n° 2002-20 Constrained Nonparametric Copulas - CREST

INSTITUT NATIONAL DE LA STATISTIQUE ET DES ETUDES ECONOMIQUES Série des Documents de Travail du CREST (Centre de Recherche en Economie et Statistique)

n° 2002-20 Constrained Nonparametric Copulas P. GAGLIARDINI1 C. GOURIEROUX2

July 2002

Les documents de travail ne reflètent pas la position de l'INSEE et n'engagent que leurs auteurs. Working papers do not reflect the position of INSEE but only the views of the authors.

1 Università della Svizzera Italiana. 2 CREST and University of Toronto, Canada.

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ru C C orj  +x m y,  orj  +x m y, / ;x> y 5 ^3> 4` 1 Cy Cy Dv dq looxvwudwlrq/ wkh ixqfwlrqv= D+y> , @ h{s +y, / D+y> , @

4  / dqg D+y> , @ +4  y, / +4 . y,

lqgxfh wkuhh idplolhv ri glvwulexwlrqv vxfk wkdw whpsrudo ghshqghqfh lv lqfuhdv0 lqj zlwk uhvshfw wr sdudphwhu / lq erwk wkh VL dqg KL vhqvh1

616

Phdvxuhv ri Ghshqghqfh1

Wkh suhylrxv glvfxvvlrq vkrzv wkdw/ iru wkh sursruwlrqdo kd}dug prgho/ wkh dssursuldwh ixqfwlrqdo ghshqghqfh phdvxuh lv qrw D lwvhoi/ exw suhihudeo|= D +y, @ 

g orj D+y,/ y 5 ^3> 4` 1 gy

Wkh surshuwlhv deryh fdq eh vxppdul}hg dv iroorzv= l1 D +y, @ 3/ ;y 5 ^3> 4` +, [w dqg [w4 duh lqghshqghqw/ w 5 Q> ll1 D +y,  3/ ;y 5 ^3> 4` +, [w lv VL dqg KL lq [w4 / w 5 Q>   lll1 D +y,  D +y,/ ;y 5 ^3> 4` +, +[w > [w4 ,  [w > [w4 / zkhuh  lv dq| ri wkh rughulqjv +VL , ru +KL , 1

617

Wdlo ghshqghqfh

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3

Wkh dqdo|wlf h{suhvvlrq ri fD>k lv qrw dydlodeoh lq jhqhudo1 Krzhyhu/ vrph ghshqghqfh surshuwlhv fdq eh ghgxfhg iurp d wkhruhp e| Idqj/ Kx dqg Mrh +4 j+[wk4 ,`  fruu ^j+[w ,> j+[wk ,`/ k 5 Q/ iru dq| prqrwrqrxv wudqvirupdwlrq j vxfk wkdw wkhvh fruuhodwlrqv h{lvw1 Sursrvlwlrq < Ohw +[w > w 5 Q, eh d vwdwlrqdu| Pdunry surfhvv zlwk sursru0 wlrqdo kd}dug dqg ghshqghqfh sdudphwhu D1 Li D lv ghfuhdvlqj/ wkhq [w lv vwrfkdvwlfdoo| lqfuhdvlqj lq [wk / k 5 Q/ dqg fruu ^j+[w ,> j+[wk4 ,`  fruu ^j+[w ,> j+[wk ,` / k 5 Q/ iru dq| prqrwrqrxv wudqvirupdwlrq j vxfk wkdw wkh fruuhodwlrqv h{lvw1 Wkxv/ zkhq D lv ghfuhdvlqj/ ghshqghqfh lv srvlwlyh dw dq| odj/ dqg ghfuhdvhv zlwk wkh krul}rq1

7

Hujrglflw| Surshuwlhv1

Wkh dlp ri wklv vhfwlrq lv wr vwxg| wkh hujrglflw| surshuwlhv ri vwdwlrqdu| Pdunry surfhvvhv zlwk sursruwlrqdo kd}dug1 47

714

Jhrphwulf hujrglflw|1

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F

4 4 k l  f / iru 3 ? |  U4 / U4 | D 4  3 h{s +D+y,|, gy 4 4 l  f|/ iru |  U5 1  k U4 | D 4  3 h{s +D+y,|, gy

Surri1 Vhh Dsshqgl{ 71

48

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.4^>

LL

; A 3 = olpy$4 +4D+yy,, @ .4>

LLL

; A 3 = olpy$4 +4D+yy,, @ 31

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lv wkh frqglwlrqdo h{shfwdwlrq ri wkh wudqviruphg surfhvv ~| frqglwlrqdo kd}dug1 49 Wkh v|pphwulf fdvh  ' f lv dqdorjrxv1

