Plqlpxp yhklfoh hhw vl}h xqghu wlph zlqgrzfrqvwudlqwv dw d frqwdlqhu whuplqdoLulv I1D1 Ylv4

Iuhh Xqlyhuvlw| Dpvwhugdp/ Vfkrro ri Hfrqrplfv dqg Exvlqhvv Dgplqlvwudwlrq/Gh Erhohoddq 4438/ Urrp 6D064/ 43;4 KY Dpvwhugdp/ Wkh QhwkhuodqgvUhqì +P1, E1P1 gh Nrvwhu

Hudvpxv Xqlyhuvlw| Urwwhugdp/ Urwwhugdp Vfkrro ri Pdqdjhphqw2Idfxowhlw EhgulmivnxqghS1R1 Er{ 4:6;/ 6333 GU Urwwhugdp/ Wkh QhwkhuodqgvPduwlq Z1S1 Vdyhovehujk

Jhrujld Lqvwlwxwh ri Whfkqrorj|/ Vfkrro ri Lqgxvwuldo dqg V|vwhpv Hqjlqhhulqj/ Dwodqwd JD/XVD

Dsulo/ 5336Devwudfw

Surgxfwv fdq eh wudqvsruwhg lq frqwdlqhuv iurp rqh sruw wr dqrwkhu1 Dw d frqwdlqhu whuplqdowkhvh frqwdlqhuv duh wudqvvklsshg iurp rqh prgh ri wudqvsruwdwlrq wr dqrwkhu1 Fudqhv uhpryhfrqwdlqhuv iurp d vkls dqg sxw wkhp dw d fhuwdlq wlph +l1h1 uhohdvh wlph, lqwr d ex�hu duhdzlwk olplwhg fdsdflw|1 D yhklfoh oliwv d frqwdlqhu iurp wkh ex�hu duhd dw wkh vkls xowlpdwho|ehiruh wkh ex�hu duhd lv ixoo +l1h1/ lq gxh wlph, dqg wudqvsruwv wkh frqwdlqhu iurp wkh vkls wr

wkh vwrudjh duhd1 Dw wkh vwrudjh duhd wkh frqwdlqhu lv sodfhg lq d ex�hu duhd1 Wkh dgydqwdjh rixvlqj wkhvh ex�hu duhdv lv wkh uhvxowdqw ghfrxsolqj ri wkh xqordglqj dqg wudqvsruwdwlrq

surfhvvhv1 Zh vwxg| wkh fdvh lq zklfk hdfk frqwdlqhu kdv d wlph0zlqgrz ^uhohdvh wlph/ gxhwlph` lq zklfk wkh wudqvsruwdwlrq vkrxog vwduw1

Wkh remhfwlyh lv wr plqlplvh wkh yhklfoh hhw vl}h vxfk wkdw wkh wudqvsruwdwlrq ri hdfk4fruuhvsrqglqj dxwkru/ .640530777939:/ id{1 .6405307779338/ lylvCihzhe1yx1qo

4

frqwdlqhu vwduwv zlwklq lwv wlph0zlqgrz1 Qr olwhudwxuh kdv ehhq irxqg vwxg|lqj wklv uhohydqwsureohp1 Zh kdyh ghyhorshg dq lqwhjhu olqhdu surjudpplqj prgho wr vroyh wkh sureohp righwhuplqlqj yhklfoh uhtxluhphqwv xqghu wlph0zlqgrz frqvwudlqwv1 Zh xvh vlpxodwlrq wr

ydolgdwh wkh hvwlpdwhv ri wkh yhklfoh hhw vl}h e| wkh dqdo|wlfdo prgho1 Zh whvw wkh delolw| riwkh prgho xqghu ydulrxv frqglwlrqv1 Iurp wkhvh qxphulfdo h{shulphqwv zh frqfoxgh wkdw wkhuhvxowv ri wkh dqdo|wlfdo prgho duh forvh wr wkh uhvxowv ri wkh vlpxodwlrq prgho1 Ixuwkhupruh/zh frqfoxgh wkdw wkh dqdo|wlfdo prgho shuirupv zhoo lq wkh frqwh{w ri d frqwdlqhu whuplqdo1Lqwurgxfwlrq

Surgxfwv duh wudqvsruwhg lq frqwdlqhuv doo ryhu wkh zruog1 Vklsv duh xvhg wr wudqvsruwfrqwdlqhuv ryhu vhdv1 Wudqvsruw ryhu odqg fdq eh fduulhg rxw e| wudlqv ru wuxfnv1 Wr wudqvvklsfrqwdlqhuv iurp rqh prgh ri wudqvsruwdwlrq wr dqrwkhu/ sruwv dqg frqwdlqhu whuplqdov duh xvhg1

Vklsrzqhuv dvn iru vkruw ehuwk wlphv ri wkhlu vklsv dw wkh sruw wr hqvxuh wkdw wkhlu frqwdlqhuvduh gholyhuhg rq wlph1 Ixuwkhupruh/ d vkls dw d sruw lv pxfk pruh h{shqvlyh wkdq d vkls dw vhd1Wr uhvsrqg wr wkhvh ghpdqgv/ frqwdlqhu whuplqdov qhhg wr shuirup doo wkh xqordglqj/ ordglqj dqgwudqvvklsphqw rshudwlrqv txlfno| dqg h!flhqwo|1 Frqvhtxhqwo|/ dq h!flhqw whuplqdo pxvw hqvxuhwkdw frqwdlqhu vklsv duh xqordghg dqg ordghg txlfno|1 Gxh wr wkh lqwurgxfwlrq ri pxfk odujhuvklsv/ lw lv qhfhvvdu| wr ghyhors qhz whfkqltxhv dqg phwkrgv zklfk doorz h�hfwlyh frruglqdwlrqri doo wkh dfwlylwlhv dw d whuplqdo1 Lq ghvljqlqj d frqwdlqhu whuplqdo/ d qxpehu ri ghflvlrqsureohpv kdyh wr eh vroyhg1 Wkhvh sureohpv lqfoxgh wkh ghyhorsphqw ri sodqqlqj dqg frqwurofrqfhswv iru wkh gl�huhqw v|vwhpv lq wkh whuplqdo1 Dq h{whqvlyh ryhuylhz ri ydulrxv ghflvlrqsureohpv dw d frqwdlqhu whuplqdo lv jlyhq lq Ylv dqg Gh Nrvwhu +5336,1

Prvw ri wkh whuplqdov pdnh xvh ri pdqqhg htxlsphqw/ vxfk dv vwudggoh fduulhuv dqg pxowlwudlohu v|vwhpv1 Krzhyhu/ d ihz whuplqdov/ iru h{dpsoh/ dw wkh sruw ri Urwwhugdp/ duh vhpl0dxwrpdwhg1 Dxwrpdwhg jxlghg yhklfohv fdq eh xvhg iru wkh lqwhuqdo wudqvsruw ri frqwdlqhuv dw

5

vhpl0dxwrpdwhg whuplqdov1 Wkh xqordglqj dqg ordglqj surfhvv ri d vkls dw d+q, +dxwrpdwhg,frqwdlqhu whuplqdo fdq eh ghvfulehg dv iroorzv +vhh dovr Iljxuh 4,1

LQVHUW ILJXUH 4Zkhq d vkls duulyhv dw wkh sruw/ frqwdlqhuv kdyh wr eh uhpryhg iurp wkh vkls1 Wklv lv grqh

e| pdqqhg Txd| Fudqhv +TFv,/ zklfk uhpryh wkh frqwdlqhuv iurp wkh vkls*v krog dqg ghfndffruglqj wr dq xqordglqj sodq1 Fxuuhqwo|/ irxu fudqhv shuirup d frpprq xqordglqj rshudwlrq1Wkhvh irxu fudqhv rshudwh rq rqh vkls wr kdqgoh doo wkh xqordglqj dqg ordglqj ri frqwdlqhuv riwkdw vkls1 Xqordglqj d frqwdlqhu wdnhv d fhuwdlq xqordglqj wlph/ zklfk ydulhv shu frqwdlqhu1Ehwzhhq wkh vkls dqg wkh vwdfn +l1h1/ wkh sodfh zkhuh frqwdlqhuv duh vwruhg,/ yhklfohv vxfk dvvwudggoh fduulhuv/ ru Dxwrpdwhg Jxlghg Yhklfohv +DJYv, wudqvsruw frqwdlqhuv1 Wkhuh duh wzrsrvvlelolwlhv iru wkh wudqvihu ri frqwdlqhuv iurp wkh fudqh wr wkh wudqvsruw yhklfoh1 Iluvw/ wkhTF fdq sodfh wkh frqwdlqhu gluhfwo| rq wkh yhklfoh1 Vhfrqg/ wkh TF fdq sodfh wkh frqwdlqhulq d ex�hu duhd dqg dw d fhuwdlq prphqw ehiruh wkh ex�hu duhd lv ixoo d oliwlqj yhklfoh oliwv wkhfrqwdlqhu dqg wudqvsruwv lw wr wkh vwdfn1 Lq wklv sdshu zh revhuyh wklv vhfrqg srvvlelolw|1 Doliwlqj yhklfoh fduu|lqj rqh frqwdlqhu fdq gulyh ryhu dqrwkhu frqwdlqhu zklfk lv sodfhg rq wkhjurxqg1 Diwhu oliwlqj wkh frqwdlqhu/ wkh yhklfoh wudqvsruwv wkh frqwdlqhu iurp wkh vkls wr wkhvwrudjh +l1h1/ vwdfn,1 Zh dvvxph wkdw yhklfohv wudyho rq wudfnv lq d orrs ehwzhhq vwdfn dqgvkls1 Hdfk fudqh kdv lw rzq wudfn +vhh dovr Iljxuh 4,1 Li d yhklfoh duulyhv dw wkh vwdfn lw vhwvgrzq wkh frqwdlqhu lq wkh ex�hu duhd1 Vwdfnlqj fudqhv vwruh wkh frqwdlqhuv dw wkh vwdfn1 Diwhud fhuwdlq shulrg/ fudqhv uhwulhyh wkh frqwdlqhuv iurp wkh vwdfn dqg yhklfohv wudqvsruw wkhp wrrwkhu wudqvsruwdwlrq prghv/ vxfk dv edujhv1 Wr ordg h{sruw frqwdlqhuv rq d vkls wklv zkrohsurfhvv lv h{hfxwhg lq uhyhuvh rughu1

Zh frqvlghu d frqwdlqhu whuplqdo lq zklfk oliwlqj yhklfohv dqg ex�hu duhdv zlwk olplwhgfdsdflw| duh xvhg1 Frqvhtxhqwo|/ wkh +xq,0ordglqj surfhvv dqg wudqvsruwdwlrq surfhvv fdq eh

