Nonparametric tests for positive quadrant dependence

(1)

Report

Reference

Nonparametric tests for positive quadrant dependence

DENUIT, Michel, SCAILLET, Olivier

Abstract

We consider distributional free inference to test for positive quadrant dependence, i.e. for the probability that two variables are simultaneously small (or large) being at least as great as it would be were they dependent. Tests for its generalisation in higher dimensions, namely positive orthant dependence, are also analysed. We propose two types of testing procedures.

The first procedure is based on the specification of the dependence concepts in terms of distribution functions, while the second procedure exploits the copula representation. For each specification a distance test and an intersection-union test for inequality constraints are developed depending on the definition of null and alternative hypotheses. An empirical illustration is given for US and Danish insurance claim data. Practical implications for the design of reinsurance treaties are also discussed.

DENUIT, Michel, SCAILLET, Olivier. Nonparametric tests for positive quadrant dependence. 2002

Available at:

http://archive-ouverte.unige.ch/unige:5817

Disclaimer: layout of this document may differ from the published version.

1 / 1

(2)

QRQSDUDPHWULF WHVWV IRU SRVLWLYH TXDGUDQW GHSHQGHQFH

Plfkho Ghqxlw Lqvwlwxw gh Vwdwlvwltxh

Xqlyhuvlwì Fdwkroltxh gh Orxydlq Yrlh gx Urpdq Sd|v/ 53 E0467; Orxydlq0od0Qhxyh/ Ehojlxp

ghqxlwCvwdw1xfo1df1eh

Rolylhu Vfdloohw KHF Jhqëyh dqg IDPH

Xqlyhuvlwì gh Jhqëyh Eg Fduo Yrjw/ 435 FK 0 4544 Jhqëyh 7/ Vxlvvh

rolylhu1vfdloohwCkhf1xqljh1fk

Wklv yhuvlrq= Pdufk 5335 +uvw yhuvlrq = Mdqxdu| 5334,

(3)

Devwudfw

Zh frqvlghu glvwulexwlrqdo iuhh lqihuhqfh wr whvw iru srvlwlyh txdgudqw ghshqghqfh/ l1h1 iru wkh suredelolw| wkdw wzr yduldeohv duh vlpxowdqhrxvo| vpdoo +ru odujh, ehlqj dw ohdvw dv juhdw dv lw zrxog eh zhuh wkh| ghshqghqw1 Whvwv iru lwv jhqhudolvdwlrq lq kljkhu glphqvlrqv/ qdpho|

srvlwlyh ruwkdqw ghshqghqfhv/ duh dovr dqdo|vhg1 Zh sursrvh wzr w|shv ri whvwlqj surfhgxuhv1 Wkh uvw surfhgxuh lv edvhg rq wkh vshflfdwlrq ri wkh ghshqghqfh frqfhswv lq whupv ri glvwulexwlrq ixqfwlrqv/ zkloh wkh vhfrqg surfhgxuh h{sorlwv wkh frsxod uhsuhvhqwdwlrq1 Iru hdfk vshflfdwlrq d glvwdqfh whvw dqg dq lqwhuvhfwlrq0xqlrq whvw iru lqhtxdolw| frqvwudlqwv duh ghyhorshg ghshqglqj rq wkh ghqlwlrq ri qxoo dqg dowhuqdwlyh k|srwkhvhv1 Dq hpslulfdo looxvwudwlrq lv jlyhq iru XV dqg Gdqlvk lqvxudqfh fodlp gdwd1 Sudfwlfdo lpsolfdwlrqv iru wkh ghvljq ri uhlqvxudqfh wuhdwlhv duh dovr glvfxvvhg1

Nh| zrugv dqg skudvhv= Qrqsdudphwulf/ Vwrfkdvwlf Rughulqj/ Srvlwlyh Txdgudqw Ghshq0 ghqfh/ Srvlwlyh Ruwkdqw Ghshqghqfh/ Frsxod/ Lqhtxdolw| Frqvwudlqw Whvw/ Ulvn Pdqdjhphqw/

Orvv Vhyhulw| Glvwulexwlrq1

MHO Fodvvlfdwlrq= F45/ G;4/ J43/ J54/ J551

DPV 5333 Vxemhfw Fodvvlfdwlrq= 93H48/ 95J43/ 95J63/ 95S38/ <4E5;/ <4E631

(4)

4 L qwurgxfwlrq

Wkh ghyhorsphqw dqg dqdo|vlv ri txdqwlwdwlyh prghov iru orvvhv lq odujh sruwirolrv ri lqvxudqfh frqwudfwv ru qdqfldo dvvhwv kdv ehhq dq duhd ri lqwhuhvw iru sudfwlwlrqhuv/ uhjxodwruv dqg dfdghplfv iru vhyhudo |hduv1 Wkhvh prghov dlp dw fdswxulqj wkh orvvhv gxh wr ghidxow hyhqwv ru dgyhuvh pryhphqwv ri dvvhw sulfhv1 Lq idfw/ prvw qdqfldo lqvwlwxwlrqv duh qrz urxwlqho| xvlqj ulvn pdqdjhphqw v|vwhpv wr dghtxdwho| frqwuro wkhlu ulvnv ru wr vxlwdeo| doorfdwh wkhlu fdslwdo1 Wklv kdv ehhq gulyhq e| hlwkhu lqwhuqdo uhtxluhphqwv +h!flhqw xvh ri fdslwdo lqyhvwhg e|

vkduhkroghuv/ ghyhorsphqw ri qhz exvlqhvv olqhv, ru h{whuqdo frqvwudlqwv +Fdslwdo Dghtxdf|

Uhtxluhphqw ri wkh Edvoh Frpplwwhh rq Edqnlqj Vxshuylvlrq/ suxghqwldo uxohv lpsrvhg e|

Hxurshdq ru Dphulfdq uhjxodwruv rq qdqfldo lqvwlwxwlrqv,1 Fohduo|/ wkh ghshqghqfh ehwzhhq qdqfldo lqvwuxphqwv pdwhuldoo| dhfwv ulvn phdvxuhv dqg dvvhw doorfdwlrqv uhvxowlqj iurp rswlpdo sruwirolr vhohfwlrqv1 Wkh dqdo|vlv ri wkh ghshqghqfh vwuxfwxuh fdqqrw eh qhjohfwhg dqg uhyhdov pxfk ri wkh gdqjhu dvvrfldwhg wr d jlyhq srvlwlrq1

Lq dfwxduldo vflhqfh/ wkh lqfuhdvlqj frpsoh{lw| ri lqvxudqfh dqg uhlqvxudqfh surgxfwv kdv ohg wr lqfuhdvhg lqwhuhvw lq wkh prgholqj ri ghshqghqw ulvnv +wklqn ri wkh hphujhqfh ri pxowl0olqh surgxfwv zklfk uhtxluh vrsklvwlfdwhg ulvn hydoxdwlrq phfkdqlvpv/ vhh Slqtxhw +4<<;, iru vlpsoh dqg srzhuixo prghov dssolfdeoh wr sdfndjhg surgxfwv lq fdu lqvxudqfh,1 Dovr/ G|qdplf Ilqdqfldo Dqdo|vlv +vhh Ndxipdqq/ Jdgphu dqg Nohww +4<<<, iru dq lqwurgxfwlrq, surylghv dfwxdulhv zlwk dq lqwhjudwhg ulvn pdqdjhphqw whfkqltxh1 Lwv pdlq fkdudfwhulvwlf lv wr lqwhjudwh wkh lqyhvwphqw dqg xqghuzulwlqj ulvn wr zklfk dq lqvxuhu lv h{srvhg1 GID qhfhvvlwdwhv d prgho wkdw frpelqhv lqirupdwlrq rq pdujlqdo glvwulexwlrqv wrjhwkhu zlwk lghdv rq lqwhughshqghqflhv1

Xqiruwxqdwho|/ frqwhpsrudu| whfkqltxhv wrr riwhq uhyroyh durxqg wkh xvh ri olqhdu fruuhod0 wlrq wr ghvfuleh d ghshqghqfh ehwzhhq ulvnv dqg lpsolflwo| dvvxph qrupdoo| glvwulexwhg ulvnv +pdlqo| iru pdwkhpdwlfdo frqyhqlhqfh,1 Exw zkdw grhv srvlwlyh fruuhodwlrq uhdoo| phdqB Lq wkh qrupdo zruog/ srvlwlyh fruuhodwlrq hqwdlov vwurqj srvlwlyh ghshqghqfh qrwlrqv/ vhhWrqj +4<<3,1 Krzhyhu/ dv looxvwudwhg e| Hpeuhfkwv hw do1 +5333,/ ghshqghqfh surshuwlhv ri wkh qrupdo zruog qr pruh krog lq wkh qrq0qrupdo zruog1 Prghuq ulvn pdqdjhphqw fdoov iru dq xqghuvwdqglqj ri vwrfkdvwlf ghshqghqfh jrlqj eh|rqg vlpsoh olqhdu fruuhodwlrq1 Lq wkdw uhvshfw/ ghshqghqfh frqfhswv olnh frprqrwrqlflw|/ pxowlyduldwh wrwdo srvlwlylw|/ frqglwlrqdo lqfuhdvlqjqhvv lq vhtxhqfh/ dvvrfldwlrq dqg srvlwlyh txdgudqw ghshqghqfh +dqg lwv pxowlydul0 dwh h{whqvlrqv, duh ri sulph lpsruwdqfh dqg vkrxog fruuhfwo| eh xqghuvwrrg e| sudfwlwlrqhuv1 Lq wkh pdqdjhphqw ri odujh sruwirolrv/ wkh pdlq ulvn lv wkh rffxuuhqfh ri pdq| mrlqw ghidxow hyhqwv ru vlpxowdqhrxv grzqvlgh hyroxwlrq ri sulfhv1 D ghhs nqrzohgjh ri wkh ghshqghqfh ehwzhhq qdqfldo dvvhwv ru fodlpv lv fuxfldo wr ehwwhu dvvhvv wklv ulvn ri orvv foxvwhulqj dqg wkhuhiurp dfklhyh d shuirupdqw ulvn pdqdjhphqw lq qdqfh dqg lqvxudqfh1 Wklv foxvwhulqj ehkdylrxu fdq eh ghvfulehg e| d xvhixo frqfhsw nqrzq dv srvlwlyh txdgudqw ghshqghqfh +STG, iru elyduldwh glvwulexwlrqv +Ohkpdqq +4<99,, dqg srvlwlyh ruwkdqw gh0 shqghqfhv +SRG, iru glphqvlrqv kljkhu wkdq wzr1 Wklv w|sh ri ghshqghqfh whoov xv krz wzr/

vpdoo lv dw ohdvw dv juhdw dv lw zrxog eh zhuh wkh| lqghshqghqw1

Rqh ri wkh pdlq lqwhuhvw ri wklv ghshqghqfh vwuxfwxuh lv wkdw lw doorzv wkh ulvn pdq0 djhu wr frpsduh wkh vxp ri STG udqgrp yduldeohv zlwk wkh fruuhvsrqglqj vxp xqghu wkh