49

l/

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+ D f/

' Mf EL| / | M Q/ zlwk frqvwdqw

srvlwlyh dqg qlwh4: 1 Ixqfwlrqv lq fodvv LL +uhvs1 LLL, grplqdwh +uhvs1 duh grplqdwhg e|, dq| ixqfwlrq lq fodvv L/ zkhq y $ 41 Sursrvlwlrq 44 Zkhq ixqfwlrq D lv hlwkhu lq fodvv L/ ru lq fodvv LL vxfk wkdw F / iru y forvh wr 4/ wkhq wkh vhfrqg uhvwulfwlrq iru vrph F A 3= D+y,  orj+4 y, lq frqglwlrq ll1 ri Sursrvlwlrq 43 lv vdwlvhg1 Surri1 Vhh Dsshqgl{ 81

715

Pl{lqj surshuwlhv

E| glvfxvvlqj pl{lqj surshuwlhv ri d vwrfkdvwlf surfhvv zh duh frqfhuqhg e| wkh ghfd| udwh ri wkh ghshqghqfh ehwzhhq wkh 0hogv xs wr wlph v/  +[w > w  v,/ dqg iurp wlph v . k rqzdug/  +[w > w  v . k,/ dv wkh krul}rq k jrhv wr lqqlw| +vhh h1j1 Ervt ^4w

k,

mS +F,  S +F m [3 ,m / k 5 Q/

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wkdw

dsshqgl{ 8 lw lv vkrzq wkdw ixqfwlrqv  lq fodvv L lpso| dxwruhjuhvvlyh ixqfwlrqv ) vxfk <  dv + < n"1

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8

H{dpsohv1

Lq wklv vhfwlrq zh glvfxvv ydulrxv h{dpsohv ri vwdwlrqdu| Pdunry surfhvvhv zlwk sursruwlrqdo kd}dug1 Wkh dvvrfldwhg g|qdplf prghov fdq eh sdudphwulf ru qrq0 sdudphwulf1 Lw lv lpsruwdqw wr qrwh wkdw l, vx!flhqw hujrglflw| frqglwlrqv duh hdvlo| zulwwhq/ ll, wkh lqyduldqw glvwulexwlrq +wkdw lv wkh xqlirup glvwulexwlrq, lv nqrzq1 Wklv lv dq lpsruwdqw dgydqwdjh ri wkhvh prghov frpsduhg wr wkh g|qdplf gxudwlrq prghov suhylrxvo| lqwurgxfhg lq wkh olwhudwxuh +vxfk dv DFG prghov, iru zklfk qhlwkhu wkh hujrglflw| frqglwlrqv/ qru wkh vwdwlrqdu| glvwul0 exwlrq duh nqrzq1

814

Frqvwdqw phdvxuh ri ghshqghqfh1

Zkhq wkh phdvxuh ri ghshqghqfh D lv frqvwdqw/ zh jhw= D +y, @ 

g orj D+y, @ / ;y 5 ^3> 4` @, D+y, @ h{s +y . f, / y 5 ^3> 4` / gy

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  4 orj z/ z 5 @ h > 4 / 

4 / z 5 1 Wkh lqyhuvh ri wkh edvholqh fxpxodwhg dqg dgplwv wkh ghqvlw| z kd}dug K3 lv rewdlqhg e| frpsxwlqj wkh Odsodfh wudqvirup ri 4  D4 = ] 4 4 h{s +}z, gz K34 +}, @ 4   h{s+, z ] 4 } h{s +|, g|1 @ 4  } h{s+, |

815

Dqdo|wlf h{dpsohv1

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FD +x> y, @ y  +4  x,y 4x / x> y 5 ^3> 4` / zlwk ghqvlw|= fD +x> y, @ 

x 4 +orj y, y 4x / x> y 5 ^3> 4` 1 5 +4  x,

Wkh dvvrfldwhg sursruwlrqdo kd}dug surfhvv lv jhrphwulfdoo| hujrglf1 Lqghhg= D+y, @  orj y @  orj ^4  +4  y,`  4  y/ iru y  4/ +vhh Sursrvlwlrq 44,/   D K34 +|, @  orj



| 4.|

   orj |/ dv | $ 3/

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4 +4  x, 4
4` 1

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z4@ /  +4 . 4@,

D+y, @ S 4 +4@> 4  y,   +4 . 4@, +4  y, / y  41

Lw iroorzv iurp Sursrvlwlrq ; wkdw wkh surfhvv ihdwxuhv srvlwlyh wdlo ghshqghqfh1 lll, Srzhu glvwulexwlrqv1 Zkhq=

4  D4 +z, @ z  / z 5 ^3> 4` / 4

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+49,

Qrwh wkdw wkh Fr{ prgho +Fr{ ^44` / zh jhw= K34 +}, @ 4 

4  h{s +}, / }  31 }

Wkh ixqfwlrqdo phdvxuh ri ghshqghqfh lv jlyhq e|= D +y, @  +y> , @

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dqg 4  D4 fruuhvsrqgv wr d srvlwlyh 0vwdeoh glvwulexwlrq zlwk sdudphwhu  @ f @p 1 Lq sduwlfxodu/ wkh odujhu lv sdudphwhu  +wkdw lv wkh odujhu wkh pdvv ri wkh glvwulexwlrq 4  D4 lq d qhljkerxukrrg ri 3,/ wkh odujhu lv wkh gxudwlrq ghshqghqfh lq wkh pdujlqdo glvwulexwlrq zlwk uhvshfw wr wkdw lq wkh frqglwlrqdo glvwulexwlrq1

