Extreme Points of the N-Dimensional Elliptope: Application to Universal Copulas

Transcription

1 Theoetcal Mathematcs & Applcatos vol. 5 o ISSN: pt) ole) Scepess Ltd 05 Eteme Pots of the N-Dmesoal Ellptope: Applcato to Uvesal Copulas Wee Hülma Abstact Based o the eplct paametezato of the set of postve sem-defte coelato matces we deve smple sphecal coodates fo the eteme pots. A applcato to the costucto of uvesal copulas s cluded. Mathematcs Subect Classfcato: 5A4; 5B99; 6H0; 65F30 Kewods: Coelato mat; Postve sem-defte popet; Caocal paametezato; Eteme pots; Uvesal copula; Lea ccula copula Itoducto A postve sem-defte mat whose dagoal etes ae equal to oe s called a coelato mat. The compact cove set of coelato Swss Mathematcal Socet. E-mal: whulma@bluew.ch Atcle Ifo: Receved: Decembe Revsed: Jaua Publshed ole : Jul 0 05.

2 5 Eteme pots ellptope matces ) s called ellptope fo ellpsod ad poltope) a temolog coed b Lauet ad Polak []. The stud of the ellptope stuctue has geeated ma teestg ad patl dffcult poblems. Fo eample the eteme pots of the ellptope have ot bee eplctl detemed though the ak oe ad two eteme pots ae kow ad thee est chaactezato esults o them b Ycat [0] L ad Tam [4] ad Pathasaath [7]. Cleal the ellptope s uquel detemed b the set of ) uppe dagoal elemets ) < deoted b E. The autho [6] Theoem 3. costucts a eplct paametezato of the coelato mat whch maps bectvel a ) ) [ ] to ) E. These so-called Catesa coodates deped ve smpl o as well as o poducts of poducts whch addtoall volve the fuctoal quattes k ad sums ) ) )..) The otato.) wll be used thoughout wthout futhe meto. The equed pelma esults ae summazed Secto. Combg the elatvel smple chaactezato of the eteme pots of coelato matces b Ycat [0] wth ou Catesa coodates a full eplct fuctoal paametezato of the eteme pots s deved Secto 3. It should be most useful poblems depedg o the kowledge of eteme pots of coelato matces. We llustate wth a applcato to the costucto of some uvesal -copulas a poblem whch emas usolved geeal e.g. Devoe ad Letac [3] autho [7] Letac [3]). Pelma esults Thee ae two questos elated to eteme pots of the ellptope amel the estece of eteme pots ad the costucto. The estece questo depeds

3 W. Hülma 53 upo the ak of a coelato mat ad has bee settled b ma authos e.g. Ycat [0] Poposto 6 Goe et al. [5] Theoem L ad Tam [4] Coolla Pathasaath [7] Coolla.). Fo fte dmesoal coelato matces the questo s cosdeed Kukas ad Pellopaa [9]. Some eale wok o eteme pots cove sets of smmetc ad Hemta matces cludes Chstese ad Vetestom [] ad Loew [5]. Theoem. Estece of eteme pots). Thee est eteme pots of ak m f ad ol f the dmeso of the ellptope satsfes the equalt m m + ). Up to ow the effectve costucto of all eteme pots has bee a ope poblem. Dffeet chaactezato codtos fo them have bee obtaed b Ycat [0] Theoem L ad Tam [4] Theoem b) ad Pathasaath [7] Theoem.. The chaactezato b Ycat [0] s most appopate to the peset eeds. Theoem. Ycat s chaactezato of eteme pots). A coelato mat ) E < of ak m such that m m + ) s a eteme pot f ad ol f thee est a s [ ]... s... m whch solve the quadatc sstem of equatos m m asa s < as.....) I Secto 3 we combe Ycat s chaactezato wth the followg so-called caocal paametezato to obta a paametezato of the eteme pots. Theoem.3 Catesa coodates of -dmesoal ellptope). Thee ests a bectve mappg betwee the cube ) [ ] ad E whch maps the Catesa coodates ) to ) such that