6

ghfrxsohg1 Txd| fudqhv uhpryh wkh frqwdlqhuv iurp wkh vkls dffruglqj wr dq xqordglqj sodqdqg sodfh wkhp lq ex�huv dw d fhuwdlq wlph1 Zh uhihu wr wklv wlph dv uhohdvh wlph1 Dv d uhvxow/wkh ruljlq txd| fudqh iru wudqvsruw iru hdfk frqwdlqhu lv nqrzq lq dgydqfh1 Oliwlqj yhklfohv oliwwkh frqwdlqhuv r� wkh jurxqg dqg wudqvsruw wkhp wr wkh vwdfn1 Dw wkh vwdfn/ ex�hu duhdv duhdovr orfdwhg dw wkh iurqw ri hdfk urz ri frqwdlqhuv1 Wkh yhklfoh sodfhv wkh frqwdlqhu lq wkh ex�hu1Diwhu d fhuwdlq shulrg ri wlph wkh vwdfnlqj fudqh vwruhv wkh frqwdlqhu lq wkh vwdfn1 Gxh wr wkhghfrxsolqj ri wkh surfhvvhv lq wkh whuplqdo/ wkh lpphgldwh wudqvsruwdwlrq ri frqwdlqhuv lv qrwuhtxluhg1 Krzhyhu/ gxh wr wkh olplwhg fdsdflw| ri wkh ex�huv hdfk frqwdlqhu kdv d �qdo wlphri wudqvsruw +l1h1/ lq gxh wlph,1 Frqvhtxhqwo|/ hdfk frqwdlqhu kdv d wlph0zlqgrz ^uhohdvh wlph/gxh wlph` lq zklfk wkh wudqvsruwdwlrq vkrxog vwduw1 Wkh sureohp lv ghwhuplqlqj wkh plqlpxpoliwlqj yhklfoh hhw vl}h xqghu wkhvh wlph0zlqgrz frqvwudlqwv1 Lq Vhfwlrq 4 zh jlyh dq ryhuylhz riolwhudwxuh rq ghwhuplqlvwlf prghov xvhixo iru ghwhuplqlqj yhklfoh uhtxluhphqwv1 Lw ehfrphv fohduwkdw qr prghov dqg dojrulwkpv h{lvw wr vroyh wkh sureohp ri ghwhuplqlqj yhklfoh uhtxluhphqwvxqghu wlph0zlqgrz frqvwudlqwv dw d frqwdlqhu whuplqdo1 Vhfwlrq 5 ghvfulehv dq lqwhjhu olqhdusurjudpplqj prgho wr ghdo zlwk wklv sureohp1 Wr vwxg| wkh shuirupdqfh ri wklv lqwhjhu olqhdusurjudpplqj prgho lq sudfwlfh zh kdyh fduulhg rxw vhyhudo qxphulfdo h{shulphqwv lq Vhfwlrq61 Iluvw/ zh fduu| rxw lq Vhfwlrq 614 d vwxg| zlwk wkh dqdo|wlfdo prgho wr ghwhuplqh yhklfohuhtxluhphqwv gxulqj wkh xqordglqj surfhvv dw d frqwdlqhu whuplqdo1 Wr ydolgdwh wkh uhvxowv riwkh dqdo|wlfdo prgho zh fduu| rxw d vlpxodwlrq vwxg| lq Vhfwlrq 6151 Lq wklv vlpxodwlrq prgho zhlqfrusrudwh doo wkh surfhvvhv zklfk rffxu gxulqj wkh xqordglqj ri d vkls1 Vhfwlrq 616 frpsduhvwkh uhvxowv iurp wkh dqdo|wlfdo dqg wkh vlpxodwlrq prgho1 Ilqdoo|/ Vhfwlrq 7 jlyhv frqfoxglqjuhpdunv1

7

4 Yhklfoh uhtxluhphqwvWkh plqlpxp qxpehu ri yhklfohv uhtxluhg lq wkh v|vwhp kdv wr eh ghwhuplqhg ehiruh wkh vwduwri wkh rshudwlrq1 Wr hqvxuh wkdw doo wkh wdvnv duh shuiruphg zlwklq wlph/ vx!flhqw yhklfohvkdyh wr eh dydlodeoh1 Krzhyhu/ iru hfrqrplfdo uhdvrqv wkh qxpehu ri yhklfohv vkrxog qrw ehryhuhvwlpdwhg1 Ixuwkhupruh/ wrr pdq| yhklfohv lq wkh v|vwhp ohdg wr pruh frqjhvwlrq1

Vwrfkdvwlf lqihuhqfhv duh suhvhqw lq DJY v|vwhpv1 Wkhuhiruh/ vwrfkdvwlf prghov fdq ehxvhg wr ghwhuplqh yhklfoh uhtxluhphqwv1 Krzhyhu/ lq wkh olwhudwxuh d qxpehu ri ghwhuplqlvwlfdssurdfkhv fdq dovr eh irxqg1 Wr vroyh wkh sureohp ri wkh ghwhuplqdwlrq ri wkh plqlpxpqxpehu ri yhklfohv/ ghwhuplqlvwlf dvvxpswlrqv duh pdgh rq wkh udqgrpqhvv ri wkh duulydo sdwwhuqri frqwdlqhuv dqg vshhg ri yhklfohv1 Lq wklv zd|/ phwkrgv fdq eh ghyhorshg wr ghwhuplqh wkhplqlpxp qxpehu ri yhklfohv lq dq rswlpdo zd| dqg lq sro|qrpldo wlph1 Ehvlghv wkhvh wzrpdwkhpdwlfdo dssurdfkhv/ vlpxodwlrq vwxglhv fdq eh fduulhg rxw wr vroyh wklv sureohp1 Xvlqj dvlpxodwlrq prgho/ wkh hqwluh v|vwhp fdq eh revhuyhg1 Krzhyhu/ wklv lv d wlph0frqvxplqj mre1Zh zloo glvfxvv vrph olwhudwxuh frqfhuqlqj wkh ghwhuplqlvwlf fdvh1 Iru d pruh h{whqvlyh ryhuylhzri olwhudwxuh rq doo wkuhh fdwhjrulhv/ zh uhihu wr Ylv +5335,1

Wkh ghwhuplqlvwlf fdvh ri wkh ghwhuplqdwlrq ri wkh plqlpxp qxpehu ri yhklfohv lv vwxglhglq Udmrwld hw do1 +4<<;,1 Wkh| glvfxvv d pl{hg lqwhjhu surjudpplqj prgho1 Frqvlghulqj ordgkdqgolqj wlph/ hpsw| wudyho wlph dqg zdlwlqj dqg frqjhvwlrq wlph/ wkh remhfwlyh ri wkh prgholv wr plqlplvh hpsw| wulsv pdgh e| wkh yhklfohv1 Wkh uhvxowv ri wkh prgho duh frpsduhg zlwkwkh uhvxowv ri d vlpxodwlrq vwxg|1 Lw lv frqfoxghg wkdw wkh yhklfoh hhw vl}h lv xqghuhvwlpdwhgxvlqj wkh dqdo|wlfdo phwkrgv1 Lq Ghoo*Dplfr hw do1 +4<<6, wkhuh lv d vwxg| ri wkh sureohplq zklfk d jlyhq vhw ri frqwdlqhuv zlwk �{hg vwduw lqvwdqwv kdyh wr eh dvvljqhg wr yhklfohv dwgl�huhqw ghsrwv1 D khxulvwlf lv jlyhq zklfk jxdudqwhhv �qglqj wkh plqlpxp qxpehu ri yhklfohvlq sro|qrpldo wlph1 Hjehox +4<;:, frpsduhv wkh vroxwlrqv iru yhklfoh uhtxluhphqwv rewdlqhg

8

e| irxu qrq0vlpxodwlrq dssurdfkhv wr wkh vroxwlrqv rewdlqhg e| ghwdlohg vlpxodwlrq h{shulphqwvxqghu gl�huhqw yhklfoh glvsdwfklqj phwkrgv1 Iurp wkh vwxglhv lq wklv sdshu lw fdq eh frqfoxghgwkdw wkhvh qrq0vlpxodwlrq dssurdfkhv xqghuhvwlpdwh wkh uhtxluhphqwv ri prvw ri wkh whvwhgglvsdwfklqj vwudwhjlhv1 Gl�huhqfhv ehwzhhq dqdo|wlfdo dqg vlpxodwhg hvwlpdwhv xs wr 83( kdyhehhq irxqg1 Hjehox +4<;:, vwdwhv wkdw dqdo|wlfdo whfkqltxhv iru hvwlpdwlqj yhklfoh uhtxluhphqwvpxvw uhfrjql}h wkh uhodwlrqvkls ehwzhhq uhtxluhphqwv dqg glvsdwfklqj srolflhv1 Ylv hw do1 +5334,vwxg| wkh sureohp ri ghwhuplqlqj yhklfoh uhtxluhphqwv dw d frqwdlqhu whuplqdo1 Wkh| suhvhqw dplqlpxp rz dojrulwkp wr vroyh wkh sureohp lq sro|qrpldo wlph1

Lq wklv sdshu/ zh frqvlghu wkh sureohp ri ghwhuplqlqj wkh plqlpxp qxpehu ri yhklfohvuhtxluhg wr wudqvsruw doo wkh frqwdlqhuv zlwklq wkhlu wlph0zlqgrzv1 D uhodwhg sureohp lv wkhyhklfoh urxwlqj sureohp zlwk wlph zlqgrzv1 Frughdx hw do1 +5335, suhvhqw d vxuyh| ri ds0sur{lpdwlrq dqg rswlpdo dssurdfkhv wr vroyh wkh yhklfoh urxwlqj sureohp zlwk wlph zlqgrzv1Krzhyhu/ kdugo| dq| dwwhqwlrq kdv ehhq sdlg lq olwhudwxuh wr wkh fdvh ri ghwhuplqlqj yhklfohuhtxluhphqwv li hdfk mre pxvw eh wudqvsruwhg zlwklq d wlph0zlqgrz1 Wkhuhiruh/ lq wklv sdshuzh ghyhors d qhz vroxwlrq dssurdfk wr vroyh wklv sureohp15 Prghoolqj d whuplqdo zlwk ex�hu duhdvEx�hu srvlwlrqv fdq eh fuhdwhg xqghu wkh fudqh rq d wudfn1 Wkh pd{lpxp fdsdflw| ri d ex�huhtxdov wkh qxpehu ri wudfnv xqghu wkh fudqhv1 Frqwdlqhuv duh qrw srvlwlrqhg qh{w wr hdfk rwkhulq wkh ex�hu wr hqvxuh wkdw wkh fudqh grhv qrw kdyh wr pdnh h{wud pryhv/ zlwk h{wud wudyhowlphv/ wr wkh ohiw ru uljkw1

Dw wkhvh ex�huv/ frqwdlqhuv fdq zdlw iru wudqvsruw1 Zkhq wkh ex�hu lv ixoo dqg d qhzfrqwdlqhu vkrxog eh sodfhg lq wkh ex�hu/ dw ohdvw rqh ri wkhp kdv wr eh wudqvsruwhg �uvw1 Wrfuhdwh vsdfh lq wkh ex�hu dqg wr dyrlg frqjhvwlrq/ zh dvvxph wkdw wkh frqwdlqhu duulylqj hduolhvw

9

kdv wr eh wudqvsruwhg �uvw1 Frqvhtxhqwo|/ wkhuh lv/ qh{w wr d �uvw wlph srlqw ri wudqvsruw +l1h1wkh prphqw wkh frqwdlqhu lv sodfhg lq wkh ex�hu,/ d prphqw dw zklfk wkh frqwdlqhu kdv wr ehwudqvsruwhg dw wkh odwhvw1 Wklv ghdgolqh htxdov wkh uhohdvh wlph ri wkh qhz frqwdlqhu wkdw vkrxogeh sodfhg lq wkh ixoo ex�hu1 Dv d uhvxow/ hdfk frqwdlqhu wkdw kdv wr eh wudqvsruwhg iurp rqhslfn0xs dqg gholyhu| srlqw wr dqrwkhu rqh kdv d wlph0zlqgrz +uhohdvh wlph dqg gxh wlph, lqzklfk wkh frqwdlqhu vkrxog eh wudqvsruwhg1

D frqwdlqhu vkrxog eh wudqvsruwhg lpphgldwho| li wkh ex�hu duhd lv ixoo/ vlqfh wkh zdlwlqjwlphv ri fudqhv duh pxfk pruh h{shqvlyh wkdq wkh zdlwlqj wlphv ri wkh yhklfohv1 Frqvhtxhqwo|/d yhklfoh vkrxog eh uhdg| wr wudqvsruw d frqwdlqhu rqfh wkh ex�hu lv ixoo1 Lq rwkhu zrugv/ zhzdqw wr jxdudqwhh d vhuylfh0ohyho iru fudqhv ri 433(1 Doorzlqj d fudqh wr zdlw iru d yhklfoh zloouhvxow lq kljkhu xqordglqj wlphv ri d vkls dqg/ dv d uhvxow/ lq d orqjhu vwd|lqj wlph ri d vkls dwd sruw1 Wkh wlph0zlqgrz ri hdfk frqwdlqhu fdq eh ghwhuplqhg lq dgydqfh/ ehfdxvh wkh uhohdvhwlphv ri hdfk frqwdlqhu duh nqrzq lq dgydqfh1 Lq wkh prgho/ wkh rughu lq zklfk frqwdlqhuv duhwudqvsruwhg fdq eh fkrvhq iuhho| dv orqj dv hdfk frqwdlqhu lv wudqvsruwhg dw wkh odwhvw dw lwv gxhwlph1 Lq wklv zd|/ wkh vlwxdwlrq zloo qrw rffxu wkdw d txd| fudqh qhhgv wr zdlw wr sodfh d qhzfrqwdlqhu lq wkh ex�hu1

