISTRIBUTIONS Condtons for Max-Infntely Dvsble Multvarate Dstrbutons In ths secton, we obtan condtons for max-nfnte dvsblty and apply them to the famles n Joe [8] and those n Secton 2 of ths artcle. A necessary and suffcent condton for a bvarate dstrbuton K to be max-d s that ts copula K(x, y) stp 2 n (x, y) (cf. [16, Theorem 3.4]). If the densty of K s TP 2, then K s TP 2 (the proof smlar to that n [2, p. 143]). For a multvarate dstrbuton K, a necessary condton s that all bvarate margns are TP 2. Hence the condton of max-d s a postve dependence condton. A general condton for max-d, that generalzes the above bvarate result to any dmenson m, s gven next. Theorem 4.1. Let m2. Suppose K(u 1,..., u m ) s m-varate dstrbuton and let }=log K. For a subset S of [1,..., m], let } S denote the partal dervatve of } wth respect to u, # S. A necessary and suffcent condton for K to be max-d s that } S 0 for all (nonempty) subsets S of [1,..., m]. Proof. We look at the dervatves of H=K # =e #k wth respect to u 1,..., u m, =1,..., m, and then permute ndces. All of the dervatves must be nonnegatve for all #>0 f K s max-d. The dervatves are: Hu 1 =#H} 1, 2 Hu 1 u 2 =# 2 H} 1 } 2 + #H} 12, 3 Hu 1 u 2 u 3 = # 3 H} 1 } 2 } 3 +# 2 H[} 1 } 23 +} 2 } 13 +} 3 } 12 ]+#H} 123, etc. For the nonnegatvty of S H> # S u for #>0 arbtrarly small, a necessary condton s that } S 0. From the form of the dervatves above, t s clear that } S 0 for all S s a suffcent condton. K For multvarate dstrbutons whch have specal forms, smpler condtons can be obtaned. For permutaton symmetrc Archmedean copulas, a condton, from Joe [8], nvolves LTs that correspond to (sum-)nfntely dvsble dstrbutons. For the condton, we need the defnton of L* n =[ : [0,)[0, ) (0)=0, ()=, (&1) &1 ( ) 0, =1,..., n], for n=1, 2,...,. L* s smlar to the condton of completely monotone; an nfntely dfferentable functon,: [0, ) [0, 1] s completely monotone f (&1), ( ) 0, =1, 2,.... Let F(x 1,..., x m )=( m =1 &1 (x )) be a permutaton symmetrc Archmedean copula and let /= &1. Then F # (x 1,..., x m )= exp[#_( m =1 /(x ))], where _=log. Note that _$=$, _"= ("& 2 ) 2, _$$$=[2($) 3 &3$"+ 2 $$$] 3, _ (4) =[&6($) 4 +

15 254 JOE AND HU 12($) 2 "&4 2 $$$$&3 2 (") (4) ] 4. Wth /$ =/$(x ), mthorder mxed dervatves h 1}}}m of F # for m=2, 3,..., are h 12 =e #_ /$ 1 /$ 2 [# 2 _$ 2 +#_"], h 123 =e #_ /$ 1 /$ 2 /$ 3 [# 3 _$ 3 +3# 2 _$_"+#_$$$], etc. From the pattern of the dervatves, F # s max-d for up to dmenson m f &_ # L* m, and F # s max-d for all m f &_ # L*. From Joe [8], the property of &log # L* holds for the famles LT1, LT2, LT3, LT4, of LTs n Secton 5, so that the condton s not too strong for applcatons. Asde. A consequence of the results here s that f s completely monotone, &log # L* and s completely monotone, then C(u 1,..., u m )=, \&log _ : m =1 &1 (e &,&1 (u ) ) &+ (4.1) s an Archmedean copula wth functon '(s)=,(&log (s)). Snce ths functon can be used for a permutaton symmetrc copula for any dmenson m, ' s completely monotone; that s, ' s a LT. Ths s a result on page 441 of Feller [3]. Hence for n one of the four famles LT1, LT2, LT3, LT4, '=,(&log ) s a LT whenever, s a LT. Next we go to max-d for partally symmetrc copulas n Secton 4 of Joe [8]. For the trvarate case, let F(u 1, u 2, u 3 )=( &1 b,(, &1 (u 1 )+, &1 (u 2 ))+ &1 (u 3 )) = def ( (`1(u 1 )+`2(u 2 ))+`3(u 3 )). Let H=F # and let _=log, so that H(u 1, u 2, u 3 )=exp[#_( (`1(u 1 )+`2(u 2 ))+`3(u 3 ))]. Suppose # L* 2 and &log # L* 3. Then the mxed dervatves up to thrd order are nonnegatve snce each term of the dervatves s nonnegatve. The dervatves are: Hu 3 =#H_$`$ 3, 2 Hu 1 u 3 =# 2 H_$ 2 $`$ 1`$ 3 + #H_" $`$ 1`$ 3, 3 Hu 1 u 2 u 3 =# 3 H_$ 3 $ 2`$ 1`$ 2`$ 3 +3# 2 H_$_" $ 2`$ 1`$ 2`$ 3 + # 2 H_$ 2 "`$ 1`$ 2`$ 3 +#H_$$$ $ 2`$ 1 `$ 2 `$ 3 +#H_" "`$ 1`$ 2`$ 3. For hgher dmensonal copulas n ths class, wrte H(u 1,..., u m )=exp[#_( 1 b 2 b }}} b k (}}})+}}})] and let `$ =`$ (u ). Suppose &log =&_#L* m and 's are n L* n for suffcently large n (greater than or equal to number of terms n the argument of ). Then the copula s max-d. As above, dfferentaton of a term wll lead to terms that are each nonnegatve. For example, dfferentaton of H n a term wth respect to u leads to a factor lke #H_$ $ 1 }}} $ k`$ 0, dfferentaton of [_ ( ) ] l n a term leads to a factor l[_ ( ) ] l&1 _ ( +1) $ 1 }}} $ k`$ whch has the same sgn as [_ ( ) ] l, and dfferentaton of ( ) l n a term leads to a factor ( +1) l $ l+1 }}} $ k`$ whch has the same sgn as ( ) l. More generally, consder F(u 1,..., u m )=(&log K(u 1,..., u m )), where K s max-d and &log # L* m. We use Theorem 4.1 to prove that F s also

16 MULTIVARIATE DISTRIBUTIONS 255 max-d. Let _=log and }=log K as above, so that F=(&log K)= exp[_(&})]. Let H=exp[#_(&})]. Then Hu 1 =&#H_$} 1 0, 2 Hu 1 u 2 =# 2 H_$ 2 } 1 } 2 +#H_"} 1 } 2 &#H_$} 12 0, etc., snce each term s nonnegatve. The pattern of dervatves of each term beng nonnegatve contnues. For example, dfferentaton of H n a term wth respect to u leads to a factor lke &#H_$} 0, dfferentaton of [_ ( ) ] l n a term leads to a factor &l[_ ( ) ] l&1 _ ( +1) } whch has the same sgn as [_ ( ) ] l, and dfferentaton of } S n a term leads to a nonnegatve factor. The LT famles ( };{) gven below n Secton 5 are such that &log # L*. Therefore the use of n one of these famles wll lead to max-d multvarate dstrbutons n (2.8). If the resultng (2.8) s substtuted as H nto (2.7) wth a dfferent n one of the famles, then another max-d multvarate dstrbuton obtans. 