816

Hqgrjhqrxv vzlwfklqj uhjlphv1

Ohw xv frqvlghu d vwhszlvh ixqfwlrqdo ghshqghqfh sdudphwhu= D+y, @

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dm L+xm >xm.4 ` +y,/ y 5 ^3> 4` /

54

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M [ m @3

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M [ m @3

h{s ^dm K3 +x,` +xm.4  xm , / ;x 5 ^3> 4` 1

+4;,

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M [ m @3 M [

h{s +dm }, +xm .4  xm , m h{s +dm }, / }  3/

+4 ===> M1 Htxdwlrq +4 ===> M/ zlwk suredelolwlhv m / m @ 3> ===> M1 l, Xqlirup vhulhv1 Dvvxph M @ Q  4 ? .4/ dqg dm @ Q  m/ m @

4 / m @ 3> 4> ===> Q  41 Q

Wkxv wkh ixqfwlrq D lv uhjxoduo| ghfuhdvlqj dqg= K34 +}, @ 4 

4 4  h{s +Q}, / }  31 Q h{s +},  4

ll, Srzhu Vhulhv1 Zkhq= 4  K34 +}, @ 4  ^4  h{s +},`  / }  3/ 4

55

zlwk   4/ wkh fruuhvsrqglqj edvholqh fxpxodwhg kd}dug lv=   K3 +x, @  orj 4  x / x 5 ^3> 4` 1 E| xvlqj wkh elqrpldo vhulhv h{sdqvlrq/ zh jhw +vhh Mrh ^4 ===1 dm @ m . 4/ m @ m .4  +m . 4,$ n@4

Wklv ghqhv dq lqfuhdvlqj vwhs ixqfwlrq +4:,/ zlwk wkuhvkrogv dw= xm .4 @

m [

o / m @ 3> 4> ===1

o@3

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4  h+4x, / x 5 ^3> 4` / 4  h

uhvshfwlyho|1 Wkh fruuhvsrqglqj glvfuhwh glvwulexwlrq lv irxqg e| h{sdqglqj wkh orjdulwkplf vhulhv lq +53, wr jhw +vhh Mrh ^4 ===1  +m . 4,

Djdlq/ d ghfuhdvlqj vwhs ixqfwlrq/ zlwk wkh vdph edvholqh fxpxodwhg kd}dug/ lv rewdlqhg e| frqvlghulqj y :$ D+4  y,1 56

817

Sursruwlrqdo kd}dug lq uhyhuvhg wlph1

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x> y 5 ^3> 4`/ iru vrph ixqfwlrqv D dqg D 1 Lq Dsshqgl{ 9 lw lv vkrzq wkdw wkh ixqfwlrqdo ghshqghqfh sdudphwhu ri d vwdwlrqdu| Pdunry surfhvv zlwk sursruwlrqdo kd}dug lq erwk wlph gluhfwlrqv lv fkdudfwhul}hg e|= D+y, @