Fohduo|/ wkh prphqw dw zklfk wkh frqwdlqhuv vkrxog �qdoo| eh wudqvsruwhg ghshqgv rq wkh vl}hri wkh ex�hu duhd1 Wkh odujhu wkh ex�hu/ wkh odwhu wkh prphqw wkh ex�hu lv ixoo1 Frqvhtxhqwo|/wkh �qdo wudqvsruw prphqw lv odwhu wkdq zkhq xvlqj d vpdoohu ex�hu1 Dv d uhvxow/ lw pljkweh srvvleoh wr wudqvsruw pruh frqwdlqhuv e| wkh vdph yhklfoh1 Wklv frxog uhvxow lq wkh xvh riihzhu yhklfohv1 Krzhyhu/ e| xvlqj d odujhu ex�hu wkh frpsxwdwlrq wlph ri lqvwdqfhv ri wkh prgholqfuhdvhv1 Lq sudfwlfh wkh sk|vlfdo fdsdflw| ri wkh ex�hu duhd lv dovr olplwhg1 Zh dvvxph dfrqwdlqhu whuplqdo zlwk irxu wudfnv dqg irxu fudqhv1 Frqvhtxhqwo|/ wkh pd{lpxp fdsdflw| riwkh ex�hu htxdov irxu1 Frqvlghulqj doo wkhvh dvshfwv/ wkh fkrlfh ri wkh ex�hu vl}h vkrxog eh pdgh

:

fduhixoo| +vhh dovr Vhfwlrq 61415,1Wkh sureohp wkdw kdv wr eh vroyhg/ lv wkh ghwhuplqdwlrq ri wkh plqlpxp qxpehu ri yhklfohv

uhtxluhg wr wudqvsruw doo wkh frqwdlqhuv zlwklq wkhlu wlph0zlqgrzv1 Wkh qxpehu ri yhklfohvrewdlqhg vhuyh dv lqsxw iru wkh rshudwlrq1 Zklfk yhklfoh wudqvsruwv zklfk frqwdlqhu dorqjzklfk urxwh vkrxog eh ghflghg gxulqj wkh rshudwlrq1 Wkhuhiruh/ wkh srvvlelolw| wr dgmxvw wkhqxpehu ri yhklfohv/ gxh wr xqh{shfwhg hyhqwv/ olnh euhdngrzqv/ lv vwloo rshq1 Zh fdq hvwlpdwhwkh qxpehu ri yhklfohv uhtxluhg e| dsso|lqj wkh prgho ri Vhfwlrq 5141 E| ydu|lqj wkh uhohdvhwlphv ri d frqwdlqhu/ urexvw hvwlpdwhv fdq eh pdgh ri wkh qhfhvvdu| qxpehu ri yhklfohv1

Lq wkh dqdo|wlfdo prgho zh dvvxph wkh iroorzlqj=41 Q frqwdlqhuv qhhg wr eh wudqvsruwhg1 Wkh uhohdvh wlph ri hdfk frqwdlqhu +l1h1/ wkh wlph dw

zklfk wkh frqwdlqhu lv sodfhg lq wkh ex�hu dqg lv dydlodeoh iru wudqvsruw e| wkh yhklfoh,fdq eh ghwhuplqhg zlwk dq hpslulfdo glvwulexwlrq dqg lv nqrzq lq dgydqfh1

51 Wkhuh duh N fudqhv zlwk d slfn0xs dqg gholyhu| srlqwv +s)g srlqwv, iru frqwdlqhuv1 Dwwkh s)g srlqwv wkhuh duh vhsdudwh ex�huv iru duulylqj dqg ohdylqj frqwdlqhuv1 Wkxv/ wkhuhlv qr lqwhuihuhqfh ehwzhhq duulylqj dqg ohdylqj frqwdlqhuv1

61 Wkh gxh wlph +l1h1/ wkh wlph dw zklfk wkh frqwdlqhu xowlpdwho| kdv wr eh wudqvsruwhg e|wkh yhklfoh, lv nqrzq lq dgydqfh1

71 Wkh wudyho wlphv ehwzhhq vkls dqg vwdfn duh ghwhuplqlvwlf dqg nqrzq lq dgydqfh1 Wkhvhwudyho wlphv duh ghulyhg lq wkh iroorzlqj zd|= li zh zdqw wr eh vxuh wkdw wkh yhklfoh lvrq wlph dw wkh ruljlq ri d qh{w frqwdlqhu/ zh vkrxog wdnh lqwr dffrxqw wkh zruvw fdvh/qdpho|/ wkh odujhvw wudyho wlph uhtxluhg iru wkh fruuhvsrqglqj wudmhfwru|1 Wr rewdlq wklvghwhuplqlvwlf ydoxh zh fdq wdnh wkh xsshu erxqg ri d vwrfkdvwlf glvwulexwlrq uhsuhvhqwlqjwkh wudyho wlph iru wkh orqjhvw h{lvwlqj urxwh ehwzhhq wkh ruljlq dqg ghvwlqdwlrq ri wkh

;

yhklfoh181 Wkh ghvwlqdwlrq ri hdfk frqwdlqhu lv nqrzq lq dgydqfh191 Iru duulylqj frqwdlqhuv vx!flhqw vsdfh lv dozd|v dydlodeoh lq wkh ex�hu ri wkh gholyhu|

srlqw1514 PrghoLq wklv vhfwlrq/ zh ghyhors d prgho wr vroyh wkh sureohp ri ghwhuplqlqj wkh plqlpxp qxpehuri yhklfohv dw d frqwdlqhu whuplqdo lq zklfk wkh wudqvsruwdwlrq surfhvv lv xqfrxsohg iurp rwkhusurfhvvhv e| ex�hu duhdv1

Iru hdfk frqwdlqhu l zh gh�qh wkh iroorzlqj sdudphwhuv=dls uhohdvh wlph/ wkh srlqw lq wlph dw zklfk frqwdlqhu l +4 � l � Q, lv dydlodeoh iru wudqvsruw

dw wkh ex�hu duhd ri ruljlq slfn0xs srlqw s +4 � s � N, e| d oliwlqj yhklfoh1els gxh wlph/ wkh srlqw lq wlph dw zklfk frqwdlqhu l +4 � l � Q, xowlpdwho| kdv wr eh

wudqvsruwhg iurp ruljlq slfn0xs srlqw s +4 � s � N, e| d oliwlqj yhklfoh1zlsg wudyho wlph ri d ixoo oliwlqj yhklfoh iurp wkh ruljlq slfn0xs srlqw s +4 � s � N, ri frqwdlqhu

l +4 � l � Q, wr wkh ghvwlqdwlrq gholyhu| srlqw g +4 � g � N, ri frqwdlqhu l1 Wkh wlph wroliw dqg sxw grzq d ordg lv lqfrusrudwhg lq wklv wudyho wlph1

wlg duulydo wlph/ wkh srlqw lq wlph dw zklfk frqwdlqhu l duulyhv dw lwv ghvwlqdwlrq gholyhu| srlqwg +4 � g � N,1 Wklv srlqw lq wlph ghshqgv rq wkh wudyho wlph ri wkh yhklfoh iurp wkhruljlq ri l wr wkh ghvwlqdwlrq ri l dqg wkh fkrvhq vwduw wlph ri frqwdlqhu l1 Dv d uhvxow/ dqhduolhvw dqg odwhvw duulydo wlph ri hdfk frqwdlqhu h{lvw/ qdpho|=

wlig hduolhvw +�uvw, duulydo wlph dw ghvwlqdwlrq gholyhu| srlqw g +4 � g � N, wlig @ dls .zlsg1<

wlog odwhvw duulydo wlph dw ghvwlqdwlrq gholyhu| srlqw g +4 � g � N, wlog @ els .zlsg1ulgms wudyho wlph ri dq hpsw| yhklfoh/ iurp wkh ghvwlqdwlrq gholyhu| srlqw g +4 � g � N, ri

frqwdlqhu l wr wkh ruljlq slfn0xs srlqw s +4 � s � N, ri frqwdlqhu m +ulgms @ 3 li ghvwlqdwlrqri l htxdov wkh ruljlq ri m,1

Ixuwkhupruh/ zh gh�qh wkh iroorzlqj ghflvlrq yduldeoh iru hdfk frqwdlqhu l +4 � l � Q,=vls vwduw wlph/ wkh srlqw lq wlph dw zklfk d yhklfoh oliwv frqwdlqhu l dw ruljlq slfn0xs srlqw s

+4 � s � N, dqg vwduwv wudqvsruwlqj wkh frqwdlqhu wr lwv ghvwlqdwlrq= dls � vls � elsDv d uhvxow/ wkh duulydo wlph wlg htxdov= wlg @ vls .zlsg1Rqh yhklfoh vkrxog wudqvsruw vhyhudo frqwdlqhuv diwhu hdfk rwkhu/ wr xvh dv ihz yhklfohv dv

srvvleoh dw wkh frqwdlqhu whuplqdo1 Li wzr frqwdlqhuv fdq eh wudqvsruwhg e| wkh vdph yhklfoh/ zhfdoo wkhp frpsdwleoh1 Frqwdlqhu l dqg m duh frpsdwleoh li=

;AA?AA=wlg . ulgms � vmsdms � vms � emsFrqwdlqhuv duh frpsdwleoh/ li wkh duulydo wlph ri frqwdlqhu l soxv wkh wudyho wlph wr wkh ruljlq

ri frqwdlqhu m lv vpdoohu ru htxdo wr wkh fkrvhq vwduw wlph ri frqwdlqhu m1 Wklv vwduw wlph vkrxogeh fkrvhq zlwklq wkh jlyhq wlph0zlqgrz iru frqwdlqhu m1

Frqvhtxhqwo|/ wzr ghflvlrqv kdyh wr eh pdgh/ qdpho|=� Zklfk frqwdlqhuv duh wudqvsruwhg e| wkh vdph yhklfohB� Dw zklfk vwduw wlph lv frqwdlqhu l +; l , wudqvsruwhgBWr prgho wkh sureohp hdfk wlph0zlqgrz ^dl> el` fdq eh glvfuhwlvhg dv vkrzq lq Iljxuh 51

Wkh uhohdvh wlph htxdov wkh prphqw wkh frqwdlqhu lv sodfhg lq wkh ex�hu1 Wklv lv dovr wkh �uvw43

srvvleoh vwduw lqvwdqw ri wkh frqwdlqhu1 Frqvhtxhqwo|/ wkh vhfrqg srvvleoh wudqvsruw prphqw riwkh frqwdlqhu htxdov wkh �uvw srvvleoh wudqvsruw prphqw soxv wkh glvfuhwlvdwlrq vwhs1 Wklv lvuhshdwhg xqwlo wkh qhzo| fuhdwhg srvvleoh wudqvsruw prphqw lv htxdo wr ru juhdwhu wkdq wkh gxhwlph ri wkh frqwdlqhu1 Wkh odvw srvvleoh vwduw lqvwdqw lv vhw wr wkh gxh wlph ri wkh frqwdlqhu1Frqvhtxhqwo|/ iru hdfk frqwdlqhu zh gh�qh d qxpehu ri srvvleoh vwduw lqvwdqwv1

LQVHUW ILJXUH 5Wkxv/ e| glvfuhwlvlqj wkh wlph0zlqgrz zh kdyh rewdlqhg pl frslhv ri srvvleoh vwduw wlphv1

Hdfk frqwdlqhu l +4 � l � Q, fdq eh h{hfxwhg dw wlph srlqwv jln +4 � n � pl,1 Ehfdxvh ri wkhidfw wkdw iru hdfk frqwdlqhu d gl�huhqw wlph0zlqgrz +zlwk lwv rzq vl}h, lv gh�qhg/ pl ydulhv shufrqwdlqhu1