4

(5)

lqghshqghqfh dvvxpswlrq1 Wkh frpsdulvrq lv lq wkh vhqvh ri glhuhqw vwrfkdvwlf rughulqjv h{suhvvlqj wkh frpprq suhihuhqfhv ri udwlrqdo ghflvlrq0pdnhuv +lq wkh iudphzrun ri wkh fodvvlfdo yrq Qhxpdq0Prujhqvwhuq h{shfwhg xwlolw| wkhru|/ dv zhoo dv lq rwkhu wkhrulhv/ olnh

\ddul +4<;:,*v rqh,1

Ilqdoo| ohw xv uhpdun wkdw qdqfldo vhfxulw| v|vwhpv duh jhqhudoo| frpsoh{/ dqg wkhlu rxwfrphv xvxdoo| lqyroyh vhyhudo glphqvlrqv1 Ghvfulelqj uhodwlrqvklsv dprqj glhuhqw gl0 phqvlrqv lv d edvlf whfkqltxh iru h{sodlqlqj wkh ehkdylru ri ulvn frqwuro phfkdqlvpv wr frqfhuqhg exvlqhvv dqg sxeolf srolf| ghflvlrq0pdnhuv1 Lq wkdw uhvshfw/ frsxod ixqfwlrqv fdq eh ri juhdw xvhixoqhvv iru ulvn pdqdjhuv dqg dfwxdulhv1 Wkh frqfhsw ri frsxodv% ru frsxod ixqfwlrqv% dv qdphg e|Vnodu+4<8<, ruljlqdwhv lq wkh frqwh{w ri suredelolvwlf phwulf vsdfhv1 Wkh lghd ehklqg wklv frqfhsw lv wkh iroorzlqj= iru pxowlyduldwh glvwulexwlrqv/ wkh xqlyduldwh pdujlqdov dqg wkh ghshqghqfh vwuxfwxuh fdq eh vhsdudwhg dqg wkh odwwhu pd| eh uhsuhvhqwhg e| d frsxod1 Wkh zrug frsxod lv d odwlq qrxq wkdw phdqv frxsoh%/ dqg lv xvhg lq judppdu dqg orjlf wr ghvfuleh wkdw sduw ri d sursrvlwlrq zklfk frqqhfwv wkh vxemhfw dqg suhglfdwh1 Lq vwdwlvwlfv/ lw qrz ghvfulehv wkh ixqfwlrq wkdw frxsohv% rqh0glphqvlrqdo glvwulexwlrq ixqf0 wlrqv wr irup pxowlyduldwh rqhv/ dqg pd| vhuyh wr fkdudfwhul}h ghshqghqfh frqfhswv vxfk dv STG dqg SRG1

Wkh sdshu lv rujdql}hg dv iroorzv1 Lq Vhfwlrq 5/ zh uhylhz vhyhudo vwrfkdvwlf rughu uh0 odwlrqv1 Lq Vhfwlrq 6/ zh uhfdoo wkh ghqlwlrq ri frsxod ixqfwlrqv/ dv zhoo dv wkh fodvvlfdo Vnodu*v uhsuhvhqwdwlrq wkhruhp iru pxowlyduldwh glvwulexwlrqv1 Vshflfdwlrq ri k|srwkhvhv lq whupv ri glvwulexwlrq ixqfwlrqv +vshflfdwlrq lq whupv ri orvv ohyhov, ru frsxodv +vshflfdwlrq lq whupv ri suredelolw| ohyhov, zloo ohdg wr glhuhqw lqihuhqfhv1 Vhfwlrq 7 jlyhv wkh ghqlwlrq ri STG dqg ri vrph ri lwv pxowlyduldwh h{whqvlrqv1 Lq Vhfwlrq 8/ zh looxvwudwh wkh lqwhuhvw ri wkhvh srvlwlyh ghshqghqfh qrwlrqv zlwk wkh khos ri ydulrxv xvhixo vwrfkdvwlf lqhtxdolwlhv1 Zh surylgh vrph uhohydqw h{dpsohv frplqj iurp phdvxuhphqw ri lqhtxdolw| dqg sryhuw|/ dv zhoo dv iurp olih lqvxudqfh1 Lq Vhfwlrq 9 zh ghvfuleh wkh qxoo dqg dowhuqdwlyh k|srwkhvhv zh duh lqwhuhvwhg lq/ dqg ghyhors whvwlqj surfhgxuhv iru vxfk sxusrvh1 Wkhvh surfhgxuhv duh forvho| uhodwhg wr wkh lqihuhqfh wrrov iru wudglwlrqdo uvw rughu dqg vhfrqg rughu vwrfkdvwlf grplqdqfh/ zklfk dovr uho| rq glvwdqfh dqg lqwhuvhfwlrq0xqlrq whvwv iru lqhtxdolw| frqvwudlqwv +vhh Gdylgvrq dqg Gxforv +5333, dqg wkh uhihuhqfhv wkhuhlq,1 Dq hpslulfdo looxvwudwlrq rq XV dqg Gdqlvk lqvxudqfh fodlp gdwd lv ghvfulehg lq Vhfwlrq :1 Wkhuhlq zh surylgh d frp0 sdulvrq ri suhplxpv frpsxwhg xqghu glhuhqw ghshqghqfh dvvxpswlrqv/ dqg glvfxvv hhfw ri STG rq wkh sulflqj ri uhlqvxudqfh wuhdwlhv1 Vhfwlrq ; frqfoxghv1 Surriv duh jdwkhuhg lq dq dsshqgl{1

Lw lv zruwk phqwlrqlqj wkdw zh ghsduw iurp wkh dfwxduldo olwhudwxuh e| dvvljqlqj d qhjdwlyh vljq wr orvvhv lq wklv sdshu1 Wklv lv lq olqh zlwk wkh djuhhphqw lq irufh lq qdqfh iru dvvhw uhwxuqv1

5 Vwrfkdvwlf rughulqjv

Vwrfkdvwlf rughulqjv duh elqdu| uhodwlrqv ghqhg rq fodvvhv ri suredelolw| glvwulexwlrqv1 Wkh|

dlp wr pdwkhpdwlfdoo| wudqvodwh lqwxlwlyh lghdv olnh ehlqj odujhu% ru ehlqj pruh yduldeoh%

iru udqgrp txdqwlwlhv1 Wkh| wkxv h{whqg wkh fodvvlfdo phdq0yduldqfh dssurdfk wr frpsduh ulvnlqhvv1

5

(6)

Ohw xv ghqh wkh iroorzlqj xwlolw| fodvvhv1 Ohw L frqwdlq doo qrq0ghfuhdvlqj xwlolwlhv G U $ U1 O hw L₂ eh wkh uhvwulfwlrq ri L wr lwv frqfdyh hohphqwv1 Pruh jhqhudoo|/ iru

& / ohw L& eh E& wlphv frqwlqxrxvo| glhuhqwldeoh xwlolw| ixqfwlrqv vxfk wkdw

*4%<n"E%En4lv qlwh/ *4%<n"^EE% ' f iru ' c c &dqgE^&3^E&3

lv qrq0ghfuhdvlqj1

Lq zkdw iroorzv/ zh dvvxph wkdw ghflvlrq0pdnhuv pd{lpl}h d yrq Qhxpdqq0Prujhqvwhuq h{shfwhg xwlolw| +exw zh phqwlrq wkdw uhvxowv lqyroylqj L dqg L2 vwloo krog lq gxdo wkhr0 ulhv iru fkrlfh xqghu ulvn/ vhh h1j1 Ghqxlw/ Gkdhqh dqg Ydq Zrxzh +4<<<, iru ixuwkhu lqirupdwlrq,1

Ohw t dqg t2 eh wzr udqgrp yduldeohv vxfk wkdw .Et .Et2 krogv iru doo 5L

+uhvs1 5L2,/ surylghg wkh h{shfwdwlrqv h{lvw1 Wkhqt lv vdlg wr eh vpdoohu wkdqt2 lq wkh vwrfkdvwlf grplqdqfh +uhvs1 lqfuhdvlqj frqfdyh rughu,/ ghqrwhg dvt g t2+uhvs1t lfy t2,1 AIurp wkh yhu| ghqlwlrqv ri _g dqg _lfy/ zh vhh wkdw wkhvh vwrfkdvwlf rughulqjv h{suhvv wkh frpprq suhihuhqfhv ri wkh fodvvhv ri surw0vhhnlqj ghflvlrq0pdnhuv/ dqg ri surw0vhhnlqj ulvn0dyhuwhuv/ uhvshfwlyho|1 Wklv surylghv dq lqwxlwlyh phdqlqj wr udqnlqjv lq wkhg0 rulfy0 vhqvh1

Li t _lfy t₂ dqg.t '.t₂/ wkhq zh zulwht _fy t₂1 Lq wklv fdvh.Et.Et₂iru doo wkh frqfdyh xwlolwlhv / vr wkdwt2 lv suhihuuhg ryhu t e| doo ulvn0dyhuwhuv1 Ixuwkhupruh/

li .Et .Et2 iru doo 5 L&/ surylghg wkh h{shfwdwlrqv h{lvw/ wkhq t lv vdlg wr eh vpdoohu wkdq t₂ lq wkh &0lqfuhdvlqj frqfdyh rughu/ ghqrwhg dv t _&3lfy t₂1 E| frqyhqwlrq zh dvvxph wkdw &3lfy uhgxfhv wr lfy dqgg iru & ' 2dqg& ' / uhvshfwlyho|1

Iru d pruh ghwdlohg h{srvlwlrq ri vwrfkdvwlf rughulqjv/ vhh h1j1 wkh uhylhz sdshuv e|

Zh vxppdul}h khuhdiwhu wkh pdlq fkdudfwhul}dwlrqv ri _g/ _lfy dqg _fy1 O hw 8 dqg 8₂ eh wkh uhvshfwlyh fgi*v iru t dqg t1 O hw 8³ dqg 8₂³ ghqrwh wkh fruuhvsrqglqj txdqwloh wudqvirupdwlrqv/ ghqhg dv

8³ER ' ?ui%5Um8E%Rjc R5dfcoc ' c2 Wkhruhp 5141

+l, t _g t₂ /8E%8₂E% iru doo %>

+ll, t g t2 /8³ER8₂³ER iru doo R5Efc>

+lll, t lfy t2 /U_%

3"8E_U_%

3"82E_ iru doo %>

+ly, t _lfy t₂ /U_R

f 8³E_ U_R

f 8₂³E_ iru doo R5Efc>

+y, t lfy t2 / wkhuh h{lvwv d udqgrp yduldeoh ] vxfk wkdw t fy ~ g t2>

+yl, t &3lfy t2 / ] _%

3"

] _%_&3

3"

] _%₂

3"8E%_%_%2 _%&3

] _%

3"

] _%_&3

3"

] _%₂

3"82E%_%_%2 _%&3 iru doo %c 6

(7)

surylghg wkh lqwhjudov duh qlwh +wkh lqwhjudov duh qlwh li8 dqg82 kdyh qlwhE&wk prphqwv,1

