AN ASYMMETRIC GENERALIZED FGM COPULA AND ITS PROPERTIES

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1 Pa. J. Statst. 015 Vol. 31(1), AN ASYMMETRIC GENERALIZED FGM COPULA AND ITS PROPERTIES Berzadeh, H., Parham, G.A. and Zadaram, M.R. Deartment of Statstcs, Shahd Chamran Unversty, Ahvaz, Iran. Corresondng author Emal: ABSTRACT In ths aer, we ntroduce a new famly of Farle Gumbel Morgenstern (FGM) coulas. Ths famly s an asymmetrc generalzaton famly of Farle Gumbel Morgenstern (FGM) coulas and ncludes some of ts recent extensons. Some general formulas for well-nown assocaton measures of these coulas are obtaned and varous roertes of the roosed model are studed. KEYWORDS FGM Coula; Assocated Measures; Admssble Range; Concets of Deendence. 1. INTRODUCTION A coula s a functon that lns unvarate margnal dstrbutons nto a jont multvarate one, secally, nto a bvarate one. As Slar (1959) stated, these functons are bvarate dstrbuton functons by suort [0,1], whose margns are unform n [0,1]. He showed that f H s a bvarate dstrbuton functon wth margnal dstrbutons F( X ) and ( ) H( X, Y) C F( X ), G( Y);, where s ntroduced as deendence arameter. Ths strategy rovdes an aroach for modelng a bvarate dstrbuton functon. However, f unvarate margnal dstrbuton functons and an arorate coula are determned, t s easy to comute the jont dstrbuton, functon (More detals can be found n Joe, 1997 and Nelson, 006). One of the most oular arametrc famles of coulas, whch were studed by Farle (1960), Gumbel (1960) and Morgenstern (1956), s the Farle-Gumbel-Morgenstern (FGM) coula. Because of ther smle analytcal form, FGM coula s wdely used n modelng and studyng the effcency of nonarametrc rocedures. However, ths coula has been shown to be somewhat lmted. In detal for coula deendence arameter [ 1,1], the GY, then there must exst a coula C such that Searman s Rho and Kendall's Tau are S / 3 [ 1/ 3,1/ 3] and /9 [ /9,/9], resectvely. Snce the correlaton doman of FGM coula s lmted, more general coulas have been ntroduced wth the am of mrovng the correlaton range. Huang and Kotz (1999) develoed Polynomal-tye sngle-arameter extensons of FGM coula. They showed that S can be ncreased u to aroxmately whle the lower bound remans Baramov and Kotz (00) further extended the famly gven by Huang and Kotz (1999) to the assocated Searman's S [ 0.48, ]. La and Xe (000) 015 Pastan Journal of Statstcs 95

2 96 An Asymmetrc Generalzed FGM Coula and ts Proertes gave condtons for ostve quadrant deendence and studed a class of bvarate unform dstrbuton wth ostve quadrant deendence roerty by generalzng the unform reresentaton of a well-nown FGM coula. By a smle transformaton, they also obtaned famles of bvarate dstrbutons wth re-secfed margnals. An alternatve aroach to generalze the FGM famly of the symmetrc sem-arametrc s defned by Fscher and Klen (004). It was extensvely studed n Amblard and Grard (00). Cuadras and Daz (01) ntroduced an extended FGM famly n two dmensons and studed how to aroxmate any dstrbuton to ths famly. Berzadeh, et al. (01) roosed a new class of generalzed FGM coula and showed that ther generalzaton can mrove the correlaton doman of FGM coula. In ths regard, ths aer roose another generalzaton of FGM coula, whch ncludes some of extended coulas ntroduced n recent years and can mrove the correlaton range.e. the roosed famly covers some of the ntroduced famly n the lterature and ts correlaton range s more effcent. From another ersectve, ths resented famly, s an asymmetrc extenson of the generalzed FGM coula dscussed n Berzadeh, et al. (01). The man contrbuton of ths aer ncludes the followngs: frst, an extenson of FGM coula and some fne roertes are resented. Second, asymmetry roertes and the general formulas for assocaton measures of ths famly are studed. The man feature of ths famly s caablty for modelng a wder range of deendence. Ths ermts us to extend the range of otental alcatons of the famly n varous branches of scences. The rest of the resent aer s as follows: the new asymmetrc extenson and ther basc characterstcs are descrbed n Secton. The assocated Searman s Rho and Kendall's tau are studed n Secton 3. Secton 4 s dedcated to the deendence structure roertes of ths new famly of coulas. Fnally, the results of ths research are stated n Secton 5.. A NEW CLASS OF FGM COPULA The coula s mostly defned as a functon boundary condtons A1. C( u,0) C(0, u) 0 and C( u,1) C(1, u) u, u [0,1], A. 4 u1, u, v1, v [0,1], such that 1 u C :[0,1] [0,1] that satsfes the u and v1 v, C u, v C u, v C u, v C u, v 0. Eventually, for twce dfferentable, -ncreasng roerty (A) can be relaced by the condton c( u, v) C( u, v) 0, (.1) uv where c( u, v ) s the so-called coula densty. A coula C s symmetrc f C( u, v) C( v, u), for every ( uv, ) [0,1], otherwse C s asymmetrc.