5. New Parametrc Famles of Copulas In ths secton, we gve examples of nterestng parametrc famles for (2.7) and (2.8), etc., whch result from takng a parametrc famly for the LTs and a parametrc famly for the bvarate copulas K. Before dong ths, we lst one-parameter famles of bvarate copulas and LTs and some of the tal dependence propertes assocated wth them. In (2.7), wrte H as K(H 1,..., H m ), where K s a m-varate copula and H 1,..., H m are the unvarate margns, so that (2.7) becomes F(x)= K # (H 1 (x 1 ),..., H m (x m )) dm(#)=(&log K(H 1 (x 1 ),..., H m (x m ))). 0 The m-varate copula for F n (5.1) s (5.1) C(u 1,..., u m )=(&log K(e &&1 (u 1 ),..., e &&1 (u m ) )), u #(0,1),=1,..., m. (5.2) Ths obtans by choosng H (u )=e &&1 (u ), =1,..., m. When m=2, wth varous choces of one-parameter famles for K and M (or ) n (5.1), one can get two-parameter famles. Smlarly for m2 n (2.8), wth a one-parameter famly for the K 's (that s, K (u, u )=K(u, u ; { ) for a one-parameter famly of copulas K( };{)) and a one-parameter famly for, one can get a parametrc famly wth m(m&1)2+m+1 parameters. Subcases wth fewer parameters obtan from takng parametrc famles n the examples at the end of Secton 2. In order that resultng famles have easly nterpretable parameters, we wll use famles of K's

17 256 JOE AND HU such that K( };{) s ncreasng n concordance as { ncreases. Then clearly C ncreases n concordance as { or { ncreases wth other parameters held fxed. From Joe [8] some useful choces of famles of bvarate copulas K(u, v; {) are: C1. exp[&[(&log u) { +(&log v { ] 1{ ], {1, C2. (u &{ +v &{ &1) &1{, {>0, C3. 1&((1&u) { +(1&v) { &[(1&u)(1&v)] { ) 1{, {1, C4. &{ &1 log (1&(1&e &{ ) &1 (1&e &{u )(1&e &{u )), {>0, C5. uv exp[[&log u) &{ +(&log v) &{ ] &1{ ], {>0, and some choces of LT famles (s; %) are: LT1. exp(&s 1% ), %1, LT2. (1+s) &1%, %0, LT3. 1&(1&e &s ) 1%, %1, LT4. &% &1 log[1&(1&e &% ) e &s ], %>0. The famles C1C4 are famles of Archmedean copulas, wth the respectve LTs gven n LT1LT4. C1 and C5 are famles of extreme value copulas and the other famles have TP 2 denstes (and, hence, TP 2 cdfs). (As hstorcal correcton to Joe [8], an earler reference for the use of C2 as a famly of copulas s Kmeldorf and Sampson [11]. Also a correcton to Joe [8] s that C3 s a LT mxture famly of copulas.) From drect calculatons or from use of the tal dependence results n Secton 3, tal dependence assocated wth C1C4 or LT1LT4 are as follows. Examples (Upper tal dependence for (2.11)): LT1. $(s)=&% &1 s 1%&1 exp[&s 1% ] and $(0)=&. The lmt n (3.2) s $ U =2&2 lm s 0 [$(2s)($(s)]=2&2}2 1%&1 =2&2 1%. LT2. $(s)=&% &1 (1+s) &1%&1 and $(0)=&% &1.So$ U =0. LT3. $(s)=&% &1 (1&e &s ) 1%&1 e &s and $(0)=&. The lmt n (3.2) s $ U =2&2 1%. LT4. $(s)=&% &1 (1&e &% ) e &s [1 &(1&e &% ) e &s ] and $(0)= &% &1 e % (1&e &% ). So $ U =0. Examples (Lower tal dependence for (2.11)): LT1. From (3.3), $ L = 2 lm s [$(2s)$(s)] = lm s 2 1% exp[&(2 1% &1) s 1% ]=0.