Vwdwlvwlfdo lqihuhqfh iru g/ lfy dqg &3lfy lv lqyhvwljdwhg lq ydvw ghwdlov lq Gdylgvrq dqg Gxforv +5333,/ zkhuh frqqhfwlrqv zlwk hfrqrplf dqg vrfldo zhoiduh lq glhuhqw srs0 xodwlrqv duh h{solflwhg1 Glhuhqw dssurdfkhv duh srvvleoh1 Hlwkhu hpslulfdo dqdorjv ri wkh lwhudwhg lqwhjudov ri wkh fgi*v duh frpsxwhg dqg frpsduhg dw hyhu| frxsoh ri revhuydwlrqv1 Wklv lv wkh urxwh iroorzhg h1j1 e| PfIdgghq +4<;<, ru Ndxu/Sudndvd Udr dqgVlqjk +4<<7,1 Ru/ zklfk lv pruh riwhq wkh fdvh/ d suhghwhuplqhg julg ri d pxfk vpdoohu qxpehu ri srlqwv +w|slfdoo|/ txdqwlohv ri wkh xqghuo|lqj glvwulexwlrqv, lv xvhg1 Lq wkh odwwhu fdvh/

wkh vwrfkdvwlf udqnlqj lpsolhv d vhw ri lqhtxdolwlhv ( f iru ' c2c c _/ zkhuh _ lv wkh qxpehu ri srlqwv lq wkh suhghwhuplqhg julg1 Wklv zloo eh wkh irxqgdwlrq iru wkh vwdwlvwlfdo whvwv glvfxvvhg lq Vhfwlrq 91 Ehiruh prylqj wr wkdw srlqw zh qhhg wr suhvhqw d yhu| srzhuixo wkhruhp gxh wr Vnodu +4<8<,1

6 Vnodu*v uhsuhvhqwdwlrq iru pxowlyduldwh glvwulexwlrqv

Zh frqvlghu d vhwwlqj pdgh ri l1l1g1 revhuydwlrqv i\|( | ' c c Aj ri d udqgrp yhfwru \ wdnlqj ydoxhv lq U^?1 Wkhvh gdwd pd| fruuhvsrqg wr hlwkhu revhuyhg lqglylgxdo orvvhv rq ? lqvxudqfh frqwudfwv/ wkh dprxqwv ri fodlpv uhsruwhg e| d jlyhq srolf| kroghu rq ? glhuhqw jxdudqwhhv lq d pxowlolqh surgxfw ru revhuyhg uhwxuqv ri? qdqfldo dvvhwv1

wr lwv xqlyduldwh pdujlqv 8/ fdq eh ghvfulehg e| d frsxod1 Zkloh wkh mrlqw glvwulexwlrq 8 surylghv frpsohwh lqirupdwlrq frqfhuqlqj wkh ehkdylrxu ri \/ frsxodv doorz wr vhsdudwh ghshqghqfh dqg pdujlqdo ehkdylrxu ri wkh hohphqwv frqvwlwxwlqj\1 Ehiruh ghqlqj irupdoo|

d frsxod/ zh zrxog olnh wr uhihu wkh uhdghu wr Qhovhq +4<<<, dqg Mrh +4<<:, iru pruh h{whqvlyh wkhruhwlfdo wuhdwphqwv1

D ?0glphqvlrqdo frsxod lv vlpso| +wkh uhvwulfwlrq wr dfco^? ri, dq ?0glphqvlrqdo fgi zlwk xqlw xqlirup pdujlqdov1 Wkh uhdvrq zk| d frsxod lv xvhixo lq uhyhdolqj wkh olqn ehwzhhq wkh mrlqw glvwulexwlrq dqg lwv pdujlqv wudqvsluhv iurp wkh iroorzlqj wkhruhp1

Wkhruhp 6141 +Vnodu*v Wkhruhp,

Ohw 8 eh dq ?0glphqvlrqdo fgi zlwk pdujlqdov 8c c 8_?1 Wkhq wkhuh h{lvwv dq ?0frsxod vxfk wkdw iru doo | lq U^?/

8E| '

8E+c c 8?E+?

+614,

Li 8c c 8? duh doo frqwlqxrxv/ wkhq lv xqltxho| ghqhg1 Rwkhuzlvh/ lv xqltxho| ghwhu0 plqhg rq udqjh8udqjh8?1 Frqyhuvho|/ li lv dq ?0frsxod dqg 8c c 8? duh fgi*v/

wkhq wkh ixqfwlrq 8 ghqhg e| +614, lv dq ?0glphqvlrqdo fgi zlwk pdujlqv 8c c 8_?1

Dowkrxjk frsxodv frqvwlwxwh d ohvv zhoo0nqrzq dssurdfk wr ghvfulelqj ghshqghqfh wkdq fruuhodwlrq/ wkh| rhu wkh ehvw xqghuvwdqglqj ri wkh jhqhudo frqfhsw ri ghshqghqf|1 Lq sduwlfxodu/ frsxodv vkduh wkh qlfh surshuw| wkdw vwulfwo| lqfuhdvlqj wudqvirupdwlrqv ri wkh

7

(8)

xqghuo|lqj udqgrp yduldeohv uhvxow lq wkh wudqviruphg yduldeohv kdylqj wkh vdph frsxod +zkdw lv qrw wuxh iru olqhdu fruuhodwlrq,1

Dv dq lpphgldwh fruroodu| ri Vnodu*v Wkhruhp/ zh kdyh

Ex ' 8E8³Ec c 8_?³E_? +615, iru dq| x 5 dfco^?1 Iurp H{suhvvlrq +615,/ zh pd| revhuyh wkdw wkh ghshqghqfh vwuxfwxuh hperglhg e| wkh frsxod fdq eh uhfryhuhg iurp wkh nqrzohgjh ri wkh mrlqw fgi 8 dqg lwv pdujlqv 81 Wklv zloo eh xvhg odwhu wr gholyhu hpslulfdo hvwlpdwhv ri frsxodv1

Qrwh wkdw lqghshqghqfh ehwzhhq udqgrp yduldeohv fdq eh fkdudfwhul}hg wkurxjk frsxodv1 Lqghhg/ ? udqgrp yduldeohv duh lqghshqghqw li/ dqg rqo| li/ wkhlu frsxod lvEx '^zEx ' T_?

'/ iru doo x 5 dfco^?1 ^z lv ixuwkhu uhihuuhg wr dv wkh lqghshqghqfh frsxod1 Wklv fkdudfwhul}dwlrq ri lqghshqghqfh lv wkh vwduwlqj srlqw iru wkh dqdo|vlv ri srvlwlyh txdgudqw ghshqghqfh wkurxjk wkh xvh ri frsxodv/ zklfk zh zlvk wr ghyhors qrz1

7 Ghshqghqfh qrwlrqv

714 Srvlwlyh txdgudqw ghshqghqfh

Wkh frqfhsw ri srvlwlyh txdgudqw ghshqghqfh +STG, lv lqwurgxfhg lqOhkpdqq +4<99, dqg ghvfulehv krz wzr udqgrp yduldeohv ehkdyh wrjhwkhu zkhq wkh| duh vlpxowdqhrxvo| vpdoo +ru odujh,1 Dv douhdg| phqwlrqhg/ mrlqw rffxuhqfh ri odujh orvvhv ru yhu| qhjdwlyh uhwxuqv lv ri sduwlfxodu lqwhuhvw lq ulvn pdqdjhphqw1

Vlqfh srvlwlyh% uhihuv wr d frpsdulvrq zlwk lqghshqghqfh/ ohw\^zghqrwh dq lqghshqghqw yhuvlrq ri wkh udqgrp yhfwru\/ wkdw lv/ \ dqg\^z kdyh lghqwlfdo xqlyduldwh pdujlqdov dqg

\^zkdv lqghshqghqw frpsrqhqwv1 Irupdoo|/ wzr udqgrp yduldeohvt dqgt₂ +ru wkh udqgrp frxsoh \ ' Etc t2, duh vdlg wr eh srvlwlyho| txdgudqw ghshqghqw li/ iru doo |5U²/

vpdoo lv dw ohdvw dv juhdw dv lw zrxog eh zhuh wkh| lqghshqghqw1 Ri frxuvh/ +714, lv htxlydohqw wr

d\ :|od\^z :|o 'dt : +odt2 : +2o +715, zklfk hqmr|v d vlplodu lqwhusuhwdwlrq +zlwk vpdoo% uhsodfhg zlwk odujh%,1

Frqvlghulqj +714,0+715,/ STG dsshduv dv d frpsdulvrq ri wkh mrlqw glvwulexwlrq ri \ wr wkdw ri \^z1 Lw fdq wkxv eh frqvlghuhg dv d vshfldo fdvh ri frpsdulvrqv ri sdluv ri elyduldwh glvwulexwlrqv zlwk lghqwlfdo pdujlqdov1 Wklv |lhogv wkh frqfrugdqfh rughu lqwurgxfhg e|

\dqdjlprwrdqgRndprwr+4<9<, dqg ixuwkhu vwxglhg e|Wfkhq+4<;3, dqgNlphogrui dqg Vdpsvrq +4<;:,1 STG lv lq sduwlfxodu vdwlvhg zkhq udqgrp yduldeohv duh uhjuhvvlrq ghshqghqw +vhh Gdfkudrxl dqg Glrqqh +5333, iru ghqlwlrq dqg xvh ri wklv ghshqghqfh frqfhsw iru rswlpdo sruwirolr vhohfwlrq lq suhvhqfh ri ghshqghqw ulvn| dvvhwv,1

Fohduo|/t dqgt2 duh STG li/ dqg rqo| li/}Etdqg }2Et2 duh STG iru dq| lqfuhdvlqj ixqfwlrqv } dqg }21 Wklv vkrzv wkdw STG lv d surshuw| ri wkh xqghuo|lqj frsxod dqg lv

8

(9)

qrw lq xhqfhg e| wkh pdujlqdov1 Lqhtxdolw| +714, fdq wkhq dovr eh zulwwhq lq whupv ri wkh frsxod ri wkh wzr udqgrp yduldeohv/ vlqfh +714, lv htxlydohqw wr wkh frqglwlrq wkdw/ iru doo x5dfco²/

Ex^zEx ' ₂ +716,

715 Srvlwlyh ruwkdqw ghshqghqfhv

Wkh elyduldwh qrwlrq ri STG kdv ehhq jhqhudol}hg wr kljkhu glphqvlrqv lq vhyhudo zd|v/ vhh h1j1 Qhzpdq +4<;7,1 Zh frqvlghu khuh srvlwlyh ruwkdqw ghshqghqflhv1

Srvlwlyh ruwkdqw ghshqghqfhv rhu qlfh h{whqvlrqv ri STG= lq wkuhh ru pruh glphqvlrqv/

ruwkdqwv duh vxevwlwxwhg iru txdgudqwv1 Wklv |lhogv wkh iroorzlqj ghqlwlrqv/ gluhfwo| lqvsluhg iurp +714, dqg +715,1 D udqgrp yhfwru \ lv vdlg wr eh srvlwlyho| orzhu ruwkdqw ghshqghqw +SORG/ lq vkruw, zkhq wkh lqhtxdolwlhv

d\ |od\^z |o '

\? '

dt +o +717,

krog iru dq| |5 U^?1 Lw lv vdlg wr eh srvlwlyho| xsshu ruwkdqw ghshqghqw +SXRG/ lq vkruw, zkhq wkh lqhtxdolwlhv

d\ :|od\^z:|o '