4 +y . , / y 5 ^3> 4` / 4 +,

+54,

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Vwdwlvwlfdo Lqihuhqfh

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Lq wklv sdshu zh kdyh lqwurgxfhg gxudwlrq wlph vhulhv prghov zlwk sursruwlrqdo kd}dug1 Wkhvh prghov doorz wr vhsdudwh wkh pdujlqdo fkdudfwhulvwlfv iurp wkh vhuldo ghshqghqfh surshuwlhv1 Wkh odwwhu duh ghvfulehg e| d frsxod zlwk sur0 sruwlrqdo kd}dug/ fkdudfwhul}hg e| d ixqfwlrqdo sdudphwhu D1 Wklv kdv wzr lpsruwdqw frqvhtxhqfhv iurp d prghoolqj srlqw ri ylhz1 Rq wkh rqh kdqg/ wkh pdujlqdo glvwulexwlrq ri wkh surfhvv fdq eh fkrvhq iuhho|/ dqg zh fdq wkhq ir0 fxv rq vhuldo ghshqghqfh e| frqvlghulqj ixqfwlrq D1 Rq wkh rwkhu kdqg/ vlqfh sdudphwhu D lv ixqfwlrqdo/ wklv fodvv ri prghov lv h{leoh hqrxjk iru doorzlqj ydulrxv qrqolqhdu dqg qrqjdxvvldq ghshqghqfh ihdwxuhv/ vxfk dv ghshqghqfh lq wkh h{wuhphv/ vhuldo shuvlvwhqfh/ qrquhyhuvlelolw|/ dv frquphg lq vlpxodwhg h{dpsohv1 Zh kdyh uhodwhg wkh sdwwhuq dqg vwuhqjwk ri vhuldo ghshqghqfh wr wkh vkdsh ri ixqfwlrqdo sdudphwhu D e| xvlqj zhoo0nqrzq frqfhswv iurp frsxodv* wkhru|1 Pruh suhflvho| ydulrxv fkdudfwhulvwlfv ri ixqfwlrqdo sdudphwhu D jlyh ulvh wr glhuhqw irupv ri ghshqghqfh/ lq xhqfh ghshqghqfh lq wkh wdlov/ dqg lpso| hujrglflw| frqglwlrqv1 Ilqdoo| zh kdyh glvfxvvhg wkh hvwlpdwlrq ri wkh ghshqghqfh sdudphwhu D/ erwk lq sdudphwulf dqg qrqsdudphwulf iudphzrunv1 D qrqsdudphwulf hvwlpdwru ri D fdq eh rewdlqhg e| plqlpl}lqj d fkl0vtxduh glvwdqfh ehwzhhq wkh qrqsdud0 phwulf frqvwudlqhg frsxod dqg dq xqfrqvwudlqhg nhuqho hvwlpdwru ri wkh frsxod ghqvlw|1 Wklv plqlpxp fkl0vtxduh hvwlpdwru lv frqvlvwhqw dqg dv|pswrwlfdoo| qrupdo1 Lq dgglwlrq lw uhdfkhv wkh qrqsdudphwulf h!flhqf| erxqg frpsxwhg xqghu wkh dvvxpswlrq wkdw wkh xqlirup yduldeohv Xw duh revhuyhg1

65

Dsshqgl{ 4 Ghshqghqfh Rughulqj 3

Iru x> y> y 5 ^3> 4` zh kdyh= V+x m y, @ h{s +D+y,K3 +x,, / l k 3 3 Vy>y3 +x, @ V V 4 +x m y, m y @ xD+y ,@D+y, / dqg= 3 3 Vy>y3 +x,   @ xD+y ,@D+y,D +y ,@D +y, 1  Vy>y3 +x, 3

Wkxv/ iru dq| y ? y 5 ^3> 4`= Vy>y3 +x,  3 +x, Vy>y



4/ ;x 5 ^3> 4` +,

+,

D +y , D+y ,   D+y, D +y,

+,

D+y, D+y ,   1 3  D +y , D +y,

3

Vy>y3 +x,  3 +x, ghfuhdvlqj lq x 5 ^3> 4` Vy>y

3

3

Dsshqgl{ 5 Frh!flhqw ri xsshu wdlo ghshqghqfh Zlwkrxw orvv ri jhqhudolw| zh fdq vhw F @ 41 Lw zloo eh suryhg lq Dsshqgl{ 8 ^htxdwlrq +d16,` wkdw=   ] 4  +4 . 4@, / dv | $ .41 h{s ^|D+y,` gy * D 4 | 3 Xvlqj D +y, * +4  y, / y $ 4/ lw iroorzv= ] 4 3

h{s ^|D+y,` gy *

 +4 . 4@, / dv | $ .41 | 4@

Wkxv= K34 +}> D, * 4 

 +4 . 4@, / } $ .4/ } 4@ 66

dqg K3 +x> D, * Lw iroorzv= D

4 @ olp x$4 4  x

 +4 . 4@, / x $ 41 +4  x,

] 4

h{s ^D +y, K3 +x> D,` gy % & ] 4  4   +4 . 4@, @ olp h{s  +4  y, gy x$4 4  x x +4  x, ] +4.4@, 4 @ h{s +z, z4@4 gz  +4@, 3   @ S 4@>  +4 . 4@, 1 x

Dsshqgl{ 6 Qrqolqhdu Dxwruhjuhvvlrqv Lq wklv Dsshqgl{ zh uhsruw vrph suredelolvwlf surshuwlhv ri qrqolqhdu dx0 wruhjuhvvlrqv zlwk dgglwlyh qrlvh= \w @ *+\w4 , . w / zkhuh wkh lqqrydwlrq  w lv d zklwh qrlvh/ lqghshqghqw ri \w4 / zlwk vwulfwo| srvlwlyh ghqvlw| j rq U/ dqg H ^w ` @ 31 Wkh frqglwlrqdo ghqvlw| ri \w jlyhq \w4 @ | lv jlyhq e|= i+{ m |, @ j +{  *+|,, / {> | 5 U/ dqg lv vwulfwo| srvlwlyh1 Wkxv \w / w 5 Q/ lv 0luuhgxfleoh/ 0Kduulv uhfxuuhqw +vhh Ihljlq dqg Wzhhglh ^4 4`1 E| wdnlqj wkh orjdulwkp ri erwk vlghv/ dqg ghulylqj zlwk uhvshfw wr x dqg y zh jhw= CD +y, CK3 +x, CD+x, CK3 +y, @ 1 Cx Cy Cy Cx Li Xw lv qrw wkh lqghshqghqw surfhvv/ zh kdyh= CK3 +y, CD +y, CK3 +x, CD+x, @ @ @ / ;x> y1 Cx Cx Cy Cy