3 Berzadeh, Parham and Zadaram 97 Many dfferent coulas can be found n the lterature, see for nstance Nelson (006). The FGM coula s one of these coulas whch are wdely used n alcaton. The FGM coula s defned by FGM C ( u, v) uv 1 (1 u)(1 v), [ 1,1]. These coulas are the only ones whose functonal form s quadratc n both u and v. Regardng lmtaton of correlaton range n FGM coula, along wth other generalzatons, Berzadeh, et al. (01) roosed a new class of symmetrc generalzed FGM famly whose deendence s as follows: C( u, v) uv 1 1u 1 v, 0 and 0,1,,.... In the followng defnton, an asymmetrc extenson of the above coula famly s ntroduced n order to extend the FGM coula. Defnton.1 Suose that the contnuous functons A1, A:[0,1] [0,1] are dfferentable on (0,1). A, A An asymmetrc functon C, :[0,1] [0,1] [0,1] s defned as A1, A, (, ) ( ) ( ) C u v uv A u A v, [0, ), ( uv, ) [0,1], (.) where the arameter [ 1,1] s called the assocated arameter. We assume that A1 and A do not change ther sgn on [0,1] n order to obtan unque determned deendence structure. Note that the coula s lmted to the range of [0,1] and therefore, 1 A1( u) A should be bounded on [0,1]. The followng theorem gves suffcent and necessary condtons on A 1 and A to, ensure that C s a bvarate coula. Theorem.1 Let A1, A:[0,1] [0,1] be contnuously dfferentable functons on (0,1). The, functon C s a bvarate coula f and only f A 1 and A satsfy the followng condtons: B1. A (1) 0, for 1,, B. xa( x) 1 and A ( x) xa( x) 1, for every x [0,1], and for 1,, where A( x) A ( x) x. Proof: The roof nvolves two stes:,, Frst, t s clear that C ( x,1) C (1, x) x, x [0,1] (B1).

4 98 An Asymmetrc Generalzed FGM Coula and ts Proertes Second, snce A 1 and A are contnuously dfferentable functons and [1 A ( u) A ] s bounded on [0,1] and -ncreasng functon, by (.1) the condton c, ( u, v) 0 hold, f and only f xa ( x) 1 x [0,1], and 1,, where c, ( u, v) s as follows:,, c ( u, v) C ( u, v) / u v and A ( x) xa( x) 1, for every A1 u A v [1 ( ) ( )] 1 A ( v ) A ( u ) ua ( u ) 1 A ( u ) A ( v ) va ( v ) ua( u) va. (.3) Note that s a arameter that shows deendence structure of the famly and C, 0 or 0, leads to the ndeendence of u and v. By theorem.1, the concrete amount of the arameter sace s deendent on the roertes of the functon of A 1 and A that has been nvestgated va (.3) for every amount of u and v n [0,1]. If C, A ( x) A ( x), x [0,1], then the famly s symmetrc. Remar.1 C, The famly researchers n recent years, whch are as follows: ncludes some nown famly of FGM coulas ntroduced by ) A ( x) 1 x, x [0,1], for 1,, and 1, the famly C leads to the, symmetrc FGM coula dscussed by Farle (1960), Gumbel (1960) and Morgenstern (1956). ) f A ( x) 1, x [0,1], for 1,, 0, and 1, the famly x C, leads to the symmetrc extended FGM coula ntroduced by Huang and Kotz (1999) q q ) f A ( x) x (1 x), x [0,1], for 1,, q 1 and 1, the famly C, leads to the symmetrc extended FGM coula ntroduced by La and Xe (000). v) f A ( x) (1 x ), x [0,1], for 1,, 0, 1 and 1, the famly, C leads to the symmetrc extended FGM coula ntroduced by Baramov- Kotz (00) v) f A ( x) A( x), x [0,1], for 1,, and 1, the famly C leads to the, symmetrc coula ntroduced by Rodrguez-Lallena and Ubeda-Flores (004).