\? '

dt : %o +718,

krog iru dq| |5U^?1 Ri frxuvh/ +717, dqg +718, duh qr pruh htxlydohqw zkhq ?1

Lqwxlwlyho|/ +718, phdqv wkdw tc t2c c t? duh pruh olnho| vlpxowdqhrxvo| wr kdyh odujh ydoxhv/ frpsduhg zlwk d yhfwru ri lqghshqghqw uy*v zlwk wkh vdph fruuhvsrqglqj xqlyduldwh pdujlqdov1 Lqhtxdolw| +717, lv vlploduo| lqwhusuhwhg1 Zkhq +717, dqg +718, vlpxowdqhrxvo|

krog/ wkhq\ lv vdlg wr eh srvlwlyho| ruwkdqw ghshqghqw +SRG/ lq vkruw,1 SRG lv lq sduwlfxodu ixooohg zkhq yduldeohv duh dvvrfldwhg +vhhPlojurp dqgZhehu +4<;5, iru ghqlwlrq dqg xvh ri wkh dvvrfldwlrq frqfhsw lq dxfwlrq wkhru|,1

Lq whupv ri wkh frsxod dvvrfldwhg wr wkh udqgrp yhfwru\/ +717, fdq eh zulwwhq dv

Ex

\? '

c +719,

dqg +718, dv

Ex

\? '

Ec +71:,

iru doo x 5dfco^?/ zkhuh ghqrwhv wkh vxuylydo frsxod dvvrfldwhg zlwk 1

Khqfh/ STG dqg SORG pd| eh fkdudfwhul}hg lq whupv ri hlwkhu fgi*v ru frsxodv/ dqg wkxv pd| eh fkhfnhg/ rqfh fgi*v ru frsxodv duh hpslulfdoo| nqrzq1 Lq Vhfwlrq 9 zh ghyhors lqihuhqfh wrrov iru wkdw sxusrvh1

9

(10)

Ilqdoo| ohw xv qrwh wkdw rwkhu ghshqghqfh frqfhswv vxfk dv qhjdwlyh txdgudqw ghshqghqfh +QTG, dqg qhjdwlyh ruwkdqw ghshqghqfhv +QRG, pd| dovr eh ghqhg e| uhyhuvlqj wkh vhqvh ri rqh/ ru doo lqhtxdolwlhv lq +714, dqg +717, +vhhQhovhq +4<<<,,1 Whvwlqj surfhgxuhv vlplodu wr rxuv pd| hdvlo| eh ghyhorshg iru wkhvh fdvhv1 Zh irfxv khuhdiwhu wr STG dqg SORG vlqfh zh eholhyh wkdw wkh| duh wkh prvw uhohydqw ghshqghqfh qrwlrqv lq vwdqgdug ulvn pdqdjhphqw dssolfdwlrqv1 Qhyhuwkhohvv/ wkh rwkhu frqfhswv frxog dovr eh ri lqwhuhvw iru rwkhu dssolfdwlrqv dv/ iru lqvwdqfh/ wr ghwhuplqh zkhwkhu d ulvn whqgv wr khgjh dqrwkhu rqh1

8 Dssolfdwlrqv ri srvlwlyh ghshqghqfh qrwlrqv

Zh qrz looxvwudwh wkh sudfwlfdo uhohydqfh ri wkh srvlwlyh ghshqghqfh qrwlrqv lq phdvxuhphqw ri lqhtxdolw| dqg sryhuw| dv zhoo dv olih lqvxudqfh1

814 STG

Rqh ri wkh pdlq lqwhuhvw ri STG lv iru frpsdulvrq zlwk udqgrp frxsohv zlwk lghqwlfdo pdujlqdov exw lqghshqghqw frpsrqhqwv1 Wklv frphv iurp wkh iroorzlqj uhvxow ri zklfk zh surylgh d vkruw surri lq dsshqgl{1 Lw lv d vwudljkwiruzdug dgdswdwlrq ri wkh uhvxow riGkdhqh dqgJrrydhuwv +4<<9, hvwdeolvkhg lq wkh frqyh{ dfwxduldo vhwwlqj1

Sursrvlwlrq 8141 Li t dqgt2 duh STG/ wkhq tnt2 fy t^znt₂^z/ zkhuht^znt₂^z lv wkh vxp ri wkh frpsrqhqwv ri wkh lqghshqghqw yhuvlrq \^z ri \ ghqhg lq +714,1

Wklv phdqv wkdw zkhq STG krogv/ hyhu| ulvn0dyhuwhu djuhhv wr vd| wkdw t nt₂ lv ohvv idyrudeoh wkdq wkh fruuhvsrqglqj vxp xqghu lqghshqghqfh1 Frqvhtxhqwo|/ prvw lqvxudqfh suhplxpv dqg ulvn phdvxuhv zloo eh odujhu irufnf2 wkdq iruf^znf₂^z+vlqfh wkh sulqflsohv xvhg wr fdofxodwh vxfk txdqwlwlhv duh lq dffrugdqfh zlwk wkh frpprq suhihuhqfhv ri ulvn0 dyhuwhuv,1 Iru lqvwdqfh/ vlqfh wkh ixqfwlrq%:$ E%Vn/ zlwkEn ' 4@ ifcj/ lv frqfdyh iru dq| V 5 U/ wkh lqhtxdolw| .Et^znt₂^z Vn .Et nt2 Vn krogv wuxh iru doo V1 Wkh txdqwlw| .Et nt₂ V_n lv uhihuuhg wr dv wkh vwrs0orvv suhplxp uhodwlqj wr t nt₂ lq dfwxduldo vflhqfh +V lv fdoohg wkh ghgxfwleoh,1 Lq qdqfh/ zkhq dssursuldwho| glvfrxqwhg/

lw fdq eh uhjdughg dv wkh sulfh ri d edvnhw rswlrq zlwk t dqg t2 dv xqghuo|lqj dvvhwv dqgV dv vwulnh sulfh1 Wkh frqyhqlhqw dvvxpswlrq ri lqghshqghqfh pd| wkxv ohdg wr vhulrxv xqghusulflqj ri lqvxudqfh suhplxpv ru rswlrq sulfhv1 Wklv zloo eh frquphg e| wkh hpslulfdo uhvxowv ri wklv sdshu1

Wkh Oruhq} rughu lv ghqhg e| phdqv ri srlqwzlvh frpsdulvrq ri Oruhq} fxuyhv1 Wkh odwwhu lv xvhg lq hfrqrplfv wr phdvxuh wkh lqhtxdolw| ri lqfrphv +vhh EhdfkdqgGdylgvrq +4<;6,/ Gdugdqrql dqg Iruflqd +4<<<, iru uhodwhg lqihuhqfh,1 Pruh suhflvho|/ ohw t eh d qrq0qhjdwlyh udqgrp yduldeoh zlwk fgi 81 Wkh Oruhq} fxuyh u dvvrfldwhg zlwk t lv wkhq ghqhg e|

uER ' .t

] _R

|'f8³E_c R 5dfco

Zkhq t uhsuhvhqwv wkh lqfrph ri wkh lqglylgxdov lq vrph srsxodwlrq/ u pdsv R 5 dfco wr wkh sursruwlrq ri wkh wrwdo lqfrph ri wkh srsxodwlrq zklfk dffuxhv wr wkh srruhvwffR( ri wkh srsxodwlrq1

:

(11)

Frqvlghu wzr qrq0qhjdwlyh udqgrp yduldeohv t dqg t2 zlwk qlwh h{shfwdwlrqv1 Wkhq/

t lv vdlg wr eh vpdoohu wkdq t₂ lq wkh Oruhq} rughu/ khqfhiruwk ghqrwhg e| t _Oruhq} t₂/ zkhq uER u2ER iru doo R 5 dfco1 Zkhq t Oruhq} t2 krogv/ t grhv qrw h{klelw pruh lqhtxdolw| lq wkh Oruhq} vhqvh wkdq grhv t21 D vwdqgdug uhihuhqfh iru Oruhq} lv Duqrog +4<;:,1

Iurp Wkhruhp 514+ly,/ zh vhh wkdw surylghg .t ' .t2/ t Oruhq} t2 / t2 fy t1 Khqfh/ li t dqg t2 duh STG wkhq t^znt₂^z Oruhq} t nt2 e| yluwxh ri Sursrvlwlrq 8141 Ohw xv jlyh dq lqwhusuhwdwlrq ri wklv vwrfkdvwlf lqhtxdolw|1 Ohw t +uhvs1 t2, ghqrwh wkh kxvedqgv* +uhvs1 zlyhv*, lqfrph lq vrph srsxodwlrq1 Vd|lqj wkdwt dqgt2 duh STG phdqv wkdw/ dv wkh vd|lqj jrhv/ elugv ri d ihdwkhu rfn wrjhwkhu%= phq dqg zrphq hduqlqj odujh +uhvs1 vpdoo, vdodulhv whqg wr eh dvvrfldwhg1 Vxfk d srsxodwlrq h{klelwv pruh lqhtxdolw| lq wkh Oruhq} vhqvh wkdq d srsxodwlrq zkhuh vsrxvhv* hduqlqjv duh lqghshqghqw1 Wklv w|sh ri lqhtxdolw| phdvxuhphqw edvhg rq STG pd| dovr eh dssolhg wr wrwdo krxvhkrog lqfrphv lq wzr frxqwulhv lqvwhdg ri kxvedqgv* dqg zlyhv* lqfrphv lq rqh frxqwu|1

Ohw xv qrz surylgh dq dssolfdwlrq ri STG lq olih lqvxudqfh1 Vwdqgdug dfwxduldo wkhru| ri pxowlsoh olih lqvxudqfh wudglwlrqdoo| srvwxodwhv wkh lqghshqghqfh iru wkh uhpdlqlqj olihwlphv lq rughu wr hydoxdwh wkh dprxqw ri suhplxp uhodwlqj wr dq lqvxudqfh frqwudfw lqyroylqj pxowlsoh olyhv1 Qhyhuwkhohvv/ wklv k|srwkhvlv reylrxvo| uholhv rq frpsxwdwlrqdo frqyhqlhqfh udwkhu wkdq uhdolvp1 D qh h{dpsoh ri srvvleoh ghshqghqfh dprqj lqvxuhg shuvrqv lv fhuwdlqo| d frqwudfw lvvxhg wr d pduulhg frxsoh1 Lq vxfk d fdvh/ wkh dfwxdu| kdv wr zrqghu zkhwkhu wkh lqghshqghqfh dvvxpswlrq lv uhdvrqdeoh dqg zkhwkhu lw zrxog qrw eh zlvhu wr exlog dq dssursuldwh sulfh olvw lqfrusrudwlqj srvvleoh hhfwv ri d ghshqghqfh dprqj wlph0xqwlo0ghdwk udqgrp yduldeohv1

Vshflfdoo|/ ohw A% +uhvs1 A%2, eh wkh kxvedqg*v +uhvs1 zlih*v, olihwlph/ zkhuh % +uhvs1