Lq sudfwlfh/ rqh fdq ghflgh dw hdfk prphqw lq wklv wlph zlqgrz zkhq wkh yhklfoh vwduwvwudqvsruwlqj wkh frqwdlqhu1 E| glvfuhwlvlqj wkh wlph zlqgrz zh jhqhudwh d qxpehu ri wlphsrlqwv dw zklfk d yhklfoh fdq wudqvsruw wkh frqwdlqhu1 Dv h{sodlqhg deryh/ wkh wlph ehwzhhqwzr srvvleoh dgmdfhqw vwduw srlqwv htxdov wkh glvfuhwlvdwlrq vwhs1 Lw lv fohdu wkdw wkh vl}h ri wkhglvfuhwlvdwlrq vwhs lq xhqfhv wkh qxpehu ri srvvleoh wlph srlqwv wkdw duh gh�qhg1 Dv pdq| wlphsrlqwv dv srvvleoh vkrxog eh jhqhudwhg zlwklq wkh wlph zlqgrz wr dssur{lpdwh wkh frqwlqxrxvuhdolw|1 Frqvhtxhqwo|/ wkh glvfuhwlvdwlrq vwhs lv yhu| vpdoo1 Krzhyhu/ wkh frpsxwdwlrq wlphv rilqvwdqfhv ri wkh prgho lqfuhdvh e| xvlqj pdq| wlph srlqwv1 Wkh qxpehu ri wlph srlqwv dw zklfkwkh frqwdlqhu fdq eh wudqvsruwhg zloo ghfuhdvh li zh lqfuhdvh wkh vl}h ri wkh glvfuhwlvdwlrq vwhs1E| grlqj wklv/ ihzhu frqwdlqhuv pljkw eh frpsdwleoh1 Dv d frqvhtxhqfh wkh uhvxow ri wkh prghofdq eh d odujhu qxpehu ri yhklfohv wkdq lv qhfhvvdu|1 Wkhuhiruh/ wkh fkrlfh ri wkh vl}h ri wkhglvfuhwlvdwlrq vwhs vkrxog eh pdgh fduhixoo| +vhh Vhfwlrq 61414,1

Zlwk wkhvh gdwd zh fdq frqvwuxfw d gluhfwhg judsk J @ +Y>D, zlwk

44

Y @ ij��> = = = > j�p�> = = = > jQ�> = = = > jQpQ > zkhuh hdfk qrgh uhsuhvhqwv d wlph srlqwdw zklfk d frqwdlqhu fdq eh wudqvsruwhgj

D @ i+f> g, = f> g 5 Y> frqwdlqhu zlwk vwduw wlph f lv frpsdwleoh zlwk dqrwkhu frqwdlqhu zlwkvwduw wlph gj

Ixuwkhupruh/ zh lqwurgxfh d vrxufh qrgh v dqg d vlqn qrgh w dqg dgg doo wkh dufv +v> f, dqg+f> w, ;f 5 Y 1

Judsk J lv df|folf dqg doo wkh qrghv fdq eh rughuhg1 D gluhfwhg sdwk lq wklv qhwzrunfruuhvsrqgv wr d ihdvleoh vhtxhqfh ri frqwdlqhuv zlwk wkhlu vwduw wlphv/ zklfk fdq eh h{hfxwhge| rqh yhklfoh1

LQVHUW ILJXUH 6Lq Iljxuh 6 dq h{dpsoh ri vxfk d judsk lv jlyhq1 Wkuhh frqwdlqhuv kdyh wr eh wudqvsruwhg1

Iru hdfk frqwdlqhu wzr srvvleoh vwduw wlphv duh jlyhq1 D sdwk lq wklv judsk fruuhvsrqgv wrd vhtxhqfh ri frqwdlqhuv wkdw fdq eh wudqvsruwhg e| wkh vdph yhklfoh1 E| ghwhuplqlqj wkhplqlpxp qxpehu ri gluhfwhg sdwkv zh fdq rewdlq wkh plqlpxp qxpehu ri yhklfohv wr wudqvsruwdoo wkh frqwdlqhuv1 Krzhyhu/ rqh h{wud frqglwlrq lv qhfhvvdu|= qdpho|/ iru hdfk frqwdlqhu h{dfwo|rqh vwduw wlph fdq eh fkrvhq1 Lw lv qrw srvvleoh wr wudqvsruw wkh vdph frqwdlqhu wzlfh1 Iruh{dpsoh/ wkh iroorzlqj vhw ri gluhfwhg sdwkv lv qrw d ydolg vroxwlrq iru wkh sureohp=

v $ j�� $ j22 $ wv $ j2� $ j�2 $ w

Li erwk gluhfwhg sdwkv duh fkrvhq/ frqwdlqhu 5 lv lqfoxghg lq wzr sdwkv dqg wkhuhiruh qhhgvwr eh h{hfxwhg wzlfh1 Dv d uhvxow/ wr vroyh wkh sureohp/ zh qhhg wr ghwhuplqh wkh plqlpxp

45

qxpehu ri gluhfwhg sdwkv/ vxfk wkdw hdfk frqwdlqhu lv lqfoxghg lq h{dfwo| rqh sdwk dqg wkdw iruhdfk frqwdlqhu rqo| rqh vwduw wlph +prghoohg zlwk frqglwlrq +7, lv fkrvhq1

Zh fdq irupxodwh wkh sureohp dv iroorzv dv dq Lqwhjhu Olqhdu Surjudpplqj Prgho =

Plq y[tfGEv>f�MD�{vf � [

tfGEf>v�MD�{fv @ y +4,[tfGEw>f�MD�{wf � [

tfGEf>w�MD�{fw @ �y +5,[tfGEg>f�MD�{gf � [

tfGEf>g�MD�{fg @ 3 ;g 5 Y qiv> wj +6,n'pl[n'�

[f {fjln @ 4 ;l> 4 � l � Q +7,

3 � {lm � 4 ;+l> m, 5 D +8,{lm lqwhjhu ;+l> m, 5 D +9,

Wkh h{sodqdwlrq ri wkh htxdwlrqv lv dv iroorzv=+4, wkh ydoxh ri doo rxwfrplqj rz plqxv wkh ydoxh ri doo lqfrplqj rz lq wkh vrxufh vkrxog

eh htxdo wr y +@ plqlpxp qxpehu ri gluhfwhg sdwkv,/+5, wkh ydoxh ri doo rxwfrplqj rz plqxv wkh ydoxh ri doo lqfrplqj rz lq wkh vlqn vkrxog eh

htxdo wr �y/+6, iru doo wkh rwkhu qrghv wkh lqfrplqj rz vkrxog htxdo wkh rxwfrplqj rz/+7, hdfk frqwdlqhu kdv wr eh wudqvsruwhg h{dfwo| rqfh1 Frqvhtxhqwo|/ rqo| rqh vwduw wlph fdq

eh fkrvhq/+8, wkh xsshu erxqg ri wkh rz htxdov rqh dqg wkh orzhu erxqg htxdov }hur/+9, doo rz vkrxog eh lqwhjhu1

46

Wkh plqlpxp qxpehu ri yhklfohv wr wudqvsruw doo wkh frqwdlqhuv zlwklq wkhlu wlph0zlqgrzfdq eh rewdlqhg e| xvlqj wklv Lqwhjhu Olqhdu Surjudpplqj Prgho1 Wkh prgho lv xvhg ehiruhwkh rshudwlrq1 Wkh vroxwlrq ri wkh prgho lqglfdwhv wkh plqlpxp qxpehu ri yhklfohv uhtxluhgwr shuirup dq h!flhqw rshudwlrq1 Vroylqj wkh prgho dovr lqglfdwhv dq dvvljqphqw ri yhklfohvwr frqwdlqhuv1 Krzhyhu/ wr rewdlq urexvw dqg uholdeoh hvwlpdwhv ri wkh qxpehu ri yhklfohv wkhprgho lv h{hfxwhg vhyhudo wlphv zlwk gl�huhqw uhohdvh wlphv ri frqwdlqhuv1 Wkh pd{lpxp ydoxhdqg dyhudjh ydoxh dprqj wkhvh uhsolfdwlrqv lv xvhg wr ghwhuplqh wkh qxpehu ri yhklfohv uhtxluhg1Wkhuhiruh/ wkh dvvljqphqwv wkhpvhoyhv duh qrw xvhixo1 Wkh dfwxdo dvvljqphqw qhhgv wr eh pdghgxulqj wkh uhdo rshudwlrq16 Qxphulfdo h{shulphqwvWr vwxg| wkh shuirupdqfh ri wkh Lqwhjhu Olqhdu Surjudpplqj prgho ri Vhfwlrq 514 dqg wkhxwlolw| ri wkh prgho lq sudfwlfh zh fduu| rxw wzr vwxglhv1 Zh xvh uhdolvwlf gdwd wr jdlq lqvljkwlqwr wkh sureohp ri ghwhuplqlqj yhklfoh uhtxluhphqwv dw d frqwdlqhu whuplqdo1 Zh rqo| frqvlghuwkh xqordglqj surfhvv/ vlqfh wkh zd| lq zklfk wkh qxpehu ri yhklfohv lv ghwhuplqhg lv wkh vdphiru erwk ordglqj dqg xqordglqj1 Lq sudfwlfh/ fudqhv �uvw xqordg d vkls dqg wkhuhdiwhu fudqhv ordgwkh vkls zlwk frqwdlqhuv1 Xqordglqj dqg ordglqj rshudwlrqv duh rqo| pl{hg gxulqj d yhu| vkruwshulrg ri wlph1 D yhklfoh pljkw eh deoh wr wudqvsruw dq xqordglqj dqg ordglqj frqwdlqhu lq rqhwuls gxulqj wklv vkruw shulrg1 D ghfuhdvh lq wkh qxpehu ri yhklfohv uhtxluhg pljkw eh h{shfwhggxh wr wkh rffxuuhqfh ri wkhvh gxdo rshudwlrqv1 Krzhyhu/ wr shuirup wkh ryhudoo rshudwlrqh!flhqwo|/ wklv uhgxfwlrq lq wkh qxpehu ri yhklfohv fdqqrw eh wdnhq lqwr dffrxqw1

Iluvw/ zh fduu| rxw ydulrxv h{shulphqwv zlwk wkh dqdo|wlfdo prgho1 Wkhuhdiwhu/ zh h{hfxwhh{shulphqwv zlwk d vlpxodwlrq prgho uhsuhvhqwlqj doo wkh dvshfwv ri wkh xqordglqj surfhvv dw dfrqwdlqhu whuplqdo1

47

614 Vwxg| zlwk wkh dqdo|wlfdo prghoFrqwdlqhuv dw TFv duh nqrzq lq dgydqfh dqg duh frqvlghuhg dv lqsxw1 E| dvvxplqj wkdw wkhuh lvqr lqwhuihuhqfh dw wkh TFv dqg qr lqwhuihuhqfh dw wkh vwdfnlqj fudqhv/ zh djjuhjdwh doo wkh TFvwr rqh TF dqg doo wkh vwdfnlqj fudqhv wr rqh vwdfnlqj fudqh1 Dv d uhvxow/ xqordglqj frqwdlqhuvfdq eh frqvlghuhg wr eh uhohdvhg dw rqh TF dqg eh wudqvsruwhg wr rqh vwdfnlqj fudqh1 Wkhqxpehu ri oliwlqj yhklfohv uhtxluhg iru doo wkh txd| fudqhv htxdov wkh surgxfw ri wkh qxpehu rifudqhv dqg wkh qxpehu ri oliwlqj yhklfohv uhtxluhg dw rqh fudqh1