18 MULTIVARIATE DISTRIBUTIONS 257 LT2. $ L =lm s 2(1+s[1+s] &1 ) &1%&1 =2 &1%. LT3. $ L =lm s 2(1+e &s ) 1%&1 e &s =0. LT4. $ L =lm s e &s [1&(1&e &% ) e &s ][1&(1&e &% ) e &2s ]=0. Examples. (Lower tal dependence for (3.1) wth & 1 =& 2 =& (and p 1 =p 2 =p=(1+&) &1 )): LT1. The lmt (3.4) s lm s #[&s &1 log ; + #] 1%&1 exp[&[&log ;+#s] 1% ] exp[s 1% ]=$ L.If&>0so that #>1 then $ L =0, and f #=1 (and &=0, \=1) and ;>0, then $ L =1 for %>1 and $ L =; for %=1. LT2. The lmt n (3.4) becomes lm s #[#+(1&#&log ;)(1+ s) &1 ] &1%&1 =# &1% =$ L.If&=0and \=1, then #=1 and $ L =1. If \=2 as for the case of the ndependence copula, then #=2 and $ L =2 &1%, the same as the copula C2 wth parameter %. If1\<2, then 1#<2 and there s more lower tal dependence than the copula C2 wth parameter %. For example, let K be the copula C1 wth parameter {1, then K(x, x)=x \ wth \=2 1{ so that $ L =2 &1(%{) f &=0 and p=1 (and #=\). If K s the copula C5 wth parameter {>0, then K(x, x)=x \ wth \=2&2 &1{. If K s the copula C4 wth parameter &<{<, then K(x, x)t &{ &1 log[1&{ 2 x 2 (1&e &{ )]t{x 2 (1&e &{ ) and #=2. LT3. The lmt n (3.4) s $ L =lm s #[(1&;e &#s )(1&e &s )] 1%&1 ;e &(#&1) s. Ths s 0 f #>1 and t s ; f #=1. LT4. The lmt n (3.4) s $ L =lm s ;e &(#&1) s [1&(1&e &% ) e &s ] [1&(1&e &% ) ;e &#s ]. Ths s 0 f #>1 and t s ; f #= Bvarate Copulas. In ths subsecton, we lst a few nterestng cases, from the pont of vew of tal dependence, of (5.2) wth m=2, wth the use of the above famles C1C5 and LT1LT4. From the above calculatons, the use of LT1, LT2, LT3 lead to copulas wth tal dependence; the propertes of the copulas for LT3 are smlar to those for LT1. Some multvarate generalzatons are gven n Subsecton 5.2. Suppose K s parameterzed by the parameter { and s parameterzed by the parameter % (denoted as % ). If K s ncreasng n concordance as { ncreases, then clearly C ncreases n concordance as { ncreases wth % fxed. The concordance orderng for { fxed and % varyng s harder to check. If K has the form of an Archmedean copula, then from (4.1), then C also has the form of an Archmedean copula. That s, f K(x, y; {)=, { (, &1 { (x)+, &1 { ( y)) for a famly, {, then C(u, v; %, {)= % (&log, { [, &1 { (e & &1 % (u) )+, &1 { (e & &1 % (v) )]) =' %, { (' &1 %, { (u)+'&1 %, {(v)), (5.3)

19 258 JOE AND HU where ' %, { (s)= % (&log, { (s)). For { fxed and % 2 >% 1 wth ' =' %, {, =1, 2, the concordance orderng of C(};% 1,{) and C( };% 2,{) could be establshed by showng that =' &1 2 b ' 1 s superaddtve ( (x+y) (x)+ (y) for all x, y>0). Ths condton holds f (s)s s ncreasng n s>0 or f (s) s convex n s (see [5]). Now four examples are lsted together wth some of ther propertes; these show dfferent types of upper and lower tal dependence behavour and nclude examples wth both upper and lower tal dependence. The tal dependence propertes come from results n Secton 3 and the precedng examples n Secton 5. Example 5.1. In (5.2), let K be the famly C1 and let be the famly LT2. Then the resultng copula, s C(u, v; %, {)=[1+[(u &% &1) { +(v &% &1) { ] 1{ ] &1% ='(' &1 (u)+' &1 (v)), {1, %>0, (5.4) where '(s)=' %, { (s)=(1+s 1{ ) &1%. Ths s the copula n the bvarate Webull dstrbuton n Eq. (2.5) of Lu and Bhattacharyya [12]. Some propertes of the famly of copulas (5.4) are: a. The famly C2 s a subfamly when {=1, and the famly C1 s obtaned as % 0. Hence the lmt as % 0 and { 1 s the case of ndependence, C I (u, v)=uv. The lmt as % or { corresponds to the Fre chet upper bound, C U (u, v)=mn(u, v). b. The lower tal dependence parameter s 2 &1({%), whle the upper tal dependence parameter s 2&2 1{, ndependently of %. The extreme value lmts from the lower and upper tals (from lmts of [C*(1&n &1 e &x, 1&n &1 e &y ; %, {)] n and [C(1&n &1 e &x,1&n &1 e &y ; %, {)] n respectvely, where C*(u, v; })=u+v&1+c(1&u, 1&v; } )) are the famles C5 and C1 respectvely. c. Concordance ncreases as % ncreases for fxed {. To show that (5.4) s ncreasng n %, we show that (s)s s ncreasng, where (s)=' &1 (' % 2, { % 1, {(s))=[(1+s 1{ ) \ &1] {, % 1 <% 2, and \=% 2 % 1. It s easly verfed that d( (s)s)ds0 s equvalent to (\&1) z&1+(1+z) 1&\ 0 for all z>0 wth z=s 1{. From the above, we can get a one-parameter famly C(u, v; {) by settng %={&1 n (5.4): C(u, v; {)=[1+[(u 1&{ &1) { +(v 1&{ &1) { ] 1{ ] &1(1&{), {1. (5.5)

20 MULTIVARIATE DISTRIBUTIONS 259 Ths s a new one-parameter famly of copulas wth both upper and lower tal dependence. Moreover, copula (5.5) s absolutely contnuous and ncludes both the cases of ndependence and the Fre chet upper bound at the extremes. Example 5.2. In (5.2), let K be the famly C2 and let be the famly. Then the copula s C(u, v; %, {)=exp [&[{ &1 log (e {u~ % +e {v~ %&1)] 1% ] ='(' &1 (u)+' &1 (v)), %1, {>0, (5.6) where u~ =&log u, v~ =&log v, and '(s)=' %, { s)=exp[&[{ &1 log(1+s)] 1% ]. Some propertes of the famly of copulas (5.6) are: a. The famly C2 s a subfamly when %=1, and the famly C1 s obtaned as { 0. The lmt as % or { corresponds to the Fre chet upper bound. b. The lower tal dependence parameter s 2 &1{ when %=1 and 1 when %>1, whle the upper tal dependence parameter s 2&2 1%, ndependently of {. The extreme value lmt from the upper tal s famly C1. c. Concordance ncreases as % ncreases. (5.6) s ncreasng n % f and only f &D{ &1 log (D{)+[e {x x log x+e {y y log y](e {x +e {y &1)0 for all x, y>0 and {>0, where D=log (e {x +e {y &1). Ths condton holds from numercal checks but has not been confrmed analytcally. Wth a change of parametrzaton to (%, :) wth :={ 1%, the famly of copulas has been shown to be ncreasng n concordance wth both parameters % and :. Example 5.3. In (5.2), let K be the famly C5 and let be the famly LT2. Then the copula s C(u, v; %, {)=(u &% +v % &1&[(u &% &1) &{ +(v &% &1) &{ ] &1{ ) &1%, %0, {>0, (5.7) Some propertes of the famly of copulas (5.7) are: a. The famly C2 s obtaned when { 0, and the famly C5 s obtaned as % 0. The Fre chet upper bound obtans as % as {. b. The lower tal dependence parameter s (2&2 &1{ ) &1%, whle the upper tal dependence parameter s 2 &1{, ndependently of %. The extreme value lmt from the lower tal leads to the mn-stable bvarate exponental