+d18,

Wkxv wkhvh udwlrv duh frqvwdqw/ htxdo wr  +vd|,1 Lw iroorzv/ e| xvlqj K3 +3, @ K3 +3, @ 3/ dqg wkh qrupdol}dwlrqv D+3, @ D +3, @ 458 = K3 +y, @  ^D +y,  4` / y 5 ^3> 4` / K3 +x, @  ^D+x,  4` / x 5 ^3> 4` 1 Qrwh lq sduwlfxodu wkdw D dqg D duh prqrwrqrxv1 E| uhsodflqj lq htxdwlrq +d17,/ zh jhw= D+x,d +y, h{s ^D +x, D +y,` h{s ^D +x,` @ D +y,d+x, h{s ^D +y, D+x,` h{s ^D +y,` / zkhuh d+x, @ gD+x,@gx dqg d +y, @ gD +y,@gy1 Wkxv= d +y, d+x, h{s ^D +x,` @  h{s ^D +y,` / ;x> y1 D+x, D +y, Lq sduwlfxodu/ ixqfwlrq D lv vxfk wkdw= d+x, h{s ^D +x,` @ / zkhuh  lv d frqvwdqw1 D+x, Ohw xv ghqrwh |+x, @ D +x,/ x 5 ^3> 4`1 Wkhq ixqfwlrq | vdwlvhv wkh vhsdudeoh glhuhqwldo htxdwlrq= h{s +|, g| @ 1 | gx YM  E

3 qrupdol}dwlrqv duh dgplvvleoh/ vlqfh ' k YYE lpsolhv wkdw Y*Y lv lqwh0 Y W judeoh/ dqg wkxv Ef n" / dqg vlploduo| iru  Ef n"1 58 Wkhvh

75

Ohw eh d sulplwlyh ri wkh ixqfwlrq | :$ h{s+|,@| rq U. 1 lv frqwlqxrxv/ vwulfwo| lqfuhdvlqj vxfk wkdw +.4, ? .4/ dqg wkh vroxwlrq lv= |+x, @ 4 +x . , / x 5 ^3> 4` /

zkhuh  lv vxfk wkdw=   +.4, / li   3/

+d19,

 .   +.4, / li  A 31

+d1:,

dqg Wkhuhiruh ixqfwlrq D lv vxfk wkdw= D+x, @

4 +x . , / x 5 ^3> 4` / 4 +,

dqg  @ 4 +,1 Vlqfh D vdwlvhv wkh vdph glhuhqwldo htxdwlrq dv D/ zh kdyh e| v|pphwu|= D @ D1

Zh qrz xvh uhvwulfwlrq +;, ri xqlirup pdujlqv1 Wkh ixqfwlrq K3 dqg lwv lqyhuvh duh jlyhq e|=   4 +x . ,  4 K3 +x, @  ^D +x,  4` @  4 +, @ 4 +x . ,  4 +, / x 5 ^3> 4` /

dqg= K34 +}, @

 4   4 +, . }   / }  31 

Wkxv wkh uhvwulfwlrq lv=

  ] 4  4 +y . , 4   4 +, . }   @ 4  h{s } gy/ }  31  4 +, 3

Diwhu wkh fkdqjh ri yduldeoh= @

+d1;,

4 +y . , / 4 +,

wkh lqwhjudo lq wkh UKV ehfrphv=   4 +y . , h{s } gy 4 +, 3

] 4

@ @ @

] 4 +4 .,    g + , 4 h{s  } . 4 +,  4  ] 4 +4 ., +}. 4 +,, + , h{s +, 4 g  }. 4 +,    4      4 + . ,   4  4 } . +,  } . +, 1  4 +, 76

Wkxv uhvwulfwlrq +d1;, ehfrphv=   4 + . ,  .  @ 4 + . , . } / ;}  31 4 +, Wklv htxdwlrq fdqqrw eh vdwlvhg zlwk ydoxhv ri  dqg  vxfk wkdw 4 + . , ? .4 dqg 4 +, ? .4/ exw lv vdwlvhg li hlwkhu 4 + . , @ .4 ru 4 +, @ .4 krogv1 Wkh fdvh 4 +, @ .4 lv qrw dgplvvleoh1 Zkhq 4 + . , @ .4/ frqglwlrq +d19, fdqqrw eh vdwlvhg/ zkhuhdv +d1:, lv wulyldoo| vdwlvhg1 Wkxv/ dq| sdlu ri frqvwdqwv  dqg  vxfk wkdw=   3/  .  @ +.4, / vdwlvhv wkh uhvwulfwlrq1