%2, vwdqgv iru wkh djh ri wkh kxvedqg +uhvs1 zlih, dw wkh vwduw ri wkh frqwudfw1 Lq oljkw ri folqlfdo vwxglhv/ wkh STG dvvxpswlrq iru A_% dqg A_% vhhpv uhdvrqdeoh1 Wklv kdv ehhq hpslulfdoo| lqyhvwljdwhg xvlqj r!fldo Ehojldq vwdwlvwlfv e| Ghqxlw dqg Fruqhw +4<<<, lq d Pdunryldq sdudphwulf vhwwlqj1 Ri frxuvh/ wkh vwdwlvwlfdo whvwv ghyhorshg lq wklv sdshu duh xvhixo lq wkdw uhvshfw/ vlqfh wkh| dyrlg wkh sdudphwulf dvvxpswlrq riwhq pdgh lq dfwxduldo vwxglhv/ qdpho| d Jrpshuw}0Pdnhkdp glvwulexwlrq iru wkh uhpdlqlqj olihwlphv1

Iru lqvxudqfh srolflhv vrog wr pduulhg frxsohv/ STG iru A% dqg A%2 doorzv wkh dfwxdu|

wr nqrz zkhwkhu wkh lqghshqghqfh dvvxpswlrq jhqhudwhv lpsolflw vdihw| ordglqj ru/ rq wkh frqwudu|/ ohdgv wr lqvx!flhqw suhplxp dprxqwv1 Lqghhg/ wklv vlpso| frphv iurp wkh idfw wkdw wkh STG dvvxpswlrq iru A% dqgA%2 hqvxuhv wkdw

4?iA_%^zc A_%^z₂j g 4?iA%c A%2j dqg 4@ iA%c A%2j g 4@ iA_%^zc A_%^z₂j

zklfk uhdglo| iroorz iurp +714,0+715,1 Qrz/ ohw xv frqvlghu dqqxlwlhv +l1h1 frqwudfwxdo jxdu0 dqwhhv wkdw surplvh wr surylgh shulrglf lqfrph ryhu wkh olihwlphv ri lqglylgxdov,1 Wkh?0|hdu odvw0vxuylyru +uhvs1 mrlqw0olih, dqqxlw| sd|v ' 4 dw wkh hqg ri wkh |hduv c2c c ?dv orqj dv hlwkhu vsrxvh vxuylyhv +uhvs1 erwk vsrxvhv vxuylyh,1 Wkh qhw suhvhqw ydoxh ri wkh lqvxuhu*v sd|phqwv duh reylrxvo| lqfuhdvlqj ixqfwlrqv ri 4@ iA_%c A_%₂j iru wkh odvw0vxuylyru dqqxlw|

dqg ri 4?iA%c A%2j iru wkh mrlqw0olih dqqxlw|1 Wkh qhw vlqjoh suhplxp fruuhvsrqglqj wr wkh odvw0vxuylyru +uhvs1 mrlqw0olih, dqqxlw| lv ghqrwhg @_E%_%₂_(? +uhvs1 @E%%2(?,> lw lv vlpso|

wkh pdwkhpdwlfdo h{shfwdwlrq ri qhw suhvhqw ydoxh ri wkh lqvxuhu*v sd|phqwv +vhh Jhuehu

;

(12)

+4<<8, iru ixuwkhu ghwdlov rq dfwxduldo qrwdwlrqv dqg frqfhswv,1 Ohw xv ghqrwh dv@^z_E%

%2(?dqg

@^z_E%_%₂_(? wkh fruuhvsrqglqj suhplxpv frpsxwhg rq wkh edvlv ri wkh lqghshqghqfh dvvxps0 wlrq iru wkh uhpdlqlqj olihwlphv1 Lq fdvh A_% dqgA_%₂ duh STG wkhq@_E%_%₂_(? @^z_E%

%2(? dqg

@E%%2(? @^z_E%_%₂_(? krog wuxh1 Vlplodu frqfoxvlrqv fdq eh rewdlqhg iru prvw vwdqgdug olih lqvxudqfh frqwudfwv pdnlqj wkh STG dvvxpswlrq ri sdudprxqw lpsruwdqfh1 Wklv kdv ehhq srlqwhg rxw e| Qruehuj +4<;<, dqg ixuwkhu dqdo|vhg e| Ghqxlw dqgFruqhw +4<<<,1

815 SORG

Sursrvlwlrq 814 qr pruh krogv li SORG lv vxevwlwxwhg iru STG1 Udwkhu/ wkh iroorzlqj uhvxow krogv wuxh1

Sursrvlwlrq 8151 Surylghg \ lv SORG/ wkh vwrfkdvwlf lqhtxdolw| S_?

't ?3S S_?

't^z krogv1

Frpsdulqj Sursrvlwlrqv 814 dqg 815/ zh vhh wkdw fy lv uhsodfhg zlwk ?3lfy lq glphq0 vlrq?1 Ehvlghv/ dv lw fdq eh vhhq iurp Sursrvlwlrq 815/ zh rqo| jhw zhdnhu rughulqjv lq kljkhu glphqvlrqv1 Wr jhw _fy dv lq Sursrvlwlrq 814/ zh qhhg dqrwkhu ghshqghqfh qrwlrq fdooohg srvlwlyh fxpxodwlyh ghshqghqfh +SFG lq vkruw, dqg ghqhg dv iroorzv= wkh udqgrp yduldeohv tc t2c c t? duh SFG li wkh udqgrp frxsohv ES₃

'tc t duh STG iru ' 2cc c ?1 Wkh iroorzlqj uhvxow lv lqvsluhg iurp Ghqxlw/ Gkdhqhdqg Uledv +5334,1

Sursrvlwlrq 8161 Surylghg\ lv SFG/ wkh vwrfkdvwlf lqhtxdolw|S_?

't fy S_?

't^z krogv1 AIurp +717, dqg +718,/ lw lv hdv| wr jhw wkh iroorzlqj uhvxow wkdw uhlqirufhv d vwrfkdvwlf lqhtxdolw| rewdlqhg e| Edffhooldqg Pdnrzvnl+4<;<,1

Sursrvlwlrq 8171 Ohw 7 eh d vxevhw ri ic2c c ?j1 Surylghg \ lv SRG/ wkh vwrfkdvwlf lqhtxdolwlhv 4?_M7t^z_g 4?_M7t dqg 4@ _M7t _g 4@ _M7t^z erwk krog1

Ohw xv looxvwudwh wkh lqwhuhvw ri Sursrvlwlrq 817 lq olih lqvxudqfh1 Frqvlghu ? lqglylgxdov djhg%c %₂c c %_?/ uhvshfwlyho|/ zlwk uhpdlqlqj olihwlphvA_%/A_%₂/ /A_%_?/ uhvshfwlyho|1 Wkh mrlqw olih vwdwxvE%c %2c c %?h{lvwv dv orqj dv doo lqglylgxdo vwdwxvhv h{lvw1 Wklv vwdwxv kdv uhpdlqlqj olihwlph=

A_E%_c%₂_cc%_? ' 4?iA_%c A_%₂c c A_%_?j

Wkh odvw vxuylyru vwdwxv E%c %2c c %?h{lvwv dv orqj dv dw ohdvw rqh ri wkh lqglylgxdo vwdwxv lv dolyh1 Lwv uhpdlqlqj olihwlph lv jlyhq e|

A_E%_c%₂_cc%_?' 4@ iA_%c A_%₂c c A_%_?j

Ohw xv qrz dvvxph wkdwA ' EA%c A%2c c A%? lv SRG1 Ohw xv dovr lqwurgxfh wkh iroorzlqj vwudljkwiruzdug qrwdwlrq=

A_E%^z_c%₂_cc%_? ' 4?

A_%^zc A_%^z₂c c A_%^z_?

dqg A_E%^z

c%2cc%? ' 4@

A_%^zc A_%^z₂c c A_%^z_?

<

(13)

AIurp Sursrvlwlrq 817/ lw iroorzv wkdw

A_E%^z_c%₂_cc%_?_g A_E%_c%₂_cc%_? dqgA_E%_c%₂_cc%_?_g A_E%^z_c%₂_cc%_?c zklfk lq wxuq lpsolhv wkdw

@^z_E%_c%₂_cc%_?_(?@_E%_c%₂_cc%_?_(? dqg@_E%_c%₂_cc%_?_(?@^z_E%_c%₂_cc%_?_(?c

zkhuh wkh vxshuvfulsw B% lv xvhg wr lqglfdwh wkdw wkh dqqxlw| lv edvhg rq A_E%^z_c%₂_cc%_? ru A_E%^z

c%2cc%?1 Wklv phdqv wkdw iru SRG uhpdlqlqj olihwlphv/ wkh lqghshqghqfh dvvxpswlrq +zkloh ohdylqj wkh pdujlqdo fgi*v xqfkdqjhg, ohdgv wr dq xqghuhvwlpdwlrq ri wkh qhw vlqjoh suhplxp +dqg uhvhuyhv, ri d mrlqw olih dqqxlw|1 Wkh rssrvlwh frqfoxvlrq krogv iru wkh odvw vxuylyru dqqxlw|1 Vlplodu frqfoxvlrqv fdq eh gudzq iru hqgrzphqw dqg zkroh olih lqvxudqfh1

9 K|srwkhvhv whvwlqj

Qrz wkdw wkh uhohydqw wkhruhwlfdo frqfhswv dqg dssolfdwlrqv kdyh ehhq suhvhqwhg/ zh pd|

wxuq rxu dwwhqwlrq wr lqihuhqfh1 Zh ghyhors wzr whvwlqj phwkrgv1 Wkh uvw rqh lv edvhg rq d vshflfdwlrq lq whupv ri glvwulexwlrq ixqfwlrqv/ zkloh wkh vhfrqg rqh uholhv rq frsxodv1

914 Lqihuhqfh edvhg rq glvwulexwlrq ixqfwlrqv

Dv lq wudglwlrqdo vwrfkdvwlf grplqdqfh whvwv zh xvh d yhuvlrq ri wkh frqglwlrqv ghqlqj STG dqg SORG rq d suhghwhuplqhg julg/ dqg rqo| frqvlghu d {hg qxpehu ri glvwlqfw srlqwv/ vd|

_ srlqwv1 Lq dfwxduldo vflhqfh/ wkhvh srlqwv zloo fryhu wkh zkroh udqjh ri srvvleoh orvvhv1 Wkh gluhfw lqvxuhu pd| ghvluh uhvruwlqj wr d wuxqfdwhg glvwulexwlrq zkhq uhlqvxudqfh kdv ehhq erxjkw/ zkloh wkh uhlqvxuhu pd| zdqw wr uhvwulfw lwv dwwhqwlrq wr wkh frqglwlrqdo glvwulexwlrq ri h{fhvvhv ryhu d kljk wkuhvkrog1 Li vshfldo dwwhqwlrq lv sdlg wr wkh mrlqw rffxuuhqfh ri odujhv orvvhv/ wkh julg rxjkw wr eh uhqhg lq wkhvh uhjlrqv1

Ohw xv vwduw zlwk wkh ghqlwlrq +717, ri SORG lq whupv ri fgi*v/ dqg wdnh _ srlqwv

| ' E+c c +? lq U^?1 Zh ghqh (₈ ' 8E|T_?