5 Berzadeh, Parham and Zadaram 99 C, Moreover, va the famly, some new generalzatons can be defned by ntroducng addtonal arameters for the famles ()-(v), and we can generate some, coulas of the famly C of tye (.) through functons A 1 and A. Prooston.1, Two lmted roertes of the famly C are as follows: A1, A. lm C ( u, v) lm uv 1 A ( u) A uv ( u, v), where ( uv, ) s the, 0 0 ndeendent coula.. Let, where, then A1, A lm C, ( u, v) lm uv 1 A1( u) A uv ex A ( u) A C ( u, v). (.4) The new famly of coula n (.4) can be a new asymmetrc generalzaton of Cuadras coula (009). 3. MEASURES OF DEPENDENCE Measures of deendence are common nstruments to summarze a comlcated deendence structure n the bvarate case. For a hstorcal revew of measures of deendence, see Joe (1997) and Nelsen (006). In ths secton, we comute the measures C, of deendence for the famly. Snce we cannot gve formulas for the roertes of deendence n terms of elementary functons, t s relaced by ts exanson seres on, : A ( u) A 1., Based on, the famly C n (.) for every [0, ) may also be wrtten by olynomal exanson wth resect to A 1 and A as (, ) ( ) ( ). (3.1) g, 1 C u v uv ua u va v Note that, n (3.1), we have g when s nteger, otherwse, g equals to. 3.1 Searman's rho Let X and Y be contnuous random varables whose coula s C. Then the oulaton verson of Searman s rho for X and Y s gven by 11 S 1 C( u, v) dudv 3. 00

6 100 An Asymmetrc Generalzed FGM Coula and ts Proertes Note that, dstrbutons. Prooston 3.1 S concdes wth correlaton coeffcent between the unform margnal C, Let ( XY, ) be a ar of random varables wth the famly. The Searman s rho S, for the famly C s gven by where g S 1 B1( ) B( ), (3.) 1 1 B ( ) x A ( x) dx, for 1,. 0 Proof: By usng (3.1), the Searman s can be exanded as A1, A, S 1 C ( u, v) dudv 3 1 uv[1 A1( u) A] dudv 3 11 g 1 uv A1 ( u) A dudv S 11 g 1 uv u A1 ( u) v A dudv g 1 uvdudv ua1 ( u) va g 11 1 ua1 ( u) va dudv 1 00 g 1 ua1 ( u) du va dv g 1 B1( ) B( ). 1 dudv 3 3. Kendall's tau In terms of coula, Kendall's tau s defned as (see Nelsen, 006) 11 4 c( u, v) C( u, v) dudv 1. 00

7 Berzadeh, Parham and Zadaram 101 Prooston 3. Let ( XY, ) be a ar of random varables wth the famly C and the famly, densty c. The Kendall's tau can be exanded as, g 4 B1 ( ) B ( ) 1 B 1( 1) B ( 1) 0 ( 1) B1( ) B( ) 1. (3.3) Proof: For the sae textual unty the rooston's roof deferred to the Aendx. Remar 3.1 For A ( x) 1 x, 1,, x [0,1], and 1 n (.), we have for the classcal FGM coula as dscussed n Farle (1960), Gumbel (1960) and Morgenstern (1956) that S 3 and 9. Hence, we have 0.33 S 0.33 and (as 1 1). As the remar (3.1) shows, the doman of correlaton of FGM coula s lmted and therefore t s not allowed for modelng of strong deendence. One of the advantages of, the famly C s caablty to mrove the doman of correlaton by ntroducng addtonal arameter n FGM coula and some generalzed FGM famles resented n recent years. Examle 3.1, In the famly C, let A ( x) 1 x, for 1,, and x [0,1]. Then the famly A1, A C leads to a new symmetrc generalzed FGM coula wth, Snce 1 1 B( ) xa ( x) dx, we have by usng (3.) that ( 1)( ) 0 g 1 S 1 1 ( 1)( ), max{1, }. where the uer bound of above Searman's S can be ncreased u to aroxmately as, whle the lower bound remans unchanged. Therefore, the admssble range of Searman's [ ,0.3805]. S n the new symmetrc generalzed FGM famly s