4 54 Eteme pots ellptope....) ) k k k + k... k k... 4 Poof Cosult Hülma [6] Theoem 3.. k + k k + k.4) 3 Eteme pots of coelato matces Ou ma esult s the followg caocal paametezato of the eteme pots the cove compact set of all coelato matces. Theoem 3. Eteme pots of -dmesoal ellptope). Let ) E be a eteme pot of ak ) m ma{ m m + )} caocal fom.)-.4). The thee est sphecal coodates α [0 π] k... m... k such that k Case : ak ) k ε ε ε { } k... k k 3.) Case : ak ) ) k cos α α ) k k 3.3) k Case 3: ak ) m 3 m m + ) ) + sα sα )

5 W. Hülma 55 k + k + cos α k + k sα sα m α k s km k k + m + k ks ) sα sα m k k m... sα sα ks... k k... m + k s ks s sα sα s ks 3.6) 3.7) Poof Case s show b ma authos e.g. Ycat [0] Remak p. 60 Lauet ad Polak [] Theoem.5 Pathasaath [7] Remak.4 p. 78). Fo ak ) m the dea s to epess Ycat s equatos.) sphecal coodates ad make them cosstet wth the caocal fom.)-.4). Cosde fst Case. Usg pola coodates fo the -sphee secod equatos.) settg a cosϕ a sϕ.... Iseted to the fst equatos of.) oe sees that R oe solves the a a + a a cos ϕ ) <. 3.8) ϕ O the othe had accodace wth.) thee ests α [0 π ]... such that. Settg 3.8) oe has cos ϕ ϕ ).... Ths matches the caocal paametezato f oe sets ϕ 0 ϕ α.... The emag equatos 3.8) ead cos α ) <. 3.9) α < Oe must show that 3.9) matches eactl.3)-.4). Fom.3) oe gets + sα sα Ths matches cos α α ) fom 3.9) f ad ol f oe has. Usg ths ad Defto.) oe sees that

6 56 Eteme pots ellptope ) ) 0 k k k k Isetg to.4) oe obtas 3.3) fo k k ad Case s show. To show Case 3 we dstgush betwee m 3 ad m 4. Fst let m 3. Solve the secod equatos.) usg sphecal coodates fo the -sphee 3 R such that a cosϕ a sϕ cosϑ a3 sϕ sϑ. Isetg to the fst equatos of.) oe obtas 3 a s a I patcula oe has s cosϕ cosϕ + sϕ sϕ cos ϑ ϑ ) <. 3.0) cosϕ cosϕ + sϕ sϕ cos ϑ ϑ ).... O the othe had vtue of.) oe ca set fo α [0 π ].... Ths matches the pecedg epesso f oe sets. ϕ 0 ϕ α.... The emag equatos 3.0) ead Fom.3) oe has + sα sα cos ϑ ϑ ) <. 3.) < + + sα sα wth fo some α [0 π ].... Ths matches 3.) f oe sets ϑ ϑ α.... The the emag equatos 3.) ead 0 + sα sα cos α α ) < <. 3.) Oe must show that ths matches eactl.4). Fo... 3 oe obtas sα sα { + sα sα } whch matches the coespodg et 3.) f ad ol f oe has whch shows 3.7) fo k of couse 3.6) s a vod statemet hee). Futhe