Pdqqhg TFv uhpryh frqwdlqhuv iurp wkh vkls1 Wklv wdnhv d fhuwdlq f|foh wlph1 Wkhvh f|fohwlphv ghshqg/ iru h{dpsoh/ rq wkh fdsdelolwlhv ri wkh fudqh gulyhu/ wkh vshfl�fdwlrqv ri wkh fudqh/wkh srvlwlrq ri wkh frqwdlqhu lq wkh vkls dqg vr rq1 Wkh wlph ehwzhhq wzr vxffhvvlyh frqwdlqhuvlv vpdoohu li d fudqh lv fdsdeoh ri xqordglqj frqwdlqhuv yhu| txlfno|/1 Wkh f|foh wlph +lq vhfrqgv,iroorzv dq hpslulfdo glvwulexwlrq zklfk lv uhsuhvhqwhg lq Wdeoh 4 +vhh Fhohq hw do1 +4<<:,,1Srvvleoh odujh glvwxuedqfhv/ vxfk dv euhdngrzqv ri htxlsphqw/ duh qrw wdnhq lqwr dffrxqw1 Zlwkwkhvh gdwd zh fdq jhqhudwh wkh srlqwv lq wlph dw zklfk wkh frqwdlqhuv duh dydlodeoh iru wudqvsruwdv iroorzv= udqgrpo| vhohfw d fodvv ri f|foh wlphv dffruglqj wr wkh hpslulfdo glvwulexwlrq1 Lqwklv fodvv hdfk f|foh wlph kdv dq htxdo suredelolw| wr eh fkrvhq1 Zlwklq d fodvv f|foh/ wlphv duhdvvxphg wr iroorz d frqwlqxrxv xqlirup glvwulexwlrq1

LQVHUW WDEOH 4Yhklfohv wudqvsruw doo wkh frqwdlqhuv iurp rqh TF wr rqh vwdfn fudqh1 Dw wkh vwduw ri wkh

surfhvv doo wkh yhklfohv duh gluhfwo| dydlodeoh dw wkh TF1 Wkh yhklfohv wudyho lq d orrs iurp wkhTF wr wkh vwdfn dqg ylfh yhuvd1 Wkh pd{lpxp vshhg ri d ixoo yhklfoh lv 7=3p2v1 Wkh pd{lpxpvshhg ri dq hpsw| yhklfoh lv 8=8p2v1 Wkh glvwdqfh ehwzhhq wkh TF dqg vwdfn lv 933 phwhuv1 Lqwklv vwxg|/ zh frqvlghu wkh iroorzlqj lqvwdqfhv= 83/ ;3 ru 433 frqwdlqhuv kdyh wr eh xqordghg1Zh h{hfxwh iru hdfk lqvwdqfh d qxpehu ri uhsolfdwlrqv1 Dv ghvfulehg/ zh fdq jhqhudwh wkh srlqwv

48

lq wlph dw zklfk wkh frqwdlqhuv duh dydlodeoh iru wudqvsruw zlwk wkh gdwd iurp Wdeoh 41 Dfudqh qhhgv wr zrun rq dyhudjh iru rqh krxu wr kdqgoh 83 frqwdlqhuv dw d odujh whuplqdo lq wkhQhwkhuodqgv1 Zlwk doo wkhvh gdwd/ zh fdq vroyh wkh sureohp ri wkh ghwhuplqdwlrq ri wkh qxpehuri yhklfohv uhtxluhg shu xvhg TF e| xvlqj wkh phwkrg ghvfulehg lq Vhfwlrq 51

Iru d qxpehu ri lqghshqghqw uhsolfdwlrqv ri hdfk ghvfulehg lqvwdqfh/ zh vroyh wkh sureohp1Wr rewdlq d ydolg hvwlpdwlrq ri wkh qxpehu ri yhklfohv shu xvhg Txd| Fudqh zh vkrxog ghwhuplqhwkh qhfhvvdu| qxpehu ri uhsolfdwlrqv iru hdfk vlpxodwlrq h{shulphqw1

Dffruglqj wr Odz dqg Nhowrq +4<<4,/ dq dssur{lpdwlrq iru wkh plqlpxp qxpehu ri uhsol0fdwlrqv l/ vxfk wkdw wkh uhodwlyh huuru lv vpdoohu wkdq � +3 ? � ? 4, zlwk d suredelolw| ri 4� �htxdov=

l � V2+l,^}�3�@2@��[+l,`2 +:,zkhuh V2+l, lv wkh vdpsoh yduldqfh/ }�3�@2 wkh 4��@5 shufhqwloh ri wkh qrupdo glvwulexwlrq/

[+l, wkh vdpsoh phdq dqg �� @ �@+4 . �,1 Zh ghwhuplqh wkh plqlpxp qxpehu ri uhsolfdwlrqve| vroylqj htxdwlrq +:, zlwk gdwd iurp d olplwhg qxpehu ri uhsolfdwlrqv1 Wr rewdlq d uhodwlyhhuuru � vpdoohu wkdq 5( zlwk d suredelolw| ri <8(/ d uhsolfdwlrq vl}h ri 433 lv vx!flhqw irudoo wkh h{shulphqwv lq wklv sdshu1 Wkhuhiruh/ iru hyhu| vlpxodwlrq h{shulphqw zh jhqhudwh 433uhsolfdwlrqv1

Dv d uhvxow/ e| h{hfxwlqj/ iru h{dpsoh/ d vlpxodwlrq h{shulphqw iru 83 frqwdlqhuv/ zh jhqhudwh433 wlphv d uhohdvh wlph iru hdfk frqwdlqhu e| xvlqj wkh gdwd iurp Wdeoh 41 Zh jhqhudwh 433prghov e| xvlqj wkhvh uhohdvh wlphv/ wudyho wlphv dqg ex�hu vl}h1 Zh rewdlq iru hyhu| uhsolfdwlrqwkh plqlpxp qxpehu ri yhklfohv qhfhvvdu| wr wudqvsruw doo wkh frqwdlqhuv e| dsso|lqj wkhphwkrg ghvfulehg lq Vhfwlrq 5 dqg xvlqj wkh vroyhu FSOH[ 9181 Dv pdwul{ jhqhudwru zh kdyhxvhg JDPV1 Wr rewdlq dq hvwlpdwlrq ri wkh qxpehu ri yhklfohv wkdw vkrxog eh xvhg wr wudqvsruw

49

wkh Q xqordglqj frqwdlqhuv/ zh fkrrvh wkh pd{lpxp ydoxh rewdlqhg iurp wkh 433 uhsolfdwlrqv1Ixuwkhupruh/ zh ghwhuplqh wkh dyhudjh qxpehu ri yhklfohv uhtxluhg dprqj wkh 433 uhsolfdwlrqv1Wr xvh wkh prgho/ �uvw zh kdyh wr ghwhuplqh dssursuldwh vl}hv ri wkh glvfuhwlvdwlrq vwhsv dqgex�hu161414 Fkrlfh ri wkh vl}h ri wkh glvfuhwlvdwlrq vwhsDv ghvfulehg lq Vhfwlrq 5 dqg Vhfwlrq 614 d wlph zlqgrz h{lvwv iru hdfk frqwdlqhu lq zklfk lwvkrxog eh wudqvsruwhg1

Zh kdyh h{hfxwhg wkh iroorzlqj vlpxodwlrq h{shulphqw wr ghwhuplqh wkh dssursuldwh vl}h riwkh glvfuhwlvdwlrq vwhs iru wkh rwkhu h{shulphqwv1 Oliwlqj yhklfohv kdyh wr wudqvsruw 83 frqwdlqhuv1Zh jhqhudwh wkh f|foh wlphv ri wkh frqwdlqhuv e| xvlqj wkh glvwulexwlrq uhsuhvhqwhg lq Wdeoh 41Wr whvw wkh prgho lq d sudfwlfdo frqwh{w/ zh xvh wkhvh gdwd rewdlqhg iurp vwxglhv iurp sudfwlfh1Dv d uhvxow/ iru hdfk frqwdlqhu wkh uhohdvh wlph lv nqrzq1 Wkh vl}h ri wkh ex�hu lv vhw wr rqh1Frqvhtxhqwo|/ wkh gxh wlph ri hyhu| frqwdlqhu htxdov wkh uhohdvh wlph ri wkh qh{w frqwdlqhu1 Wkhgxh wlph ri wkh odvw xqordglqj frqwdlqhu htxdov lwv uhohdvh wlph soxv 433 vhfrqgv1 Wr xvh wkhprgho ri Vhfwlrq 5/ zh glvfuhwlvh wkh wlph zlqgrz ri hyhu| frqwdlqhu1 Wr revhuyh wkh h�hfw ri wkhglvfuhwlvdwlrq vwhs rq wkh plqlpxp qxpehu ri yhklfohv qhfhvvdu| wr wudqvsruw doo wkh frqwdlqhuvzlwklq wkhlu wlph zlqgrz zh kdyh whvwhg ydulrxv glvfuhwlvdwlrq vwhsv/ qdpho|/ 83/ 73/ 63/ 53/ dqg43 vhfrqgv1 Wkh pd{lpxp ydoxh dprqj wkh uhsolfdwlrqv lq wkh vlpxodwlrq h{shulphqw htxdov �yhyhklfohv shu fudqh iru doo gl�huhqw glvfuhwlvdwlrq vwhsv1 Wkh dyhudjh qxpehu ri yhklfohv shu fudqhghfuhdvhv voljkwo| iurp 71<4 wr 71:: +51<(, li wkh vl}h ri wkh glvfuhwlvdwlrq vwhs ghfuhdvhv iurp83 vhfrqgv wr 43 vhfrqgv1 Dv phqwlrqhg ehiruh/ wkh vpdoohu wkh glvfuhwlvdwlrq vwhs wkh odujhuwkh qxpehu ri srvvleoh wlph srlqwv dw zklfk yhklfohv fdq wudqvsruw wkh frqwdlqhu1 Wkhuhiruh/ lwpljkw eh srvvleoh wkdw zlwk d vpdoohu glvfuhwlvdwlrq vwhs d yhklfoh fdq wudqvsruw pruh frqwdlqhuv1

4:

Wklv pljkw uhvxow lq d vpdoohu qxpehu ri yhklfohv uhtxluhg wr wudqvsruw doo wkh frqwdlqhuv zlwklqwkhlu wlph zlqgrz1 Krzhyhu/ lw lv fohdu wkdw wkh uhgxfwlrq lq wkh dyhudjh qxpehu ri yhklfohvrewdlqhg e| uhgxflqj wkh vl}h ri wkh glvfuhwlvdwlrq vwhs lv uhodwlyho| vpdoo1 Ixuwkhupruh/ wkhpd{lpxp ydoxh irxqg dprqj wkh uhsolfdwlrqv lv wkh vdph iru doo vl}hv1 Dv d uhvxow/ zh fkrrvh dglvfuhwlvdwlrq vwhs ri 83 vhfrqgv iru wkh h{shulphqwv lq wklv sdshu161415 Fkrlfh ri wkh ex�hu vl}hZh h{hfxwh wkh iroorzlqj vwxg| wr ghwhuplqh dq dssolfdeoh vl}h ri wkh ex�hu duhd iru wkh rwkhuh{shulphqwv1 Oliwlqj yhklfohv kdyh wr wudqvsruw 83 frqwdlqhuv1 Zh jhqhudwh wkh f|foh wlphv riwkh frqwdlqhuv e| xvlqj wkh glvwulexwlrq uhsuhvhqwhg lq Wdeoh 41 Wkh glvfuhwlvdwlrq vwhs htxdov83 vhfrqgv +vhh Vhfwlrq 61414,1 Wr revhuyh wkh h�hfw ri wkh ex�hu vl}h rq wkh plqlpxp qxpehuri yhklfohv qhfhvvdu| wr wudqvsruw doo wkh frqwdlqhuv zlwklq wkhlu wlph zlqgrzv wkh ex�hu vl}hydulhv ehwzhhq }hur dqg irxu1