8 10 An Asymmetrc Generalzed FGM Coula and ts Proertes Examle 3., In the famly C, let A ( x) 1, for 1,, x [0,1], and 0. Then, x the famly C leads to a new symmetrc generalzed Hung-Kotz famly wth 1 1 max 1, ( ). Snce 0 / / 1 B( ) xa ( x) dx, for 1,, we have / / g S 1 1. (3.4) Tang 1.85 and 500 n (3.4), we have,max Smlarly, tang 0.1 and 500, we obtan,mn Therefore, the admssble range of S Searman's n the generalzed Hung-Kotz famly s [ 0.50,0.43]. So, the generalzed S Hung-Kotz coula mroves the amltude Remar 3. For 1, (3.4) reduces to S 3 1 S 3 max 1, 3, ( ) S of Hung-Kotz famly. S whose range s whch s the same as the one dscussed by Huang and Kotz (1999). 4. SOME CONCEPTS OF DEPENDENCE In ths secton, we focus on some concets of deendence for our resented generalzed FGM coula. Several concets of ostve (negatve) deendence and deendence stochastc orders have been ntroduced n the lterature (Nelsen, 006). In the followng defnton, we recall some of these concets and then, study deendence, structure of the famly C gven n (.). Defnton 4.1 The random varables X and Y are ) Postvely Quadrant Deendent PQD f P X x, Y y P X x P Y y, for all ( xy, ) or equvalently C( u, v) uv, ( uv, ) [0,1]. (4.1)

9 Berzadeh, Parham and Zadaram 103 ) Left Tal Decreasng or equvalently, LTD f PY y X x s non-ncreasng n x for all y, C( u, v) u (4.) u s non-ncreasng for all v. ) Left Corner Set Decreasng s nonncreasng n x 1 and y 1 for all x functon of order Prooston 4.1 LCSD f P X x, Y y X x1, Y y1 4 TP, for all,, 1 1 and y, or equvalently, C s a Totally Postve u1 u, v1 v wth u1 u and v1 v f C u, v C u, v C u, v C u, v 0. (4.3) C, Let ( XY, ) be a ar of random varables wth the famly. The random varables X and Y are a. PQD f and only f 0. b. LTD f and only f A 1 ( u) 0. c. LCSD f and only f both A1 and A be decreasng or ncreasng. Proof. a. Usng (4.1), the roof s straghtforward. 1, C A A, (, ) ( ) ( ) u v uv A u A v b. Based on (4.), u u resect to u f and only f, 1 v1 A1 ( u) A A1 ( u) va 1 A1 ( u) A 0, u on necessary and suffcent condton that A 1 ( u) 0. c. Suose u1 u and v1 v. By (4.3), we have 1 1 u v u v A u A v A u A v By smle comutatons, we have s non-ncreasng wth 1 A u A v 1 A u A v 0. A u A u A v A v 0 f and only f. (4.4) The relaton (4.4) holds, f and only f both A1 and A be decreasng or ncreasng.

10 104 An Asymmetrc Generalzed FGM Coula and ts Proertes As an examle, for A ( x) 1 x, 1, and x [0,1], we have that the classcal FGM coula are PQD, f 0 1, LTD f 1 0 and LCSD for all 1 1. Recently, Baramov and Bayramoglu (011) showed that f n the Baer s model, one uses the deendent random varables ( XY, ) wth ostve quadrant deendent ( PQD ) jont dstrbuton functon F( x, y ), nstead of ndeendent random varables, then the correlaton ncreases, and n contrast, for negatve quadrant deendent ( NQD ) F( x, y ), t decreases. La and Xe (000) stated condtons for havng PQD and studed a class of bvarate unform dstrbutons havng PQD roerty by generalzng the unform reresentaton of a well-nown FGM coula. By a smle transformaton, they also obtaned famles of bvarate coulas wth re-secfed margnals. We showed that the assocaton arameter n ths class can mrove the PQD roerty for a wder range. Thus, we C, beleve that the famly s more alcable n a greater varety of stuatons. 5. CONCLUSION In ths aer, we ntroduced a generalzaton of an asymmetrc famly of Farle Gumbel Morgenstern (FGM) coula; ncludng some of ther new extensons. Also, the general formulas for measures of assocaton of these coulas were studed. Moreover, we studed necessary and suffcent condtons for some deendence concets n ths famly. Snce the doman of correlaton of FGM coula s lmted, t s necessary to extend ths doman. By usng the generalzed FGM famly, the correlaton coeffcent of FGM coula mroved. The man feature of ths famly s caablty for modelng a wder range of deendence. Ths ermts us to extend the range of otental alcatons of the famly n varous branches of scences. ACKNOWLEDGEMENTS The authors are grateful to the referees for ther constructve comments and helful suggestons. REFERENCES 1. Amblard, C. and Grard, S. (00). Symmetry and deendence roertes wthn a sem arametrc famly of bvarate coulas. J. Nonaramet. Stat., 14, Baramov, I. and Bayramoglu, K. (011). From Huang-Kotz dstrbuton to Baer s dstrbuton. J. Mult. Anal., n ress. 3. Baramov, I. and Kotz, S. (00). Deendence structure and symmetry of Huang-Kotz FGM dstrbutons and ther extensons. Metra, 56, Berzadeh, H., Parham, G.A. and Zadaram, M.R. (01). The New Generalzaton of Farle Gumbel Morgenstern Coulas. Al. Math. Sc., 6(71), Cuadras, C.M. (009). Constructng coula functons wth weghted geometrc means. J. Statst. Plan. Inf., 139, Cuadras, C.M. and Daz, W. (01). Another generalzaton of the bvarate FGM dstrbuton wth two-dmensonal extensons. Act. et Comment. Un. Tart. De. Math., 16, 3-1.