7 W. Hülma 57 ths mples that k ) ) 0 k k. k Isetg to.4) oe obtas takg to accout the vashg compoets) the emag fomulas 3.7) fo k It emas to geealze the pecedg steps fo a fed ak m 4. Usg sphecal coodates fo the m ) -sphee m R oe solves the secod equatos.) settg a a cosϕ sϕ cosϕ sϕ sϕ sϕ sϕ...sϕm cosϕm am m a Iseted to the fst equatos.) oe obtas a 3 cosϕ... sϕ sϕ...sϕ 3 m sϕ m. cosϕ cosϕ + cos ϕ m ϕ m m3 + m + + ) sϕ sϕ s s sϕ sϕ s <. s 3.3) I vtue of.) set fo α [0 π ].... Ths matches the coespodg epesso fo 3.3) f oe sets ϕ 0 ϕ α.... The fomula 3.4) s show. Fom.3) oe gets + + sα sα wth fo some α [0 π ].... Ths matches the epesso fo 3.3) f oe sets ϕ 0 ϕ α.... The fomula 3.5) follows. Poceedg the same mae oe obtas fom.4) fo k... m... k the epessos k + k + k k + k k k k + + k k s ks sα sα + k k + sα sα s ks k whee the fact that k cos α k + fo some α k + [0 π ]... k + has bee used. Ths choce matches the coespodg equatos fo k

8 58 Eteme pots ellptope 3.3) f oe sets ϕ ϕ α k... m... k. Ths kk + 0 k + k + shows the fomula 3.6). It emas to show that the emag equatos 3.3) fo wth < < m + match eactl the coespodg epessos.4). Fst oe has m+ + { m m+ m+ m + m+ m3 + m+ m m+ + m+ + sα m sα sα sα m+ + s m+ m m+ m+ m } sα sα m+ s s m+ m+ s Ths matches the coespodg epesso 3.3) f ad ol f oe has m+ whch shows 3.7) fo k m. Futhe ths mples that k m+ m+ ) km+ ) 0 k m k. Isetg to.4) fo k m k oe obtas smlal to the above the emag fomulas 3.7). Theoem 3. s show.. Remaks 3. Case has also bee solved b Ycat [0] Coolla p. 6. I Cases ad 3 oe must esue that the coelato matces ae of ak m. Ths s fulflled povded the vectos T a a a... a )... m defed the poof of Theoem 3. ae leal depedet. It s well-kow that ths holds f ad ol f the detemat of the Gam mat G < a a > ) s o-zeo. Ths s alwas satsfed up to some degeeate cases. Fo eample f m t suffces that α 0 π fo some de {... }. 4 Applcato to the costucto of uvesal copulas A -dmesoal copula s called -uvesal f eve -dmesoal vald coelato mat ca be ealzed as a ak coelato mat.e. thee ests a

9 W. Hülma 59 -vaate ufom dstbuto wth ths ak coelato stuctue. I the lteatue -uvesal copulas ae bette kow ude the amg compehesve o clusve copulas e.g. Nelse [6]). Although the estece of 3-uvesal copulas has bee settled b seveal authos e.g. Joe [8] Eecse 4.7 pp Kuowcka ad Cooke [0] Secto p.0 Devoe ad Letac [3]) the effectve costucto of 3-uvesal copulas s moe dffcult. The autho [7] costucts a aaltcal 3-uvesal copula that s based o the bvaate lea ccula copula Pelma ad Welle [8]. The latte copula seems to have bee depedetl obtaed b Kuowcka et al. [] whch called t ellptcal copula. As poted out b Letac [3] the lea ccula copula s a specal case of pobablt dstbutos studed b Gaspe [4]. Ths -uvesal copula ca be used to costuct -uvesal copulas fo ak two etemal coelato matces. Reduced to ts essetal steps the pesetato b Letac [3] has a elemeta appeal. Let B be the ut dsk ad [ ] R C the ceteed squae. Cosde the lea ccula copula dest wth ufom [-] mags defed b U V u v) B p U V ) u v) π u v 4.) 0 u v) C B. A cucal step towads the ma esult below s the followg smple popet. Lemma 4. -uvesal ak two eteme lea ccula copula) Gve s the eteme coelato mat of ak two of the fom [ 0 ] ) cos α α )) α π. The thee est a adom vecto X X... X ) wth ufom [-] mags X... ad ak two coelato mat ). Poof Cosde the adom vecto X X... X ) defed b X cos α ) U + s α ) V...