Lq wkh fdvh wkdw qr ex�hu lv xvhg/ wkh dyhudjh qxpehu ri yhklfohv uhtxluhg htxdov 81881 Wkhpd{lpxp ydoxh dprqj wkh uhsolfdwlrqv iru wkh plqlpxp qxpehu ri yhklfohv htxdov �yh iru doowkh ex�huv zlwk fdsdflw| juhdwhu wkdq }hur1 Ixuwkhupruh/ wkh dyhudjh qxpehu ri yhklfohv uhgxfhvvoljkwo| iurp 719< wr 7184 dqg 717 e| lqfuhdvlqj wkh vl}h ri wkh ex�hu uhvshfwlyho| iurp wzr wrwkuhh ru irxu1 Frqvlghulqj wklv dqg frpsxwdwlrq wlphv ri lqvwdqfhv ri wkh prgho e| xvlqj odujhuex�hu duhdv/ zh kdyh fkrvhq wr xvh d ex�hu vl}h ri wzr iru wkh iroorzlqj h{shulphqwv161416 Uhvxowv ri wkh dqdo|wlfdo prghoLq wkh iroorzlqj vwxg| zh h{dplqh krz pdq| yhklfohv duh uhtxluhg e| lqwurgxflqj d ex�hu duhdri vl}h wzr +vhh Vhfwlrq 61415,1 Djdlq zh xvh d glvfuhwlvdwlrq vwhs ri 83 vhfrqgv1 Zh whvw wkhiroorzlqj lqvwdqfhv= 83/ ;3 dqg 433 frqwdlqhuv1 Wdeoh 5 suhvhqwv wkh uhvxowv iru wklv vlwxdwlrqzlwk/ iru hyhu| lqvwdqfh/ wkh qxpehu ri uhsolfdwlrqv zlwk wkh jlyhq qxpehu ri yhklfohv1 Wklv

4;

wdeoh dovr jlyhv wkh pd{lpxp dqg wkh dyhudjh qxpehu ri yhklfohv1 Wkh frpsxwdwlrq wlphv shuprgho rq d sf zlwk 683PK} ydu| ehwzhhq 58 dqg 533 vhfrqgv1

LQVHUW WDEOH 5Lq Wdeoh 5 lw fdq eh vhhq wkdw e| lqfuhdvlqj wkh zrunordg iurp 83 wr 433 frqwdlqhuv/ wkhuh lv

rqo| d vpdoo lqfuhdvh lq wkh dyhudjh qxpehu ri yhklfohv1 Dq h{sodqdwlrq/ wkhuhiruh/ zrxog eh wkdwwkh fudqh qhhgv vrph zdup0xs shulrg1 Dw wkh vwduw ri wkh xqordglqj surfhvv wkh fudqhv qhhg ihzyhklfohv1 Diwhu d zkloh wkh qxpehu ri yhklfohv uhtxluhg vwdelolvhv1 Dssurdfklqj wkh hqg ri wkhxqordglqj surfhvv d qxpehu ri yhklfohv gr qrw kdyh wr shuirup qhz mrev1 Frqvhtxhqwo|/ ihzhuyhklfohv duh uhtxluhg1 Wkh ohqjwk ri wkh vwdelolvdwlrq shulrg ghshqgv rq wkh qxpehu ri frqwdlqhuvwr eh xqordghg1 Krz odujh wkh qxpehu ri frqwdlqhuv/ wkh odujhu wkh shulrg ri vwdelolvdwlrq1 Zhqrwlfh d vwdelolvdwlrq lq wkh dyhudjh qxpehu ri yhklfohv li zh h{whqg wkh zrunordg ixuwkhu1 Wkhdyhudjh qxpehu ri yhklfohv dssurdfkhv 8/ exw grhv qrw h{fhhg 81

Iurp Wdeoh 5 lw lv fohdu wkdw �yh oliwlqj yhklfohv shu fudqh vkrxog eh xvhg wr wudqvsruw doo wkhfrqwdlqhuv zlwklq wkhlu wlph zlqgrz1 Frqvhtxhqwo|/ iru d whuplqdo lq zklfk vklsv duh xqordghge| xvlqj irxu fudqhv/ plqlpdo 53 oliwlqj yhklfohv duh qhfhvvdu| wr wudqvsruw doo wkh frqwdlqhuv lqwlph161417 Ydu|lqj f|foh wlphv lq wkh dqdo|wlfdo prghoLw lv fohdu wkdw pdq| idfwruv/ vxfk dv wkh qxpehu ri fudqhv/ frvwv dqg xqordglqj wlphv/ lq xhqfhwkh qxpehu ri yhklfohv wkdw zloo eh xvhg lq wkh whuplqdo1 Lq wklv vhfwlrq/ zh zloo whvw wkh lq xhqfhri wkh f|foh wlphv ri wkh frqwdlqhuv rq wkh plqlpxp qxpehu ri yhklfohv1 Doo rwkhu gdwd/ vxfk dvwkh gulylqj wlphv ri yhklfohv/ uhpdlq wkh vdph lq wkh vwxg|1

Zh revhuyh wkh h�hfwv ri d fudqh wkdw xqordgv frqwdlqhuv idvwhu dqg vorzhu rq wkh yhklfohuhtxluhphqwv1 Idvwhu dqg vorzhu lq wklv frqwh{w phdqv wkdw wkh lqwhuydov phqwlrqhg lq Wdeoh

4<

4 zloo ghfuhdvh ru lqfuhdvh e| d jlyhq qxpehu ri vhfrqgv1 Wkh iudfwlrqv/ 3138/ 3148 dqg vr rq/uhpdlq wkh vdph dv lq Wdeoh 41 Li d fudqh xqordgv/ iru h{dpsoh/ 8 vhfrqgv idvwhu/ wkh �uvwlqwhuydo zlwk iudfwlrq 3=38 ehfrphv 58� 68 vhfrqgv1 Frqvhtxhqwo|/ wkh dyhudjh f|foh wlph zlooeh 8 vhfrqgv +:=6(, ihzhu wkdq wkh dyhudjh f|foh wlph phqwlrqhg lq Vhfwlrq 6141

Zh revhuyh d fudqh wkdw xqordgv frqwdlqhuv 5/ 8/ : dqg 43 vhfrqgv idvwhu dqg vorzhu wkdqwkh glvwulexwlrq lq Wdeoh 41 Iljxuh 7 jlyhv wkh uhvxowv ri wkh vwxg|1 Wklv �jxuh dovr uhsuhvhqwvwkh uhvxowv ri wkh ruljlqdo glvwulexwlrq1

LQVHUW ILJXUH 7Iurp Iljxuh 7 lw fdq eh frqfoxghg wkdw wkh dyhudjh qxpehu ri yhklfohv ghfuhdvhv e| 4<=<( li

wkh f|foh wlphv ri wkh frqwdlqhuv lqfuhdvh e| 48(1 Wkh dyhudjh qxpehu ri yhklfohv lqfuhdvhv e|431;( li wkh f|foh wlphv ri wkh frqwdlqhuv ghfuhdvh e| 48(1 Li wkh phdq ri f|foh wlphv lqfuhdvhv/zkloh wkh yduldqfh uhpdlqv wkh vdph/ wkhq wkh shufhqwdjh ghyldwlrq iurp wkh phdq zloo ghfuhdvh1Lq rwkhu zrugv/ wkh xqfhuwdlqw| ghfuhdvhv dqg wkh ex�huv fdq hdvlhu frpshqvdwh iru wkhvh vpdoofkdqjhv lq f|foh wlphv/ wkxv/ hqdeolqj d shufhqwdjh ghfuhdvh lq yhklfohv zklfk lv hyhq odujhu wkdqwkh lqfuhdvh lq f|foh wlphv1 Wkh pd{lpxp ydoxh dprqj wkh uhsolfdwlrqv htxdov vl{ yhklfohv shufudqh li wkh f|foh wlphv ghfuhdvh e| 43 vhfrqgv1 Li wkh f|foh wlphv lqfuhdvh e| 43 vhfrqgv wkhpd{lpxp ydoxh dprqj wkh uhsolfdwlrqv htxdov �yh yhklfohv shu fudqh1 Ixuwkhupruh/ rqh fdqfrqfoxgh wkdw d fudqh fdq xqordg rqo| 5 vhfrqgv idvwhu wr nhhs wkh vdph qxpehu ri yhklfohv dvlq wkh ruljlqdo fdvh +vhh Vhfwlrq 61416,1615 Vlpxodwlrq vwxg|Zh kdyh exlow d vlpxodwlrq prgho ri d frqwdlqhu whuplqdo wr vwxg| wkh shuirupdqfh ri wkhdqdo|wlfdo prgho1 Duhqd 618 zdv xvhg dv vlpxodwlrq vriwzduh1

Lq wkh vlpxodwlrq prgho zh kdyh lqfrusrudwhg doo wkh surfhvvhv rffxuulqj gxulqj wkh xqordg053

lqj ri d vkls/ qdpho|= irxu Txd| Fudqhv uhpryh frqwdlqhuv iurp wkh vkls dqg sodfh wkhp lqwrex�hu duhdv zlwk fdsdflw| wzr1 Wkh f|foh wlphv ri wkh fudqhv duh gudzq iurp wkh hpslulfdoglvwulexwlrq iurp Wdeoh 41 Oliwlqj yhklfohv wudyho lq d orrs wr wudqvsruw doo wkh frqwdlqhuv wr wkhvwdfn1 Wkh glvwdqfhv wr eh wudyhohg duh vlplodu wr wkh rqhv xvhg lq wkh dqdo|wlfdo prgho1 Wkhhpsw| dqg ixoo wudyho vshhgv duh dovr wkh vdph wr wkh vshhgv ghvfulehg lq Vhfwlrq 6141

Dw wkh vwduw ri wkh vlpxodwlrq/ wkh yhklfohv duh suhvhqw dw wkh TF zdlwlqj lq d urz1 Wkhglvsdwfklqj ri yhklfohv wr fudqhv lv grqh dffruglqj wr wkh sulqflsoh ri qhduhvw yhklfoh �uvw +vhh/iru h{dpsoh/ Hjehox dqg Wdqfkrfr/ 4<;7,1 Wkh �uvw yhklfoh lq olqh lv glvsdwfkhg wr wkh fudqhqhhglqj d yhklfoh1 Rwkhu glvsdwfklqj uxohv fdqqrw eh xvhg lq sudfwlfh gxh wr wkh idfw wkdw wkhyhklfohv wudyho lq d orrs zlwkrxw wkh srvvlelolw| ri sdvvlqj hdfk rwkhu1 Li d ex�hu duhd lv ixoodqg qr yhklfohv duh dydlodeoh/ d fudqh qhhgv wr zdlw zlwk lwv ordg xqwlo d yhklfoh duulyhv1 Diwhuoliwlqj wkh frqwdlqhu wkh yhklfoh vwduwv wudqvsruwlqj wkh frqwdlqhu1 Wkh glvsdwfklqj ri frqwdlqhuvwr vwdfn fudqhv rffxuv gxulqj wkh surfhvv1 Zh xvh wkh iroorzlqj uxoh= wkh frqwdlqhu lv glvsdwfkhgwr wkh fudqh dw wkh iduwkhvw wudyho glvwdqfh dqg wkh fudqh zlwk wkh ohdvw qxpehu ri vfkhgxohgmrev1 Lq wklv zd|/ frqjhvwlrq dw wkh vwdfn zloo eh dyrlghg1 Vl{whhq vwdfnlqj fudqhv duh xvhg lqvlpxodwlrq prgho1

Dw wkh vwdfn/ wkh yhklfohv sodfh wkh frqwdlqhuv lq d ex�hu duhd zlwk fdsdflw| wzr1 Vwdfnlqjfudqhv qhhg d fhuwdlq dprxqw ri wlph wr vwruh frqwdlqhuv lq wkh vwdfn1 Iurp sudfwlfdo h{shulhqfhzh nqrz wkdw f|foh wlphv wr vwruh frqwdlqhuv lq wkh vwdfn htxdo rq dyhudjh wkuhh wlphv wkh f|fohwlphv wr xqordg frqwdlqhuv r� wkh vkls1 Wkhuhiruh/ zh xvh wkh f|foh wlphv ri Wdeoh 4 pxowlsolhge| wkuhh1 Wkh iudfwlrqv uhpdlq wkh vdph1