'8E+/ dqg G8 ' E(₈c c (^_₈1 Wkh qxoo k|srwkhvlv ri d whvw iru SORG pd| eh zulwwhq dv

M₈^f 'iG₈ GG₈ fjc dqg zh wdnh dv dowhuqdwlyh k|srwkhvlv=

M₈ ' iG8 GG8 xquhvwulfwhgj

Wr h{dplqh wkhvh k|srwkhvhv zh zloo xvh wkh xvxdo glvwdqfh whvwv iru lqhtxdolw| frqvwudlqwv/

lqlwldwhg lq wkh pxowlyduldwh rqh0vlghg k|srwkhvlv olwhudwxuh iru srvlwlylw| ri wkh phdq +Eduwkrorphz +4<8<d/e,,1

Zh pd| dovr frqvlghu d whvw iru qrq0SORG edvhg rq wkh qxoo k|srwkhvlv=

M7₈^f 'iG8 G(₈ f iru vrph jc 43

(14)

dqg wkh dowhuqdwlyh k|srwkhvlv=

M7₈ 'iG₈ G(₈ :f iru doo j

Wkhvh k|srwkhvhv zloo eh whvwhg wkurxjk lqwhuvhfwlrq0xqlrq whvwv edvhg rq wkh plqlpxp ri d |0vwdwlvwlf1

Erwk whvwlqj surfhgxuhv zloo eh exlow iurp wkh hpslulfdo frxqwhusduw (₈ ri (₈ rewdlqhg e| vxevwlwxwlqj wkh hpslulfdo glvwulexwlrqv iru wkh xqnqrzq glvwulexwlrqv1 Wkh mrlqw dqg lqglylgxdo hpslulfdo glvwulexwlrqv duh jlyhq e|

8E| ' A

[A

|'

\? '

Ldt| +oc ' c c _c +914,

8E+ ' A

[A

|'

Ldt| +oc ' c c _c ' c c ? +915, Ohw xv ghqh |_&2, ' E+&a+,c c +&?a+,? zkhuh @aK' 4?E@c K1 Wkhq wkh iroorzlqj sursrvlwlrq jlyhv wkh dv|pswrwlf glvwulexwlrq ri Gg₈1

Sursrvlwlrq 9141 Wkh udqgrp yhfwru s

AEGg₈ G₈ frqyhujhv lq glvwulexwlrq wr d _0 glphqvlrqdo qrupdo udqgrp yduldeoh zlwk phdq }hur dqg fryduldqfh pdwul{ Y₈ zkrvh hoh0 phqwv duh

8c&,'8E|_&2,8E|_&8E|_,c &c ,' c c _

D frqvlvwhqw hvwlpdwhYg8 ri Y8 fdq eh rewdlqhg e| uhsodflqj wkh xqnqrzq glvwulexwlrq 8 e| lwv hpslulfdo frxqwhusduw 81

915 Lqihuhqfh edvhg rq frsxodv

Ohw xv qrz surfhhg zlwk wkh dqdorjxrxv txdqwlwlhv zkhq zh xvh frsxodv/ dqg wdnh _ srlqwv x ' Ec c ?/ zlwk 5 Efc/ ' c c _/ ' c c ?1 Wkh _ srlqwv fruuhvsrqg khuh wr suredelolw| ohyhov lqvwhdg ri uhwxuq ru orvv ohyhov1 Qrwh wkdw vrph uhjxodwruv lpsrvh wr wklqn lq whupv ri suredelolw| ohyhov lq sodfh ri uhwxuq ohyhov zkhq dvvhvvlqj qdqfldo ulvnv1 Fuhglw udwlqj dqdo|vlv lv dovr gluhfwo| olqnhg wr ghidxow suredelolw| ohyhov dqg qrw orvv ohyhov1 Wklv h{sodlqv zk| zh ghyhors erwk w|shv ri lqihuhqfh lq wklv sdshu1

Zh pd| wkhq ghqh( 'ExT_?

'/ dqgG ' E(c c (^_1 Dv lq wkh suhylrxv olqhv zh pd| frqvlghu wkh qxoo k|srwkhvlv iru d whvw iru SORG=

M^f 'iG GG fjc wrjhwkhu zlwk wkh dowhuqdwlyh k|srwkhvlv=

M 'iG GG xquhvwulfwhgjc zkloh wkh whvw iru qrq0SORG fdq eh edvhg rq wkh qxoo k|srwkhvlv=

M7^f 'iG G( f iru vrph jc 44

(15)

zlwk M7 'iG G( :f iru doo jc dv dowhuqdwlyh k|srwkhvlv1

Zh dvvxph khuhdiwhu wkdw doo fgi duh frqwlqxrxv/ dqg wkdw wkh fgi 8 ri t_|/ lv vxfk wkdw wkh htxdwlrq8E+ ' dgplwv d xqltxh vroxwlrq ghqrwhg l/ ' c c _/ ' c c ?/ zkloh sEl:f dw hdfk txdqwlohl1

Lq ylhz ri +615, zh pd| wklqn ri hvwlpdwlqj Ex ' 8E e| Ex ' 8E zkhuh ' E lc cl? lv pdgh ri wkh hpslulfdo xqlyduldwh txdqwlohvl1 Wkh pdlq glhuhqfh zkhq frpsduhg zlwk +914, lv wkdw wkh ohyhov duh qr pruh jlyhq ghwhuplqlvwlf ydoxhv/ exw txdqwlohv hvwlpdwhg rq wkh edvlv ri vdpsoh lqirupdwlrq/ dqg wkxv udqgrp txdqwlwlhv1 Dv zh zloo vhh lq d prphqw wklv voljkwo| frpsolfdwhv pdwwhuv/ exw rqh riwhq suhihuv +ru lv lpsrvhg, wr zrun zlwk suhghwhuplqhg suredelolw| ohyhov lqvwhdg ri orvv ohyhov1

Ohw xv sxw_&2_,' El_&c c l_&al_,c c l_&?/_&2,c ' E_&a_,/ dqg8₂E_&2, 'dt El&al,c t2 El&2al,2o/c 2 ' c c ?c 9'21 Wkhq wkh iroorzlqj sursrvlwlrq jlyhv wkh dv|pswrwlf glvwulexwlrq riGg1

Sursrvlwlrq 9151 Wkh udqgrp yhfwru s

AEGg G frqyhujhv lq glvwulexwlrq wr d _0 glphqvlrqdo qrupdo udqgrp yduldeoh zlwk phdq }hur dqg fryduldqfh pdwul{ Y zkrvh hoh0 phqwv duh

_c&, 'e_&D_&,e_,c &c , ' c c _c zkhuh

e '

#

^{Y8 E}_Y%

sEl ^{Y8 E}_Y%_? s?El?

$

c ' c c _c dqg

D&, ' 3 EE EC

8E_&2,8E_&8E_, 8E_&2_,8E_&, 8E_&2_?_,8E_&,?

8E_,2_&8E_,_& _&2,c_&_, 8_?E_&2,_&_,?

111 111 111 111

8E_,2_?_&8E_,_&? 8_?E_&2,_,_&? _&2,c?_&?_,?

4 FF FD

Wkh dv|pswrwlf fryduldqfh pdwul{ Y lqyroyhv ghulydwlyhv ri 8 dqg wkh xqlyduldwh ghq0 vlwlhv s1 Wkhvh txdqwlwlhv pd| eh hvwlpdwhg e| vwdqgdug nhuqho phwkrgv +vhh h1j1 Vfrww +4<<5,, lq rughu wr gholyhu d frqvlvwhqw hvwlpdwh Yg ri Y1 Iru h{dpsoh zh pd| wdnh d Jdxvvldq nhuqho dqg glhuhqw edqgzlgwk ydoxhv lq hdfk glphqvlrq/ zklfk ohdgv wr=

Y8Ee

Y% ' EA ³[^A

|'

)

#t_|l

$ _?

\

,'

x

#t_,|l_, ,

$ c sE l ' EA ³

[A

|'

)

#t|l

$ c

zkhuh ) dqg x ghqrwh wkh sgi dqg fgi ri d vwdqgdug Jdxvvldq yduldeoh1 Lq wkh hpslulfdo vhfwlrq ri wkh sdshu/ zh rsw iru wkh vwdqgdug fkrlfh +uxoh ri wkxpe, iru wkh edqgzlwkv / wkdw lv fDA^3*D wlphv wkh hvwlpdwhg vwdqgdug ghyldwlrq ri t1

45

(16)

916 Whvwlqj surfhgxuh

Wkh glvwulexwlrqdo uhvxowv ri Sursrvlwlrqv 914 dqg 915 duh wkh exloglqj eorfnv ri wkh whvwlqj surfhgxuhv1 Wkh uvw whvwlqj surfhgxuh frqvlghuv M_f⁸ +uhvs1 M_f, djdlqvw M⁸ +uhvs1 M, dqg pdnhv xvh ri glvwdqfh whvwv1 Lw zloo eh uhohydqw zkhq rqh ru vhyhudo frpsrqhqwv riGg_g duh irxqg wr eh qhjdwlyh +lq vxfk d fdvh rqh zdqwv wr nqrz zkhwkhu wklv lqydolgdwhv SORG,1

Ohw Ghg/g '8c / eh vroxwlrq ri wkh frqvwudlqhg txdgudwlf plqlplvdwlrq sureohp=

?uG AEGGggYgg3

EGGgg r| G fc +916,

zkhuh Ygg lv d frqvlvwhqw hvwlpdwh riYg/ dqg sxw 1g 'AEGhgGggYgg3

EGhgGgg

Urxjko| vshdnlqj/ Gjg lv wkh forvhvw srlqw wr Ggg xqghu wkh qxoo lq wkh glvwdqfh phdvxuhg lq wkh phwulf ri Yeg/ dqg wkh whvw vwdwlvwlf 1eg lv wkh glvwdqfh ehwzhhq Gjg dqg Ggg1 Wkh lghd lv wr uhmhfw M_f^g zkhq wklv glvwdqfh ehfrphv wrr odujh1

Wkh dv|pswrwlf glvwulexwlrq ri 1g xqghu wkh qxoo +vhh h1j1 Jrxulìurx{/ Kroo| dqg Prqiruw +4<;5,/Nrggh dqgSdop +4<;9,/Zrodn +4<;<d/e,, lv vxfk wkdw iru dq| srvl0 wlyh %=

d 1_g %o '[^_

'

d² %oE_c _cYg_g nJEc

zkhuh wkh zhljkw E_c _ cYgg lv wkh suredelolw| wkdw Ghg kdv h{dfwo| _ srvlwlyh hohphqwv1

Frpsxwdwlrq ri wkh vroxwlrq Ghg fdq eh shuiruphg e| d qxphulfdo rswlplvdwlrq urx0 wlqh iru frqvwudlqhg txdgudwlf surjudpplqj sureohpv dydlodeoh lq prvw vwdwlvwlfdo vriwzduh1 Forvhg irup vroxwlrq iru wkh zhljkwv duh dydlodeoh iru _ e +Nxgr +4<96,,1 Iru kljkhu gl0 phqvlrqv rqh xvxdoo| uholhv rq d vlpsoh Prqwh Fduor whfkqltxh dv dgyrfdwhg lqJrxulìurx{/