Dsshqgl{ : Frpsxwdwlrq ri wkh glhuhqwldo ri f +x> y> D, zlwk uhvshfw wr D Wkh dlp ri wklv dsshqgl{ lv wr ghulyh glhuhqw h{suhvvlrqv ri wkh glhuhqwldo ri wkh frsxod zlwk uhvshfw wr wkh ixqfwlrqdo sdudphwhu1 Lq d uvw vwhs zh ghulyh wkh glhuhqwldo zlwk uhvshfw wr D/ e| wdnlqj lqwr dffrxqw wkdw K3 lv d ixqfwlrqdo ri D/ gxh wr wkh uhodwlrqvkls lpsolhg e| wkh frqglwlrq ri xqlirup pdujlqdo glvwulexwlrq1 Lq d vhfrqg vwhs zh surylgh lqwhusuhwdwlrqv lq whupv ri edfnzdug h{shfwdwlrqv1 Ilqdoo| wkh uhvxowv duh sduwlfxodul}hg wr wkh sdudphwulf iudphzrun1 l, Wkh jhqhudo h{suhvvlrq1 Ohw xv ghulyh wkh uvw rughu h{sdqvlrq ri wkh frsxod orj ghqvlw|= orj f +x> y> D, @ orj D +y, . orj k3 +x> D,  D +y, K3 +x> D, / zlwk uhvshfw wr ixqfwlrqdo sdudphwhu D1 Zh jhw= orj f +x> y> D . D, @ orj ^D +y, . D +y,` . orj k3 +x> D . D,  ^D +y, . D +y,` K3 +x> D . D, D +y, . kG orj k3 +x> D, > Dl * orj f +x> y> D, . D +y, K3 +x> D, D +y,  D +y, kGK3 +x> D, > Dl 4  D +y, K3 +x> D, D+y, @ orj f +x> y> D, . D +y, . kG orj k3 +x> D, > Dl  D +y, kGK3 +x> D, > Dl / +d1 D, Zh kdyh= K 4 +}> D . D, 3

@ 4 * 4

] 4 3

] 4 3

h{s ^D +y, }  D +y, }` gy ^4  D +y, }` h{s ^D +y, }` gy

@ K34 +}> D, . khqfh=

 GK34 +}> D, > D @

] 4 3

] 4 3

}D +y, h{s ^D +y, }` gy/

} h{s ^D +y, }` D+y,gy1

H{suhvvlrq ri GK3 +x> D, E| dsso|lqj wkh lpsolflw ixqfwlrq wkhruhp zh jhw=

 kGK3 +x> D, > Dl @ k3 +x> D, GK34 +K3 +x> D, > D, > D ] 4 @ k3 +x> D, K3 +x> D, h{s ^D +y, K3 +x> D,` D+y,gy 3

+d143,

H{suhvvlrq ri G orj k3 +x> D, Zh jhw= $4   g 4 K +}> D, k3 +x> D, @ g} 3 } @K3 +x>D, ] 4 4 @ D+y, h{s ^D+y,K3 +x> D,` gy 1 #

3

Ohw xv lqwurgxfh wkh ixqfwlrqdo= t +x> D, 

] 4 4 @ D+y, h{s ^D+y,K3 +x> D,` gy/ k3 +x> D, 3 78

dqg ghulyh lwv uvw rughu h{sdqvlrq1 Zh jhw= ] 4 t +x> D . D, @ ^D+y, . D+y,` h{s i ^D+y, . D+y,` K3 +x> D . D,j gy 3 ] 4 * t+x> D, . D+y, h{s ^D+y,K3 +x> D,` gy 3

K3 +x> D,

] 4 3

D+y,D+y, h{s ^D+y,K3 +x> D,` gy

 kGK3 +x,> Dl * t+x> D, .

] 4

] 4

3

D+y,5 h{s ^D+y,K3 +x> D,` gy

D+y, ^4  D+y,K3 +x> D,` h{s ^D+y,K3 +x> D,` gy ] 4  kGK3 +x,> Dl D+y,5 h{s ^D+y,K3 +x> D,` gy1 3

3

Lw iroorzv= kG orj k3 +x> D, > Dl @ k3 +x> D, kGt +x> D, > Dl ] 4 D+y, ^4  D+y,K3 +x> D,` h{s ^D+y,K3 +x> D,` gy @ k3 +x> D, 3 ] 4  .k3 +x> D, D+y,5 h{s ^D+y,K3 +x> D,` gy kGK3 +x,> Dl 3

@ k3 +x> D,

] 4

D+y, ^4  D+y,K3 +x> D,` h{s ^D+y,K3 +x> D,` gy  ] 4 k3 +x> D,5 D+y,5 h{s ^D+y,K3 +x> D,` gy 