Fohduo|/ wklv vlpxodwlrq prgho jlyhv d jrrg uhsuhvhqwdwlrq ri d uhdo whuplqdo1 Wkh dvvxps0wlrqv pdgh lq wkh dqdo|wlfdo prgho duh qrw xvhg lq wkh vlpxodwlrq prgho1 Uhohdvh wlphv rifrqwdlqhuv duh qrw nqrzq lq dgydqfh1 Wkhvh duh gudzq iurp wkh hpslulfdo glvwulexwlrq iurp

54

Wdeoh 4 gxulqj wkh vlpxodwlrq1 Gxh wlphv ri frqwdlqhuv duh qrw nqrzq lq dgydqfh1 D gxh wlph rifrqwdlqhu { ghshqgv rq wkh uhohdvh wlphv ri wkh frqwdlqhuv xqordghg r� wkh vkls diwhu frqwdlqhu{1 Wkhvh uhohdvh wlphv ehfrph dydlodeoh gxulqj wkh vlpxodwlrq wkh prphqw d fhuwdlq frqwdlqhulv xqordghg1 Wkh wudyho wlphv ri yhklfohv duh vwrfkdvwlf1 Wkh glvwdqfhv dqg vshhgv ri wkh yhklfohduh nqrzq1 Krzhyhu/ gxh wr/ iru h{dpsoh/ wkh suhvhqfh ri rwkhu yhklfohv/ wkh wudyho wlphv duh qrwnqrzq lq dgydqfh1 Wkh ghvwlqdwlrq ri hdfk frqwdlqhu lv qrw nqrzq lq dgydqfh/ exw ghwhuplqhggxulqj wkh vlpxodwlrq dffruglqj wr wkh glvsdwfklqj uxoh zh kdyh ghvfulehg1 Lw pljkw rffxu wkdwwkh yhklfoh qhhgv wr zdlw ehiruh lw fdq gurs r� d frqwdlqhu dw wkh vwdfn1 Frqwudu| wr wkh dqd0o|wlfdo prgho/ zh kdyh lqfrusrudwhg wkh surfhvvhv dw wkh vwdfn1 Wkhuhiruh/ wkh dvvxpswlrq wkdwvx!flhqw vsdfh lv dozd|v dydlodeoh lq wkh ex�hu duhdv ri vwdfnlqj fudqhv grhv qrw krog1

Lq wkh dqdo|wlfdo prgho wkh remhfwlyh zdv wr wudqvsruw doo wkh frqwdlqhuv vxfk wkdw qr ghod|vrffxu dw wkh txd| fudqhv1 E| hqvxulqj wkdw qr ghod|v rffxu dw wkh txd| fudqhv wkh xqordglqj wlphri wkh vkls fdq eh plqlplvhg1 Wkh remhfwlyh ri wkh vlpxodwlrq vwxg| lv/ wkhuhiruh/ wr h{dplqhkrz pdq| yhklfohv duh uhtxluhg wr wudqvsruw doo wkh frqwdlqhuv vxfk wkdw wkh xqordglqj wlph riwkh vkls lv plqlplvhg1 Wkh vlpxodwlrq prgho lv h{hfxwhg zlwk d ydulrxv qxpehu ri yhklfohv wr�qg wkh plqlpxp qxpehu ri yhklfohv uhtxluhg vxfk wkdw wkh xqordglqj wlph lv plqlplvhg1 Iruhdfk h{shulphqw zh zloo lqglfdwh krz pdq| yhklfohv zh kdyh xvhg lq wkh vlpxodwlrq1

Wr frpsduh wkh uhvxowv ri wkh dqdo|wlfdo prgho dqg wkh vlpxodwlrq prgho zh fduu| rxw vlploduh{shulphqwv dv lq wkh suhylrxv vhfwlrqv1 Vhfwlrq 61514 ghvfulehv wkh uhvxowv li wkh f|foh wlphv duhghulyhg dffruglqj wr Wdeoh 41 Lq Vhfwlrq 61515 zh jlyh wkh uhvxowv lq wkh fdvh zh ydu| wkh f|fohwlphv1 Wr rewdlq d ydolg hvwlpdwlrq ri wkh qxpehu ri yhklfohv/ zh kdyh h{hfxwhg 433 uhsolfdwlrqvri hdfk h{shulphqw1 Dv d uhvxow/ wkh uhodwlyh huuru lv vpdoohu wkdq 5( zlwk d suredelolw| ri <8(1Lq Vhfwlrq 616 zh frpsduh wkh uhvxowv ri erwk prghov dqg lqglfdwh wkh xwlolw| ri wkh dqdo|wlfdoprgho1

55

61514 Uhvxowv ri wkh vlpxodwlrq vwxg|Lq wklv vhfwlrq zh h{dplqh krz pdq| yhklfohv duh uhtxluhg dw d frqwdlqhu whuplqdo zlwk ex�huduhdv ri fdsdflw| wzr1 Wkh remhfwlyh lv wr xqordg wkh vkls dv txlfno| dv srvvleoh1 Zh vlpxodwhwkh iroorzlqj lqvwdqfhv= 83/ ;3 dqg 433 frqwdlqhuv wr eh xqordghg shu Txd| Fudqh1 Rq dyhudjh/wkh frpsxwdwlrq wlphv htxdo : plqxwhv shu lqvwdqfh zlwk 433 uhsolfdwlrqv rq d 4833 PK}1 Lqwkh vlpxodwlrq vwxg| zh fduu| rxw wkh h{shulphqwv vxevhtxhqwo| zlwk 49 wr 58 yhklfohv1 Iljxuh 8suhvhqwv wkh uhvxowv ri wkh h{shulphqwv1 Iru hdfk qxpehu ri yhklfohv/ wkh dyhudjh xqordglqj wlphlq vhfrqgv ryhu kxqguhg uhsolfdwlrqv ri wkh vkls lv lqglfdwhg1 Iru hdfk lqvwdqfh d vwdelolvdwlrqlq wkh qxpehu ri yhklfohv fdq eh qrwlfhg1 Iurp Iljxuh 8 lw lv fohdu wkdw wklv vwdelolvdwlrq rffxuvdw 55 yhklfohv1 Wklv vwdelolvdwlrq lq xqordglqj wlph lv fdxvhg e| wkh rffxuuhqfh ri frqjhvwlrq liwkh qxpehu ri yhklfohv lqfuhdvh1

LQVHUW ILJXUH 8Wkh xqordglqj wlph ri 83 frqwdlqhuv htxdov 747; vhfrqgv/ wkh xqordglqj wlph ri ;3 frqwdlqhuv

htxdov 9575 vhfrqgv dqg wkh xqordglqj wlph ri 433 frqwdlqhuv htxdov :97; vhfrqgv1 Iurp wkhvhgdwd lw fdq eh qrwlfhg wkdw irxu fudqhv hdfk kdqgolqj 83 frqwdlqhuv qhhg wr zrun/ rq dyhudjh/iru 9< plqxwhv1 D kxqguhg frqwdlqhuv shu fudqh fryhu/ rq dyhudjh/ 45: plqxwhv1 Wkhvh uhvxowvduh frpsdudeoh zlwk sodqqlqj krul}rqv iru fudqhv dw wkh suhylrxvo| phqwlrqhg whuplqdo161515 Yduldwlrq lq f|foh wlphv lq wkh vlpxodwlrq prghoZh ydu| wkh f|foh wlphv ri wkh Txd| Fudqhv lq wkh vlpxodwlrq prgho vxfk wkdw fudqhv xqordg 5/8/ : dqg 43 vhfrqgv vorzhu dqg idvwhu wkdq wkh ruljlqdo glvwulexwlrq1 Doo rwkhu sdudphwhuv/ vxfkdv f|foh wlphv ri vwdfnlqj fudqhv/ uhpdlq wkh vdph dv lq wkh ruljlqdo prgho1 Zh pljkw h{shfw/vlplodu wr wkh dqdo|wlfdo fdvh/ wkdw wkh qxpehu ri yhklfohv zloo ghfuhdvh li wkh f|foh wlphv lqfuhdvhdqg wkdw wkh qxpehu ri yhklfohv zloo lqfuhdvh li wkh f|foh wlphv lqfuhdvh1 Dovr/ zh pljkw h{shfw

56

wkdw wkh xqordglqj wlphv zloo lqfuhdvh li wkh f|foh wlphv lqfuhdvh dqg wkdw wkh xqordglqj wlphvlqfuhdvh li wkh f|foh wlphv ghfuhdvh1

E| fduu|lqj rxw vlpxodwlrq h{shulphqwv zlwk d ydu|lqj qxpehu ri yhklfohv/ zh fdq ghulyhwkh qxpehu ri yhklfohv uhtxluhg wr plqlplvh xqordglqj wlphv1 Iru doo wkh lqvwdqfhv/ vwdelolvdwlrqrffxuv lq wkh qxpehu ri yhklfohv1

Lq wkh fdvh wkdw wkh fudqhv xqordg 43/ : ru 8 vhfrqgv idvwhu/ 59 yhklfohv duh uhtxluhg lq wkhfdvh wkdw 83/ ;3 dqg 433 frqwdlqhuv qhhg wr eh xqordghg1 Li fudqhv xqordg 5 vhfrqgv idvwhu/ 56yhklfohv duh uhtxluhg lqvwhdg ri 55 yhklfohv uhtxluhg lq wkh ruljlqdo fdvh1 Rq wkh rwkhu kdqg/ lifudqhv xqordg 5 ru 8 vhfrqgv vorzhu zh vwloo qhhg 55 yhklfohv1 Rqo| 53 yhklfohv duh uhtxluhg lifudqhv xqordg : ru 43 vhfrqgv vorzhu1 Wkxv/ d ghfuhdvh ri </4( lq wkh qxpehu ri yhklfohv fdq ehqrwlfhg li wkh f|foh wlphv lqfuhdvh e| 48(1 Rq wkh rwkhu kdqg/ wkh qxpehu ri yhklfohv lqfuhdvhe| 4; ( li wkh f|foh wlphv ghfuhdvh e| 48(1 Wkh h�hfw ri ghfuhdvlqj f|foh wlphv lv pxfk kljkhuwkdq wkh h�hfw ri lqfuhdvlqj f|foh wlphv1 Wkh phdq ri wkh f|foh wlphv ghfuhdvhv lq wkh �uvw fdvhzkloh wkh yduldqfh uhpdlqv wkh vdph1 Frqvhtxhqwo|/ wkh shufhqwdjh ghyldwlrq iurp wkh phdqlqfuhdvhv zklfk uhvxowv lq dq lqfuhdvh lq xqfhuwdlqw|1 Wkh fdsdflw| ri wkh ex�hu lv wrr uhvwulfwhgdqg lw lv kdughu wr frpshqvdwh iru vpdoo fkdqjhv lq f|foh wlphv1 Wklv fdq dovr eh qrwlfhg iurpwkh idfw wkdw wkh ghfuhdvh lq xqordglqj wlphv lv vpdoohu wkdq rqh pljkw h{shfw1 Li f|foh wlphvghfuhdvh e| 43 vhfrqgv/ wkh dyhudjh f|foh wlph dovr ghfuhdvhv e| 43 vhfrqgv1 Rq dyhudjh/ fudqhvfdq xqordg d kxqguhg frqwdlqhuv 4333 vhfrqgv idvwhu1 Iurp wkh uhvxowv ri wkh vlpxodwlrq vwxg|/zh fdq frqfoxgh wkdw d ghfuhdvh lq xqordglqj wlph iurp rqo| 953 +:97;0:35;, vhfrqgv fdq ehrewdlqhg1

Lq wkh fdvh wkdw f|foh wlphv lqfuhdvh/ ex�huv fdq hdvlhu frpshqvdwh iru wkhvh fkdqjhv1 Wkhuh0iruh/ wkh lqfuhdvh lq xqordglqj wlphv lv orzhu wkdq h{shfwhg1 Iru h{dpsoh/ wkhuh lv d jds iurp 749vhfrqgv lqvwhdg ri 833 li frqwdlqhuv duh xqordghg 43 vhfrqgv vorzhu1 Wkhuh lv d joredo lqfuhdvlqj