21 260 JOE AND HU famly exp[&a(x, y)] wth A(x, y)=x+y&[x &% +y &% &(x %{ + y %{ ) &1{ ] &1% and ths s a two-parameter extenson of the famly C5. The extreme value lmt from the upper tal s famly C5. c. Concordance ncreases as % ncreases. (5.7) s ncreasng n % f and only f [x+y&1&((x&1) &{ +( y&1) &{ ) &1{ ] log [x+y&1&((x&1) &{ + ( y&1) &{ ) &1{ ]&x log x&y log y+((x&1) &{ +(y&1) &{ ) &1{&1 ((x&1) &{ x log x+(y&1) &{ y log y)0 for all x, y>1 and {>0. Ths condton holds from numercal checks but has not been confrmed analytcally. Example 5.4. In (5.2), let K be the famly C1 and let be the famly LT3. Then the copula s C(u, v; %, {)=1&(1&exp[&[(&log(1&u % )) { +(&log(1&v % )) { ] 1{ ]) 1% ='(' &1 (u)+' &1 (v)), %1, {1, (5.8) where u =1&u, v =1&v, and '(s)=' %, { (s)=1&[1&exp (&s 1{ )] 1%. Some propertes of the famly of copula (5.8) are: a. The famly C1 s obtaned when %=1, and the famly C3 s obtaned when {=1. The lmt as % + or { + corresponds to the Fre chet upper bound. b. The lower tal dependence parameter s 0, and the upper tal dependence parameter s 2&2 1(%{). The extreme value lmt from the upper tal s the famly C1. c. Concordance ncreases as % ncreases. To show that (5.8) s ncreasng n %, we show that (s) s convex n s>0, where (s)= ' &1 (' % 2, { % 1, {(s))=[_(s 1{ )] and _(s)=&log (1&[1&e &s ] \ ), \=% 2 % 1 >1. It s easly verfed that _$(s)>0, _"(s)>0 for all s>0, and hence _(t)t_$(t), where t=s 1{. Therefore, "(s)={ &1 s &2 [_(t)] {&2 [t_"(t)+ ({&1) _$(t)(t_$(t)&_(t))]0 for all s> Multvarate Copulas and Multvarate Extreme Value Dstrbutons. In ths subsecton, we lst three famles of parametrc multvarate copulas, two of whch are extensons from subsecton 5.1. Two of these famles are famles of extreme value copulas; n the thrd case, the lmtng famly of extreme value copulas s obtaned. (Extreme value copulas C satsfy the property of C(u t 1,..., ut m )=Ct (u 1,..., u m ), for all t>0.) The results are three parametrc famles of multvarate extreme value dstrbutons, wth closed form cdfs and flexble dependence structure. The famles of multvarate extreme value dstrbutons n Joe [9, 10] have flexble dependence structure but not closed form cdfs.

22 MULTIVARIATE DISTRIBUTIONS 261 Example 5.5. In (2.10), let K be the famly C1 wth parameter { and let be the famly wth parameter %. The result s the famly: m C(u 1,..., u m )=exp _ {& : ((p z % ){ +(p z % ){ ) 1{ + : & p z & 1%= %, 1<m =1 (5.9) where z = &log u, =1,..., m. Ths s a famly of extreme value copulas snce the exponent n (5.9) s homogeneous of order one as a functon of z 1,..., z m. The bvarate margns are: C (u, u )=exp[&[((p z % ){ +(p z % ){ ) 1{ +(& +m&2) p z % +(& +m&2) p z % ]1% ]. When p =p =1 or & =& =2&m, ths bvarate copula s C1. The correspondng upper tal dependence parameter s 2&[( p { +p { ) 1{ + (& +m&2) p +(& +m&2) p ] 1% ; t ncreases as { or % ncreases. In a specal case, we check for the range of dependence that s possble for the bvarate tal dependence parameters. For m=3, { 13 =1, { 12 ={ 23 ={, & 1 =& 3 =&1, & 2 =0, and p 1 =p 3 =1, p 2 =0.5, (5.9) and (2.13) become C(u 1, u 2, u 3 )=exp[&[(2 &{ z %{ 2 +z%{ 1 )1{ +(2 &{ z %{ 2 +z%{ 3 )1{ ] 1% ], (5.10) where z =&log u, =1, 2, 3. The bvarate margns are C 2 (u, u 2 ) =exp[&[(2 &{ z %{ 2 +z%{ ) 1% z% 2 ]1% ], =1, 3, and C 13 (u 1, u 3 )= exp[&(z % 1 +z% 3 )1% ]. The bvarate (upper) tal dependence parameters are $ 12 =$ 23 =2&[(2 &{ +1) 1{ +2 &1 ] 1% and $ 13 =2&2 1%. As {, $ 12 = $ 23 2&(1.5) 1%. A check for how close $ 13 s to the nonsharp lower bound max[0, $ 12 +$ 23 &1]=max[0, 3&2(1.5) 1% ] (from [10]) s gven n Table I. Table I shows that there s a lot of flexblty n how small $ 13 can get, gven $ 12 =$ 23. For the trvarate famles n Joe [8], two of the bvarate margns must have the same dependence parameter and the bvarate dependence of the thrd margn must be at least as large as the other two (hence n comparson wth the above, the bvarate tal dependence parameter for the (1, 3) margn s at least as large as that for the (1, 2) and (2, 3) margns). It can also be shown that (5.10) has a flexble range for the trple of bvarate Kendall taus. TABLE I Tal Dependence Parameters n Specal Trvarate Case % &2 1% &2(1.5) 1%

23 262 JOE AND HU Example 5.6. In (2.10), let K be the famly C5 wth parameter { and let be the famly LT2 wth parameter %. Let u^ = p ( u & % &1), =1,..., m. The result s the famly: C(u 1,..., u m )= _: m =1 u &% &(m&1)& : 1<m &{ (u^ &{ +u^ ) & &1{ &1%. (5.11) The specal case [ m =1 u&% &(m&1)] &1% arses as p 0, =1,..., m. The bvarate margns are C (u, u )=[u &% +u &% &1&(u^ & { & { +u^ ) &1{ ] &1%. When p =p =1, ths bvarate copula corresponds to that derved n Example 5.3. Ths copula has both lower and upper tal dependence. Usng the approxmatons u &% &1r%(1&u ) and u^ r p % (1&u ) as u 1, =1,..., m, the upper tal dependence parameters can be computed as ( p &{ +p &{ ) &1{. The lower tal dependence parameters are [2&(p &{ +p &{ ) &1{ ] &1%. The tal dependence parameters are ncreasng as { ncreases. The upper tal dependence parameters do not depend on %; the lower tal dependence parameters ncreases as % ncreases. The upper tal extreme value lmt (obtaned from the lmt of C n (1&n &1 e &x 1,..., 1&n &1 e &x m )) s exp[& e &x + < ( p &{ e { x + p &{ e { x ) &1{ ]. Ths s not very nterestng as t does not depend on %. The lower tal extreme value lmt s more nterestng. It s analogous to (5.9) and generalzes C5. Let S be a subset of [1,..., m] of sze 2 or more. Let C S denote the margn of C n (2.10) wth ndces n S. The functon h S (z, # S) s defned as the lmt of nc S (n &1 z, # S) as n, wth z >0, =1,..., m. It s straghtforward to verfy that h S (z, # S)= _: z &% & : # S <, #S, # S (p &{ z %{ +p &{ The lmtng multvarate extreme value copula has the form where z = &log u, C(u 1,..., u m )=exp[&a(z 1,..., z m )] &1% z %{ ) & &1{. A(z 1,..., z m )=z 1 +}}}+z m +: (&1) S +1 h S (z, #S), (5.12) wth the summaton over all subsets S of [1,..., m] of sze 2 or more, and S beng the cardnalty of S. S