11 Berzadeh, Parham and Zadaram Farle, D.G.J. (1960). The erformance of some correlaton coeffcents for a general bvarate dstrbuton. Bometra, 47, Gumbel, E.J. (1960). Bvarate exonental dstrbutons, J. Amer. Statst. Assoc., 55, Huang, J.S. and Kotz, S. (1999). Modfcatons of the Farle-Gumbel-Morgenstern dstrbutons. A tough hll to clmb. Metra, 49, Fscher, M. and Klen, I. (007). Constructng Symmetrc Generalzed FGM Coulas by means of certan Unvarate Dstrbutons. Metra, 65, Joe, H. (1997). Multvarate models and deendence concets. Chaman & Hall. London. 1. La, C.D. and Xe, M. (000). A new famly of ostve quadrant deendent bvarate dstrbutons. Statst. Prob. Letters, 46, Morgenstern, D. (1956). Enfache besele zwedmensonaler vertelungen, Mttelungsblatt für Mathematsche. Statst, 8, Nelsen, R.B. (006). An ntroducton to coulas. Srnger Seres n Statstcs. Srnger. New Yor. 15. Slar, A. (1959). Fonctons de rèartton àn dmensons et leurs marges. Publ. Inst. Statst. Unv. Pars, 8, The roof of Prooston.1: By usng (.) and (.3), we have,, APPENDIX 1 A A1 ( u) ua1 ( u) 1 A1 ( u) A va ( ) ( ) c u v C u v uv A u A v (, ) (, ) 1 ( ) ( ) u A u va v The relaton above may be wrtten through olynomal sectons wth resect to A1 and A as: g A1, A A1, A c, ( u, v) C, ( u, v) ua1 ( u) va 0 1 A [ A1 ( u) ua1 ( u)] 1 A1 ( u)[ A va] u A( u) va g ua ( u) va ua ( u) va ua ( u) v A A 1 ua ( u) va u A ( u) v A A u A ( u) A ( u) va u A ( u) A ( u) va 1 1 u A ( u) A ( u) v A A u A ( u) A ( u) v A A

12 106 An Asymmetrc Generalzed FGM Coula and ts Proertes So, the Kendall's tau s 11 A1, A A1, A,, 00 4 c ( u, v) C ( u, v) dudv 1 g 4 ua1 ( u) du va dv ua1 ( u) du va dv ua1 ( u) du v A A dv 0 0 ua1 ( u) du va dv ua1 ( u) du v A A dv u A1 ( u) A1 ( u) du va dv u A1 ( u) A1 ( u) du va dv 0 0 u) A ( u) du v A A dv 1 u A1 ( u A1 ( u) A1 ( u) du v A A dv g 4 B1 ( ) B ( ) B1 ( 1) B ( 1) B1 ( 1) B ( 1) 0 1 B1( ) B( ) B1 ( ) B ( ) B1 ( 1) B ( 1) 1 1 B ( ) B ( ) 4 B ( ) B ( ) 1 4 B1( 1) B( 1) 1 1 g 4 B1 ( ) B ( ) 1 B 1( 1) B ( 1) 0 ( 1) B1( ) B( ) 1.