Kroo|dqgPrqiruw+4<;5, +vhh dovrZrodn+4<;<d,,1 Lqghhg lw lv hqrxjk wr gudz d jlyhq odujh qxpehu ri uhdolvdwlrqv ri d pxowlyduldwh qrupdo zlwk phdq }hur dqg fryduldqfh pdwul{

Yg_g1 Wkhq xvh wkhvh uhdolvdwlrqv dv Gg_g lq wkh deryh plqlplvdwlrq sureohp +916,/ frpsxwh Ghg/ dqg frxqw wkh qxpehu ri hohphqwv ri wkh yhfwru juhdwhu wkdq }hur1 Wkh sursruwlrq ri gudzv vxfk wkdw Gh_g kdv h{dfwo| _ hohphqwv juhdwhu wkdw }hur jlyhv d Prqwh Fduor hvwlpdwh ri E_c _cYgg1 Li rqh zlvkhv wr dyrlg wklv frpsxwdwlrqdo exughq/ wkh xsshu dqg orzhu erxqg fulwlfdo ydoxhv ri Nrggh dqg Sdop +4<;9, fdq eh dgrswhg1

Ohw xv qrz wxuq rxu dwwhqwlrq wr wkh vhfrqg whvwlqj surfhgxuh dlphg wr whvw M7_f⁸/ uhvs1 M7_f/ djdlqvw M7⁸/ uhvs1 M7/ dqg uho|lqj rq wkh lqwhuvhfwlrq0xqlrq sulqflsoh1 Lw zloo eh xvhg zkhq doo wkh frpsrqhqwv riGeg duh irxqg wr eh srvlwlyh1 Wkh txhvwlrq lv wkhq zkhwkhu wklv vx!fhv wr hqvxuh SORG1

Ohw _g 's

A(_g*s

gc/ g '8c 1 Wkhq xqghu M7_g^f/ wkh olplw ri d?u _g : 53ko zloo eh ohvv ru htxdo wr kc dqg h{dfwo| htxdo wr k li (_g ' f iru d jlyhq dqg (_g^, :f iru , 9'/ zkloh lwv olplw lv rqh xqghu M7_g1 Khqfh wkh whvw frqvlvwlqj ri uhmhfwlqj M7_g^f zkhq ?u _g lv deryh wkh Ek0txdqwloh 53k ri d vwdqgdug qrupdo glvwulexwlrq kdv dq xsshu erxqg k rq wkh dv|pswrwlf vl}h +vhh h1j1 Krzhv +4<<6,/Ndxu/Sudndvd Udr dqgVlqjk +4<<7,,1

46

(17)

Srzhu lvvxhv duh h{whqvlyho| vwxglhg iru vwrfkdvwlf grplqdqfh dqg qrqgrplqdqfh whvwv lq Gdugdqrql dqg Iruflqd +4<<<, +vhh dovr wkh frpphqwv lq Gdylgvrq dqg Gxforv +5333,,1 Wkh| fduu| ryhu wr rxu fdvh1 Iluvw/ dssurdfkhv edvhg rq glvwdqfh whvwv h{sorlw wkh fryduldqfh vwuxfwxuh/ dqg duh wkxv h{shfwhg wr dfklhyh ehwwhu srzhu surshuwlhv uhodwlyh wr dssurdfkhv/ vxfk dv rqhv edvhg rq |0vwdwlvwlfv/ wkdw gr qrw dffrxqw iru lw1 Lq d vhw ri Prqwh Fduor h{shulphqwv/ wkh| qg wkdw/ lqghhg/ glvwdqfh whvwv duh zruwk wkh h{wud dprxqw ri frpsxwdwlrqdo zrun1 Vhfrqg/ lw lv srvvleoh wkdw qrquhmhfwlrq ri wkh qxoo ri grplqdqfh/ khuh SORG/ e| glvwdqfh whvwv rffxuv dorqj zlwk wkh qrquhmhfwlrq ri wkh qxoo ri qrqgrplqdqfh/

khuh qrq0SORG/ e| lqwhuvhfwlrq0xqlrq whvwv1 Wklv lv gxh wr wkh kljko| frqvhuydwlyh qdwxuh ri wkh odwwhu/ dqg zloo w|slfdoo| rffxu lq rxu vhwwlqj liGgg lv forvh hqrxjk wr }hur iru d qxpehu ri frruglqdwhv1

: Hpslulfdo looxvwudwlrqv

Wklv vhfwlrq looxvwudwhv wkh lpsohphqwdwlrq ri wkh whvwlqj surfhgxuhv ghvfulehg lq wkh suhylrxv vhfwlrq1 Zh surylgh wzr hpslulfdo dssolfdwlrqv wr lqvxudqfh1 Wkh| frqfhuq wkh ghwhfwlrq ri STG lq XV dqg Gdqlvk lqvxudqfh fodlp gdwd/ dqg lwv hhfw rq suhplxp ydoxdwlrq1

:14 XV Orvvhv dqg DODH*v

Ydulrxv surfhvvhv lq fdvxdow| lqvxudqfh lqyroyh fruuhodwhg sdluv ri yduldeohv1 D surplqhqw h{dpsoh lv wkh orvv dqg doorfdwhg orvv dgmxvwphqw h{shqvhv +DODH/ lq vkruw, rq d vlqjoh fodlp1 Khuh DODH duh w|sh ri lqvxudqfh frpsdq| h{shqvhv wkdw duh vshflfdoo| dwwulexwdeoh wr wkh vhwwohphqw ri lqglylgxdo fodlpv vxfk dv odz|huv* ihhv dqg fodlpv lqyhvwljdwlrq h{shqvhv1 Wkh mrlqw prghoolqj lq sdudphwulf vhwwlqjv ri wkrvh wzr yduldeohv kdv ehhq h{dplqhg e| Iuhhv dqg Ydogh} +4<<;,/ dqg Noxjpdq dqg Sduvd +4<<<,1 Wkh gdwd xvhg lq wkhvh hpslulfdo vwxglhv zhuh froohfwhg e| wkh XV Lqvxudqfh Vhuylfhv R!fh/ dqg frpsulvh jhqhudo oldelolw|

fodlpv udqgrpo| fkrrvhq iurp odwh vhwwohphqw odjv1 Iuhhv dqgYdogh} +4<<;, fkrrvh wkh Sduhwr glvwulexwlrq wr prgho wkh pdujlqv dqg vhohfw Jxpeho dqg Iudqn frsxodv +rq wkh edvlv ri d judsklfdo surfhgxuh vxlwdeoh iru Dufklphghdq frsxodv,1 Erwk prghov h{suhvv STG e| wkhlu hvwlpdwhg sdudphwhu ydoxhv1 Noxjpdq dqg Sduvd +4<<<, rsw iru wkh Lqyhuvh Sdudorjlvwlf iru wkh orvvhv dqg iru wkh Lqyhuvh Exuu iru DODH*v1 Wkh| xvh Iudqn*v frsxod1 Djdlq/ wkh hvwlpdwhg ydoxh ri wkh ghshqghqfh sdudphwhu hqwdlov STG iru orvvhv dqg DODH*v1 Lq wkh iroorzlqj zh uho| rq d qrqsdudphwulf dssurdfk wr dvvhvv STG1 Wklv dvvhvvphqw kdv pdq| lpsolfdwlrqv lq lqvxudqfh/ iru h{dpsoh/ iru wkh frpsxwdwlrq ri uhlqvxudqfh suhplxpv +zkhuh wkh vkdulqj ri h{shqvhv ehwzhhq wkh fhglqj frpsdq| dqg wkh uhlqvxuhu kdv wr eh ghflghg rq, dqg iru wkh ghwhuplqdwlrq ri wkh h{shqvh ohyho iru d jlyhq orvv ohyho +iru uhvhuylqj dq dssursuldwh dprxqw wr fryhu ixwxuh vhwwohphqw h{shqvhv,1

Wkh gdwd frqvlvw lq A ' ceSS xqfhqvruhg revhuyhg ydoxhv ri wkh sdlu +ORVV/DODH,1 Vrph vxppdu| vwdwlvwlfv duh jdwkhuhg lq Wdeoh :141 Wkh hvwlpdwhg ydoxhv iru Shduvrq*v o/ Nhqgdoo*v dqg Vshdupdq*v 4 duh 316;38/ 31639: dqg 31776:/ uhvshfwlyho|1 Doo ri wkhp duh mxgjhg vljqlfdqwo| srvlwlyh dw 4(1 Ehfdxvh vrph yhu| kljk ydoxhv ri wkh yduldeohv duh frqwdlqhg lq wkh gdwd vhw/ zh zloo zrun rq d orjdulwkplf vfdoh1 Wklv zloo qrw dowhu wkh uhvxowv ri rxu dqdo|vlv vlqfh +ORVV/DODH, STG / +orj+ORVV,/orj+DODH,, STG1 Qrwh wkdw wkh

47

(18)

wudqvirupdwlrq ri wkh pdujlqv uhvxowv lq d qhz Shduvrq*vo +olqhdu fruuhodwlrq frh!flhqw, ri 317646/ zkloh Nhqgdoo*v dqg Vshdupdq*v ydoxhv duh ohiw xqfkdqjhg1 Wkhvh duh qrw dhfwhg e| vwulfwo| lqfuhdvlqj wudqvirupdwlrq ri wkh yduldeohv1

ORVV DODH

Phdq 6:/43<18; 45/34:17:

Vwg Ghy1 <5/8451;3 59/:45168

Vnhz1 431<8 4313:

Nxuw1 53<195 48516<

Plq 43133 48133

Pd{ 5/4:6/8<8133 834/;96133 4vw Txduw1 6/:83133 5/64;158 Phgldq 44/37;183 8/753183 6ug Txduw1 65/333133 45/5<5133

Wdeoh :14= Vxppdu| vwdwlvwlfv iru yduldeohv ORVV dqg DODH1

Iljxuh :14 vkrzv wkh nhuqho hvwlpdwru ri wkh elyduldwh sgi ri wkh frxsoh +orj+ORVV,/orj+DODH,,/ wrjhwkhu zlwk lwv frqwrxu sorw1 Wklv hvwlpdwlrq uholhv rq d surgxfw ri Jdxvvldq nhuqhov dqg edqgzlgwk ydoxhv vhohfwhg e| wkh vwdqgdug uxoh ri wkxpe +Vfrww +4<<5,,1 Wkh judskv reylrxvo| vxjjhvw vwurqj srvlwlyh ghshqghqfh ehwzhhq erwk yduldeohv1