] 4 3

3

3

K3 +x> D, h{s ^D +y, K3 +x> D,` D+y,gy1 +d144,

H{solflw h{suhvvlrq ri wkh frsxod*v ghulydwlyh E| lqvhuwlqj +d143, dqg +d144, lqwr +d1 y> D, lv ri wkh irup= ] orj f +x> y> D . D, * orj f+x> y> D, .  3 +x> y> D,D+y, .  4 +x> y> z> D,D+z,gz/ zkhuh=  3 +x> y> D, @

4  D+y,K3 +x> D, / D+y, 79

+d145,

dqg=  4 +x> y> z> D, @ k3 +x> D, h{s ^D+z,K3 +x> D,`    ] 4 5 D+}, k3 +x> D, h{s ^D +}, K3 +x> D,` g} 1  4  K3 +x> D, D+y, . D+z,  3

Wkh h{suhvvlrq ri wkh glhuhqwldo ri orj f+x> y> D, iroorzv= ] kG orj f+x> y> D,> Dl @  3 +x> y> D,D+y, .  4 +x> y> z> D,D+z,gz1

+d146,

+d147,

ll, Frqglwlrqdo h{shfwdwlrqv lq uhyhuvh wlph1 Ydulrxv ixqfwlrqdo ghulydwlyhv zlwk uhvshfw wr D fdq eh zulwwhq dv h{shfwdwlrqv lq uhyhuvh wlph1 Iurp +d143, zh jhw= kGK3 +x> D, > Dl @ K3 +x> D, H ^D +Xw4 , @D+Xw4 , m Xw @ x` / ru htxlydohqwo|= kG orj K3w > Dl @ H ^Dw4 @Dw4 m Xw ` / zkhuh K3w @ K3 +Xw > D, dqg Dw4 @ D +Xw4 ,1 Vlploduo|/ iurp +d144, zh jhw= kG orj k3w > Dl @ H ^+4  Dw4 K3w , Dw4 @Dw4 m Xw ` H ^Dw4 K3w m Xw ` H ^Dw4 @Dw4 m Xw ` 1 Wkhq iurp +d1 Xw4 > D, > Dl @ +4  Dw4 K3w , +Dw4 @Dw4  H ^Dw4 @Dw4 m Xw `, H i+4  Dw4 K3w , +Dw4 @Dw4  H ^Dw4 @Dw4 m Xw `, m Xw j 1 +d148, lll, Wkh sdudphwulf fdvh1 Zkhq ixqfwlrq D lv sdudphwhul}hg= D+y, @ D+y> ,/ wkh vfruh ri wkh prgho lv rewdlqhg iurp +d148, zlwk= D+y, @

CD +y> ,1 C 7:

Zh jhw= C orj f +Xw > Xw4 > D +,, C    C C orj Dw4 +,  H orj Dw4 +, m Xw @ +4  Dw4 K3w , C C      C C orj Dw4 +,  H orj Dw4 +, m Xw H +4  Dw4 K3w , m Xw 1 C C

Cow +, @ C

Vlploduo|/ wkh ghulydwlyhv ri orj K3 +x> D +,, dqg orj k3 +x> D +,, zlwk uhvshfw wr  duh jlyhq e|=   C C orj K3w +, @ H orj Dw4 +, m Xw / C C dqg=

  C C orj k3w +, @ H +4  Dw4 K3w , orj Dw4 +, m Xw C C   C orj Dw4 +, m Xw 1 H ^K3w Dw4 m Xw ` H C

Dsshqgl{ ; Wkh lqirupdwlrq rshudwru l, Wkh h{suhvvlrq ri wkh lqirupdwlrq rshudwru Ohw xv ghulyh wkh lqirupdwlrq rshudwru LK 1 Iurp +d147, lq Dsshqgl{ :/ wkh glhuhqwldo G orj f +=> => D3 , dgplwv d phdvxuh ghfrpsrvlwlrq zlwk erwk d glvfuhwh dqg d frqwlqxrxv sduw ^vhh Jdjolduglql dqg Jrxulhurx{ +5335,`1 Wkhuhiruh= ] ] +j> LK k,O5 + , @ j+y,3 +y> D3 ,k+y,gy . j+z,4 +z> y> D3 ,k+y,gzgy/ +d149, iru j> k 5 K/ zkhuh= k l 3 +y> D3 , @ H3  3 +Xw > Xw4 ,5 m Xw4 @ y @ dqg= 4 +z> y> D3 , @

4 / D3 +y,5

]  3 +x> z> D3 , 4 +x> z> y> D3 ,gx ] 4 .  4 +x> |> z> D3 , 4 +x> |> y> D3 ,gxg| . +z #$ y, 1 5 7;