10 60 Eteme pots ellptope whee the adom pa U V ) has the lea ccula copula dest 4.). Cleal the vaables X... ae ufom [-] adom vaables. Moeove though applcato of the Jacoba tasfomato method oe sees that the pobablt dest of p X X ) X X ) s gve b ) ) E π ) ) ) 4.) 0 u v) C E whee the suppot { ) } + < E s the e of a ellpse ad cos α α ) cocdes wth the coelato coeffcet of the pa X X ) e.g. Kuowcka et al. [] Pelma ad Welle [8] Hülma [7] Secto 3). Theoem 4. -uvesal ak two copula) Gve s a ak two coelato mat ). The thee est a adom vecto X X... X ) wth ufom [-] mags X... ad ak two coelato mat ). Poof Ths follows though applcato of the theoem of Caathéodo [] ad Stetz [9]. A vald coelato mat of ak two) s a fte cove combato of eteme coelato matces of ak two). The esult follows fom the fact that t holds fo the eteme coelato matces of ak two b Lemma 4.. Oe otes that Theoem 4. settles the estece questo fo -uvesal copulas 345. Ideed coelato matces of dmesos 34 5 have mamum ak two b Theoem..

11 W. Hülma 6 Refeeces [] C. Caathéodo Übe de Vaabltätsbeech de Fouesche Kostate vo postve hamosche Fuktoe Red. del Ccolo Matem. Palemo 3 9) [] J.P.R. Chstese ad J. Vestestom A ote o eteme postve defte matces Math. A ) [3] L. Devoe ad G. Letac Copulas thee dmesos wth pescbed coelatos URL: [math.st] 00. [4] G. Gaspe Baach algeba fo Jacob sees ad postvt of a keel A. of Math ) [5] R. Goe S. Pece ad W. Watks Etemal coelato matces Lea Algeba Appl ) [6] W. Hülma Catesa ad pola coodates fo the -dmesoal ellptope Theoetcal Mathematcs & Applcatos 43) 04) -7. [7] W. Hülma A closed-fom uvesal tvaate pa-copula Joual of Statstcal Dstbutos ad Applcatos 7) 04) 5p. [8] H. Joe Multvaate Models ad Depedece Cocepts Moogaphs o Statstcs ad Appled Pobablt 73 Chapma & Hall Lodo 997. [9] J. Kukas ad J.-P. Pellopaa A ote o fte eteme coelato matces. Lea Algeba Appl. 48-) 008) [0] D. Kuowcka ad R. Cooke Ucetat Aalss wth Hgh Dmesoal Depedece Modellg J. Wle Chcheste 006. [] D. Kuowcka R. Cooke ad J. Msewcz Ellptcal copulae. I: G.I. Schuelle ad P.D. Spaos Eds.) Poceedgs Mote Calo Smulato pp. 09-4) Lsse Balkema 000. [] M. Lauet ad S. Polak O a postve semdefte elaato of the cut poltope Lea Algeba Appl. 3/4 995)

12 6 Eteme pots ellptope [3] G. Letac The cove set of the coelato matces ad the Jacob polomals cofeece pesetato Othogoal Polomals ad Hpegoups Toulouse Jue URL: [4] C.-K. L ad B.-S. Tam A ote o eteme coelato matces SIAM J. of Mat Aal. Appl. 53) 994) [5] R. Loew Eteme pots of a cove subset of the coe of postve sem-defte Matces Math. A ) 7-3. [6] R.B. Nelse A Itoducto to Copulas Lectue Notes Statstcs 39 d edto) Spge-Velag New Yok 006. [7] K.R. Pathasaath O etemal coelatos J. Statst. Pla. Ifeece 03 00) [8] M.D. Pelma ad J.A. Welle Squag the ccle ad cubg the sphee: ccula ad sphecal copulas Smmet 3 0) [9] E. Stetz Bedgt kovegete Rehe ud kovee Ssteme J. Ree Agew. Math ) [0] B. Ycat Eteme pots cove sets of smmetc matces Poc. Ame. Math. Soc. 954) 985)