Lq rughu wr whvw zkhwkhu STG krogv rq wkh zkroh revhuydwlrq grpdlq/ zh wdnh 7< srlqwv frplqj iurp wkh htxdoo| vsdfhg julg iSc.c c2j iSc.c c2j1 Wklv ohdgv wr d yhfwru Gg8 zlwk rqo| rqh qhjdwlyh frpsrqhqw ffff21 Zh zlvk wr fkhfn zkhwkhu wklv lqydolgdwhv STG ru qrw1 Wkh glvwdqfh ehwzhhqGj8 dqgGg8 lv irxqg wr ehSDf³²1 Orzhu erxqgv rq wkh fulwlfdo ydoxhv rewdlqhg e| Nrggh dqgSdop +4<;9, duh jlyhq lq Wdeoh :15 iru glhuhqw ohyhov k1 Qrwh wkdw wkh| gr qrw ghshqg rq wkh julg vl}h _1 Lq ylhz ri wkhvh erxqgv zh gr qrw uhmhfw wkh qxoo ri STG dw dq| uhdvrqdeoh frqghqfh ohyho1

k 58( 43( 8( 518( 4( 318( 314(

Orzhu erxqg 31788 41975 51:39 61;74 81745 91968 <1833 Wdeoh :15= Orzhu erxqgv rq fulwlfdo ydoxhv iru wkh glvwdqfh whvw1

Ohw xv qrz frqvlghu d srvlwlyh ghshqghqfh/ exw rqo| lq wkh xsshu wdlov1 Zh wdnh wkh julg ifcfcfScccSc2jifcfcfScccSc2j1 Doo 7< frpsrqhqwv riGg8

duh vwulfwo| srvlwlyh/ zklfk phdqv wkdw M_f⁸ lv dxwrpdwlfdoo| qrw uhmhfwhg1 Wkh lqwhuvhfwlrq0 xqlrq whvw pd| wkhq eh xvhg wr nqrz zkhwkhu wkh gdwd h{klelw STG1 Zh jhw4? ₈ ' fDD zklfk grhv qrw doorz xv wr uhmhfw M⁸_f lq idyru ri STG1 Wklv qrq uhmhfwlrq lv gxh wr wkh forvhqhvv ri gG8 wr }hur iru d odujh qxpehu ri frruglqdwhv1 Wklv srlqw kdv douhdg| ehhq glvfxvvhg dw wkh hqg ri Vhfwlrq 91

Ohw xv qrz wxuq wr frsxod edvhg whvwv1 Iru wkh x*v/ zh wdnh wkh ;4 ghflohv ri wkh julg ifcf2c cfbjifcf2c cfbj1 Doo frpsrqhqwv ri wkh fruuhvsrqglqjGe duh srvlwlyh/

vr wkdw M_f fdqqrw eh uhmhfwhg1 Iru wkh lqwhuvhfwlrq0xqlrq whvw/ zh rewdlq 4?e ' fbeHb 48

(19)

4 6

8 10

12 14

LogLOSS 4

6 8 10 12

LogALAE

00.020.040.060.08

Density function

LogLOSS

LogALAE

4 6 8 10 12 14

4681012

75 50 25

Iljxuh :14= Nhuqho hvwlpdwlrq ri wkh elyduldwh sgi iru +orj+ORVV,/orj+DODH,,1

49

(20)

zklfk grhv qrw doorz xv wr uhmhfw M_f lq idyru ri M1 Li zh irfxv rq wkh wdlov/ wdnlqj wkh kljk shufhqwlohv lq ifbcfb2c cfbbj ifbcfb2c cfbbj/ zh jhw wkdw doo frpsrqhqwv ri Ge duh djdlq srvlwlyh uhvxowlqj lq wkh qrq0uhmhfwlrq ri M_f1 Ixuwkhu/4?e ' fSbH/ vr wkdw M_f lv qrw uhmhfwhg/ hlwkhu1

Lw kdv wr eh srlqwhg rxw wkdw wkh fkrlfh ri wkh edqgzlwk kdv yhu| olwwoh lpsdfw rq wkh ydoxhv ri wkh whvw vwdwlvwlfv1 Wkh| kdyh ehhq frpsxwhg zlwk kdoi/ wzlfh dqg wkuhh wlphv wkh vwdqgdug fkrlfh/ dqg wklv kdv rqo| uhvxowhg lq vpdoo yduldwlrqv1

Ohw xv qrz glvfxvv sudfwlfdo lpsolfdwlrqv ri wkh suhvhqfh ri STG lq wkh suhylrxv gdwd1 Zh orrn dw wkh lpsdfw rq suhplxp ydoxdwlrq lq uhlqvxudqfh wuhdwlhv1 Zh frqvlghu d uhlqvxudqfh wuhdw| rq d srolf| zlwk xqolplwhg oldelolw| dqg lqvxuhu*v uhwhqwlrq -1 Dvvxplqj d surudwd vkdulqj ri h{shqvhv/ wkh uhlqvxuhu*v sd|phqw iru d jlyhq uhdol}dwlrq ri +ORVV/DODH, lv ghvfulehg e| wkh ixqfwlrq

}EORVV/DODH '

+ f li ORVV-c ORVV-n ORVV0-

ORVV DODH li ORVV: - Wkh sxuh suhplxp uhodwlqj wr wklv uhlqvxudqfh wuhdw| lv

Z '.d}EORVV/DODHo

Wkh uhvxowv lq Wdeoh :16 surylgh wkh suhplxpv wkh uhlqvxuhu zrxog kdyh dvvhvvhg wr fryhu frvwv ri orvvhv dqg h{shqvhv dffruglqj wr ydulrxv lqvxuhu*v uhwhqwlrq1 Wkuhh vlwxdwlrqv kdyh ehhq frqvlghuhg=

41 wkh uvw rqh dvvxphv lqghshqghqfh/ l1h1 e

Z ' A²

[A

|'

[A

|'

}EORVV|cDODH|(

51 wkh vhfrqg rqh wdnhv lqwr dffrxqw wkh ghshqghqfh h{suhvvhg e| wkh gdwd/ l1h1 e

Z ' A

[A

|'

}EORVV_|cDODH_|(

61 dqg wkh odvw rqh uhvruwv wr wkh fodvvlfdo frprqrwrqlf dssur{lpdwlrq iru +ORVV/DODH,/

l1h1

e Z '

A [A

|'

}

ORVV_|c8e₂³

8eEORVV_|

Zh vhh wkdw vxevwdqwldo plvsulflqj frxog uhvxow iurp wkh lqghshqghqfh k|srwkhvlv/ zkloh wkh frprqrqrwlf dssur{lpdwlrq lv wrr frqvhuydwlyh1 Zh vhh wkdw lqghshqghqfh jhqhudwhv orzhu suhplxpv wkdq wkrvh vxjjhvwhg e| wkh gdwd/ zklfk wkhpvhoyhv duh vpdoohu wkdq wkrvh edvhg rq wkh frprqrwrqlf dvvxpswlrq +dv wkh| duh wkhruhwlfdoo| erxqg wr eh xqghu STG,1

4:

(21)

- 43/333 83/333 433/333 833/333 4/333/333 lqghs1 66/63;1<387 4</43;16937 45/7351:848 4/;331<<;7 ;371<9;7 ghs1 69/:981;9;: 54/55:1;3:4 46/;3414<5: 4/;:8135:: ;83149;9 frprq1 6;/<9519:67 56/5:414<3; 48/73:1::;5 5/63;1346< <;816;34

Wdeoh :16= Sxuh suhplxpv iru d uhlqvxudqfh wuhdw| zlwk uhwhqwlrq-1

:15 Gdqlvk uh orvvhv

Wkhvh gdwd frpsulvh orvvhv ryhu rqh ploolrq Gdqlvk Nurqh iru wkh |hduv 4<;304<;<1 Orvv jxuhv duh fodvvlhg dv gdpdjh wr exloglqjv +yduldeoh Exloglqjv,/ dqg gdpdjh wr ixuqlwxuh dqg shuvrqdo surshuw| +yduldeoh Frqwhqwv,/ dqg frqvlvw lqA ' ceHDgdwd srlqwv1 Wrwdo orvvhv pdgh ri wkh vxp ri erwk orvvhv kdyh ehhq suhylrxvo| vwxglhg lq wkh frqwh{w ri wdlo dqdo|vlv e|

Hpeuhfkwv/NoxsshoehujdqgPlnrvfk+4<<:,/PfQhlo+4<<:,/ dqgVfdloohw+5333,1 Wdeoh :17 jlyhv vxppdu| vwdwlvwlf iru wkh yduldeohv Exloglqjv% dqg Frqwhqwv%1 Wkh hvwlpdwhg ydoxhv iru Shduvrq*v o/ Nhqgdoo*v dqg Vshdupdq*v 4 duh 318695/ 313:74 dqg 3146;8/ uhvshfwlyho|1 Doo ri wkhp duh vljqlfdqwo| srvlwlyh1 Ehfdxvh yduldeohv wdnh vrphwlph yhu| kljk ydoxhv/ zh djdlq ghflgh wr zrun rq d orjdulwkplf vfdoh1 Diwhu wklv orjdulwkplf wudqvirupdwlrq/ wkh qhz Shduvrq*v o lv 3156481

Exloglqjv Frqwhqwv Phdq 4/:64/345 4/6<4/<:<

Vwg Ghy1 5/;75/84< 6/::9/46:

Vnhz1 451435 <139;

Nxuw1 54:18;: 4551649

Plq 58/333 43/333

Pd{ 98/333/333 :5/833/333 4vw Txduw1 ;33/333 583/333 Phgldq 4/433/333 763/333 6ug Txduw1 4/::8/333 4/333/333

Wdeoh :17= Vxppdu| vwdwlvwlfv iru yduldeohv Exloglqjv dqg Frqwhqwv1

Iljxuh :15 glvsod|v d nhuqho hvwlpdwh ri wkh elyduldwh sgi ri wkh frxsoh +orj+Exloglqjv,/orj+Frqwhqwv,,/ dv zhoo dv lwv dvvrfldwhg frqwrxu sorw1 Djdlq srvlwlyh gh0 shqghqfh lv h{shfwhg lq oljkw ri wkhvh judskv1 Wkh vkdsh ri wkh ghshqghqfh lv krzhyhu glhuhqw wkdq iru orvvhv dqg DODH*v1

Wdnlqj wkh htxdoo| vsdfhg julg ic2c c.j ic2c c.j ryhu wkh zkroh revhu0 ydwlrq grpdlq zh jhw 48 qhjdwlyh frpsrqhqwv iru Gg81 Wkh glvwdqfh ehwzhhq Gj8 dqgGg8

lv irxqg wr eh f1 Khqfh zh gr qrw uhmhfw wkh qxoo ri STG dw dq| uhdvrqdeoh frqghqfh ohyho +vhh wkh erxqgv ri Wdeoh :15,1

Ohw xv qrz h{dplqh wkh suhvhqfh ri srvlwlyh ghshqghqfh lq wkh xsshu wdlov1 Zh wdnh wkh julg iDcDcDScScScSSc.j iDcDcDScScScSSc.j1 Doo frpsrqhqwv ri Gg8 duh srvlwlyh1 Zh kdyh4? ₈ ' fD lq wkh lqwhuvhfwlrq0xqlrq whvw1 Wklv grhv qrw doorz

4;

(22)

12

14

16

log(Buildings) 10

12 14 16

log(Contents)

00.050.10.150.20.250.30.35

Density function

log(Buildings)

log(Contents)

10 12 14 16 18

1012141618

75 50 25

Iljxuh :15= Nhuqho hvwlpdwlrq ri wkh elyduldwh sgi iru +orj+Exloglqjv,/orj+Frqwhqwv,,1

4<