24 MULTIVARIATE DISTRIBUTIONS 263 Some specal cases are gven next. For m=2, & 1 =& 2 =0 (and hence p 1 =p 2 =1), and {={ 12, (5.12) becomes A(z 1, z 2 )=z 1 +z 2 &[z &% 1 +z &% & 2 (z %{ 1 +z%{ 2 )&1{ ] &1% whch appears from the bvarate extreme value lmt n Example 5.3. For m=3, { 13 0, { 12 ={ 23 ={, & 1 =& 3 =&1, & 2 =0, (5.11) becomes A(z 1, z 2, z 3 )=z 1 +z 2 +z 3 &(z &% 1 +z &% 3 ) &1% &[z &% 1 +z &% 2 &(z %{ 1 +2{ z %{ 2 )&1{ ] &1% &[z &% 3 +z &% 2 &(z %{ 3 +2{ z %{ 2 )&1{ ] &1% +[z &% 1 +z &% 2 +z &% 3 &(z %{ 1 +2{ z %{ 2 )&1{ &(z %{ 3 +2{ z %{ 2 )&1{ ] &1%. The bvarate margns (by lettng one of the z 's go to zero n turn) are A 2 (z, z 2 )=z +z 2 &[z &% +z &% 2 &(z %{ +2 { z %{ 2 )&1{ ] &1%, =1, 3, and A 13 (z 1, z 3 )=z 1 +z 3 &(z &% 1 +z &% 3 ) &1%. The correspondng upper tal dependence parameters, 2&A (1, 1), are $ 2 =[2&(1+2 { ) &1{ ] &1%, =1, 3, and $ 13 =2 &1%.As{,$ 12 =$ 32 (1.5) &1%. The check, gven n Table II, shows that $ 13 s qute close to the nonsharp lower bound max[0, $ 12 +$ 23 &1]=max[0, 2(1.5) &1% &1], for part of the range of %. Example 5.7. In (2.10), let K be the famly C5 wth parameter { and let be the famly LT1 wth parameter %. Let z = &log u, =1,..., m. The result s the famly of extreme value copulas: C(u 1,..., u m )=exp {& _ : m =1 z % & : 1<m (p &{ Note that & 's appear only mplctly n the p 's. The bvarate margns are C (u, u )=exp[&[z % +z% &(p&{ z &%{ TABLE II z &%{ +p &{ Tal Dependence Parameters n Specal Trvarate Case z &%{ ) & 1%= &1{. (5.13) +p &{ z &%{ ) &1{ ] 1% ]. (5.14) % &1% max[0, 2(1.5) &1% &1]

25 264 JOE AND HU The upper tal dependence parameter for (5.14) s 2&[2&( p &{ + p &{ ) &1{ ] 1%. It s ncreasng as { or % ncreases. For m=3, { 13 0, { 12 ={ 23 ={, & 1 =& 3 =&1, & 2 =0, p 1 =p 3 =1, p 2 =0.5, (5.13) becomes C(u 1, u 2, u 3 )=exp[&[z % 1 +z% 2 +z% 3 &(z&%{ 1 +2 { z &%{ 2 ) &1{ &(z &%{ 3 +2 { z &%{ 2 ) &1{ ] 1% ]. (5.15) The bvarate margns of (5.15) are C 2 (u, u 2 )=exp[&[z % +z% & 2 (z &%{ +2 { z &%{ 2 ) &1{ ] 1% ], =1, 3, and C 13 (u 1, u 3 )=exp [&[z % 1 +z% 3 ]1% ]. The tal dependence parameters are $ 2 =2&[2&(1+2 { ) &1{ ] 1%, =1, 3, and $ 13 =2&2 1%.As{,$ 2 2&(1.5) 1%. The remanng analyss s the same as n Example Dscusson In ths artcle, many multvarate dstrbutons and copulas are derved based the mxture of powers approach. The potental usefulness of the multvarate models may be based on the closure property of weghted maxma mnma, or on dependence and other propertes of the copulas. The famles of copulas here have postve dependence only; extensons to allow for some negatve dependence wll come n a subsequent artcle. Future research wll consst of applcatons to multvarate ordnal data and comparsons wth the multvarate probt model and the model n Molensbergh and Lesaffre [17]. As well, the new multvarate extreme value famles wll be appled to data sets of extreme values. Other mxture or transform approaches could be used to derve famles of multvarate dstrbutons and copulas, for example, mxtures of convolutons or usng a property of (sum)-nfntely dvsble, nstead of maxnfntely dvsble. However, we have not been successful n gettng closed form parametrc famles of copulas usng other approaches. Acknowledgments Ths research has been supported by a NSERC Canada grant. The second author s grateful to the Department of Statstcs, Unversty of Brtsh Columba, for the hosptalty durng a vst n Thanks to the referees and assocate edtor for comments leadng to an mproved presentaton.