Ohw xv qrz ghulyh dq h{suhvvlrq iru LK k/ k 5 K1 Iurp +d149, zh jhw=   ] ] g j+z, LK k+z, +z,  3 +z> D3 ,k+z,  4 +z> y> D3 ,k+y,gy gz @ 3/ ;j 5 K1 g Wkxv wkhuh h{lvwv d frqvwdqw n vxfk wkdw= LK k+z,

g +z, @ 3 +z> D3 ,k+z, . g

] 4 +z> y> D3 ,k+y,gy . n1

U Frqvwdqw n lv ghwhuplqhg e| wkh frqglwlrq LK k 5 K/ wkdw lv= LK k+z,gz @ 31 Zh jhw= ] 3 +z> D3 , 4 +z> y> D3 , k+z, . k+y,gy LK k+z, @ g@g +z, g@g +z,   ] 4 ]  ] gz 3 +z> D3 ,k+z, 4 +z> y> D3 ,k+y, . gy gz  g@g +z, g@g +z, g@g +z, 4  1 g@g +z, +d14:, Wkxv LK dgplwv wkh uhsuhvhqwdwlrq= LK k+z, @

3>K +z> D3 , k+z, . g@g +z,

]

4>K +z> y> D3 , k+y,gy/ vd|/ g@g +z,

zlwk 3>K @ 3 1 ll, Erxqghgqhvv dqg lqyhuwlelolw| ri LK Zh dvvxph wkdw/ iru dq| D/ wkhuh h{lvwv d srvlwlyh ghqlwh pdwul{ K +=> D, vxfk wkdw= ] ] 4>K +z> y> D,5 gzgy ? .41 K +z> D,K +y> D, Ixuwkhu ohw xv lqwurgxfh wkh phdvxuh  vxfk wkdw=   4 g ;D =  +y> D, / ;y1 K g D+y,5 Wkhq/ iurp Sursrvlwlrq 55 lq Jdjolduglql/ Jrxulhurx{ +5335,/ LK lv d erxqghg rshudwru iurp K lq lwvhoi1 Ohw xv qrz frqvlghu wkh lqyhuwlelolw| ri LK 1 Zh uvw vkrz wkdw wkh glhuhqwldo G orj f +=> => D3 , kdv d }hur qxoo vsdfh1 Lqghhg ohw xv frqvlghu d ixqfwlrq k 5 K vxfk wkdw= kG orj f +Xw > Xw4 > d3 , > kl @ 3 d1v1 7
y> D3 , > kl gxgy/ ;k 5 O5 +, 1 +k> #W ,O5 + , @ Iurp Jdjolduglql/ Jrxulhurx{ +5335, zh jhw= ] ] ] g +z,#W +z, @ e fW +z> y, 3 +z> y, gy . e fW +x> y, 4 +x> y> z, gxgy1 g +d14;, ll, Wkh uvw rughu frqglwlrq1 Iurp Jdjolduglql/ Jrxulhurx{ +5335, wkh uvw rughu frqglwlrq lv jlyhq e|= eW * SK #W / LK  D

83

zkhuh SK lv wkh ruwkrjrqdo surmhfwlrq rq wkh wdqjhqw vsdfh K/ zklfk lv jlyhq e|=  ] 4 ] 4 gz k+z,gz 1 SK k+y, @ k+y,  g@g +z, g@g +y, Iurp +d14:, dqg +d14;, zh jhw= ] eW +z, . 4 +z> y, D eW +y,gy 3 +z, D ]  4 ]  ] gz 3 +z, 4 +z> y, e eW +z, .  D  DW +y,gy gz g@g +z, g@g +z, g@g +z, ] ] ] e fW +x> y, 4 +x> y> z, gxgy * e fW +z> y, 3 +z> y, gy . ] 4 ] ] gz 4 e fW +z> y, 3 +z> y, gy.  g@g +z, g@g +z,  ] ] e fW +x> y, 4 +x> y> z, gxgy gz/ +d14 y, gy1 W kW  D Iurp +59, lw iroorzv=

  ] s g 4 N 5+z,gz / 0d1v1 lq y 5 ^3> 4` 1 eW +y, $ W kW  D Q 3> 3 +y,

Wkh frpsohwh surri ri wklv uhvxow lv jlyhq lq Sursrvlwlrq 56 ri Jdjolduglql/ Jrxulhurx{ +5335,1 ly, Dv|pswrwlf glvwulexwlrq ri olqhdu ixqfwlrqdov ri D U Wkh dv|pswrwlf glvwulexwlrq ri olqhdu ixqfwlrqdov j+y,D+y, +gy,/ j 5 O5 +, lv ghulyhg iurp Sursrvlwlrq 56 ri Jdjolduglql/ Jrxulhurx{ +5335,1

84

Dsshqgl{ 43 Wkh h!flhqf| erxqgv iru wkh vwhszlvh prgho l, Ghwhuplqdwlrq ri wkh sdudphwulf h!flhqf| erxqg Zh kdyh=

#

L +3 , @

3

hh lgQ  Q

gldj +d3

$

,4

4 H

3 Y3 ^ w +]w4

gldj +d3 , #

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