57

wuhqg lq xqordglqj wlphv li wkh f|foh wlphv lqfuhdvh1616 Frpsdulvrq uhvxowv dqdo|wlfdo prgho dqg vlpxodwlrqIurp wkh uhvxowv lq Vhfwlrqv 61416 dqg 61514 lw fdq eh frqfoxghg wkdw wkh dqdo|wlfdo prghovoljkwo| xqghuhvwlpdwhv wkh qxpehu ri yhklfohv uhtxluhg wr wudqvsruw doo wkh frqwdlqhuv vxfk wkdwwkh xqordglqj wlphv ri wkh vkls duh plqlplvhg1 Dffruglqj wr wkh dqdo|wlfdo prgho/ 53 yhklfohvvkrxog eh xvhg wr xqordg 83/ ;3 dqg 433 frqwdlqhuv zlwk 7 fudqhv1 Dffruglqj wr wkh vlpxodwlrqprgho 55 yhklfohv duh uhtxluhg wr plqlplvh xqordglqj wlphv1 Lq rwkhu zrugv/ d 43( gl�huhqfh lqyhklfohv h{lvwv1 Wr vwxg| wkh h�hfw ri xvlqj ihzhu yhklfohv/ zh revhuyh wkh xqordglqj wlphv iruxvlqj uhvshfwlyho| 53 dqg 55 yhklfohv1 Wkh uhvxowv iru 55 yhklfohv duh jlyhq lq Vhfwlrq 615141 Lqwkh fdvh wkdw 53 yhklfohv kdg ehhq xvhg/ lw zrxog kdyh uhvxowhg lq wkh iroorzlqj xqordglqj wlphv=7734 vhfrqgv iru 83/ 9:9< vhfrqgv iru ;3 dqg ;6:8 vhfrqgv iru 433 frqwdlqhuv1

Wkh gl�huhqfh lq xqordglqj wlphv lv vpdoo1 Lq wkh fdvh wkdw wkh fudqhv kdqgoh 433 frqwdlqhuvhdfk/ wkh vkls kdv wr ehuwk 45 plqxwhv orqjhu li rqo| 53 yhklfohv duh xvhg1 E| xvlqj ihzhuyhklfohv fudqhv qhhg wr zdlw orqjhu iru yhklfohv1 Wkhvh h{wud zdlwlqj wlphv uhvxow lq dq lqfuhdvhlq xqordglqj wlphv1 Zh fdq frqfoxgh wkdw d ghfuhdvh lq wkh qxpehu ri yhklfohv ri 43( uhvxowv lqd vpdoohu lqfuhdvh lq xqordglqj wlphv +;1:(,1 Wklv fdq eh h{sodlqhg dv iroorzv= iurp wkh uhvxowvlq Iljxuh 8 dqg uhvxowlqj xqordglqj wlphv lw fdq eh frqfoxghg wkdw wkh odz ri glplqlvklqj uhwxuqvdssolhv wr wkh xvh ri pruh yhklfohv1 Wkh ghfuhdvh lq xqordglqj wlphv lv pxfk kljkhu lq wkh fdvhwkdw zh xvh 4: yhklfohv lqvwhdg ri 49 wkdq xvlqj 55 lqvwhdg ri 54 yhklfohv1 Lw fdq eh vhhq wkdwxvlqj 4 ru 5 yhklfohv ohvv wkdq wkh qxpehu ri yhklfohv rewdlqhg iurp wkh vlpxodwlrq vwxg| uhvxowvlq d vpdoo lqfuhdvh lq xqordglqj wlphv ri wkh vkls1 Wkxv/ xqordglqj wlphv duh ohvv vhqvlwlyh wr wkhqxpehu ri yhklfohv lq wkh fdvh wkdw wkh qxpehu ri yhklfohv dssurdfkhv wkh rswlpdo ydoxh1 Wkhpdlq remhfwlyh lv/ wkhuhiruh/ wr dssurdfk wkh rswlpdo ydoxh ri wkh qxpehu ri yhklfohv lqvwhdg

58

ri ghwhuplqlqj lw h{dfwo|1 Wkh dqdo|wlfdo prgho suhvhqwhg lq wklv sdshu dssurdfkhv wkh ydoxhvrewdlqhg zlwk vlpxodwlrq1 Dv h{sodlqhg/ vpdoo xqghuhvwlpdwhv ri wkh qxpehu ri yhklfohv kdyh dvpdoo lpsdfw rq wkh xqordglqj wlphv ri wkh vkls1 Frqvhtxhqwo|/ wkh dqdo|wlfdo prgho lv xvhixodqg surylghv d jrrg hvwlpdwh ri wkh qxpehu ri yhklfohv uhtxluhg1

Ixuwkhupruh/ rqh fdq qrwlfh wkdw wkh gl�huhqfh lq xqordglqj wlphv lv vpdoohu lq wkh fdvhwkdw ihzhu frqwdlqhuv duh kdqgohg1 Lq wkh ehjlqqlqj ri wkh vlpxodwlrq/ d zdup0xs shulrg fdq ehqrwlfhg1 Doo wkh yhklfohv duh suhvhqw dw wkh fudqhv dqg gluhfwo| uhdg| wr wudqvsruw frqwdlqhuv1Li d yhklfoh kdv wudqvsruwhg d frqwdlqhu/ lw wudyhov edfn wr wkh hqg ri wkh urz ri yhklfohv dw wkhfudqh1 Li doo wkh yhklfohv duh shuiruplqj wkhlu wdvnv/ lw pljkw rffxu/ frqwudu| wr wkh vlwxdwlrqdw wkh ehjlqqlqj ri wkh vlpxodwlrq/ wkdw qrw hqrxjk yhklfohv duh suhvhqw dw wkh fudqhv1 Lq wkdwfdvh/ fudqhv qhhg wr zdlw iru yhklfohv1 Wkhuhiruh/ wkh h�hfw ri xvlqj ihzhu yhklfohv fdq eh voljkwo|odujhu lq wkh fdvh wkdw pruh frqwdlqhuv duh kdqgohg1

Wkh voljkw gl�huhqfh lq wkh qxpehu ri yhklfohv dqg wkh vpdoo lpsdfw ri xvlqj ihzhu yhklfohv rqxqordglqj wlphv srlqw wr wkh xvhixoqhvv ri wkh dqdo|wlfdo prgho1 Wkh dqdo|wlfdo prgho surylghvuhvxowv forvh wr wkh uhvxowv ri wkh vlpxodwlrq prgho1 Wkh fkrlfh ri prghoolqj wkh sureohp dv dghwhuplqlvwlf sureohp/ e| xvlqj zruvw fdvhv iru vwrfkdvwlf hohphqwv lq wkh prgho/ lv wkhuhiruh/mxvwl�hg e| wkh uhvxowv suhvhqwhg lq wklv vhfwlrq1

Iru ixuwkhu hydoxdwlrq ri wkh xwlolw| ri wkh dqdo|wlfdo prgho/ zh kdyh ydulhg wkh f|foh wlphvri wkh txd| fudqhv lq Vhfwlrqv 61417 dqg 615151 Lq wkh fdvh wkdw d fudqh xqordgv 5 vhfrqgv idvwhu/wkh dyhudjh qxpehu ri yhklfohv lqfuhdvhv lq wkh dqdo|wlfdo prgho1 Wkh pd{lpxp ydoxh dprqjwkh uhsolfdwlrqv lv vwloo 81 Wkhuhiruh/ wkh qxpehu ri yhklfohv uhtxluhg lq wkh dqdo|wlfdo prghouhpdlqv wkh vdph1 Iurp Wdeoh 7 zh fdq frqfoxgh wkdw wkh uhvxowv ri wkh vlpxodwlrq prgholqglfdwh wkdw zh qhhg wr xvh 56 yhklfohv lqvwhdg ri 55 yhklfohv lq wkh ruljlqdo fdvh1 Erwk prghovlqglfdwh dq lqfuhdvh lq wkh qxpehu ri yhklfohv1 Krzhyhu/ wklv lqfuhdvh fdq rqo| eh qrwlfhg lq wkh

59

dqdo|wlfdo prgho e| revhuylqj wkh dyhudjh qxpehu ri yhklfohv1 Li zh xvh 53 yhklfohv wr xqordg83 frqwdlqhuv zh pljkw h{shfw d ghod| lq wkh dyhudjh xqordglqj wlph ri wkh vkls ri 7:4 vhfrqgv+43(, dffruglqj wr wkh vlpxodwlrq vwxg|1 Dq ghfuhdvh lq wkh f|foh wlphv ri 43 vhfrqgv uhvxowv lqdq lqfuhdvh iurp 53 wr 57 yhklfohv +53(, dffruglqj wr wkh uhvxowv ri wkh dqdo|wlfdo prgho1 Wkhvlpxodwlrq uhvxowv lqglfdwh dq lqfuhdvh iurp 55 wr 59 yhklfohv +4;(,1 Rqh fdq frqfoxgh wkdw erwkprghov lqglfdwh d vlplodu lqfuhdvh lq wkh qxpehu ri yhklfohv1 E| xvlqj 57 yhklfohv lqvwhdg ri 59wr xqordg 433 frqwdlqhuv wkh vlpxodwlrq uhvxowv lqglfdwh wkh rffxuuhqfh ri d ghod| ri wkh vkls ri717(1 D gl�huhqfh ri ;16( lq wkh qxpehu ri yhklfohv kdv d vpdoohu lpsdfw rq wkh gl�huhqfh lqxqordglqj wlphv1 Iru erwk prghov zh kdyh vkrzq d vlplodu sdwwhuq lq wkh qxpehu ri yhklfohv e|ghfuhdvlqj f|foh wlphv1

Lq wkh fdvh wkdw f|foh wlphv ri fudqhv lqfuhdvh e| 43 vhfrqgv/ zh kdyh vkrzq lq Vhfwlrq 61417d ghfuhdvh lq wkh dyhudjh qxpehu ri yhklfohv lq wkh dqdo|wlfdo prgho ri 4<1<(1 Wkh qxpehu riyhklfohv uhtxluhg shu fudqh vwloo htxdov 71 Frqvhtxhqwo|/ ghfuhdvlqj f|foh wlphv lq wkh dqdo|wlfdoprgho rqo| lpsdfwv rq wkh dyhudjh qxpehu ri yhklfohv1 Dffruglqj wr wkh uhvxowv lq Wdeoh 7/ wkhqxpehu ri yhklfohv uhtxluhg ghfuhdvhv iurp 55 wr 53 +<14(, li fudqhv xqordg 43 vhfrqgv vorzhu1E| lqfuhdvlqj f|foh wlphv d joredo ghfuhdvlqj sdwwhuq lq wkh qxpehu ri yhklfohv uhtxluhg fdq ehqrwlfhg e| erwk prghov1

Wkh uhvxowv ri wkh dqdo|wlfdo prgho duh forvh hvwlpdwhv ri wkh qxpehu ri yhklfohv uhtxluhgwr xqordg d vkls dv txlfno| dv srvvleoh frpsduhg wr wkh uhvxowv ri wkh vlpxodwlrq prgho1 LqVhfwlrq 4 zh ghvfulehg phwkrgv iru ghwhuplqlqj yhklfoh uhtxluhphqwv zlwkrxw wlph0zlqgrzv1Wkh shufhqwdjh gl�huhqfhv ehwzhhq dqdo|wlfdo dqg vlpxodwlrq prghov lq wkhvh sdshuv duh odujhuwkdq wkh gl�huhqfhv irxqg lq wklv sdshu1 Frqfoxglqj/ zh kdyh ghyhorshg dq dqdo|wlfdo prghoiru d fdvh rffxuulqj lq sudfwlfh dqg qrw vwxglhg lq olwhudwxuh iru zklfk zh kdyh vkrzq wkdw lwshuirupv zhoo lq wkh frqwh{w ri d frqwdlqhu whuplqdo1 Frqvhtxhqwo|/ lw fdq eh frqfoxghg wkdw wkh

5:

dvvxpswlrqv pdgh lq Vhfwlrq 5 duh qrw xquhdolvwlf lq wkh prgho1 Wkh shuirupdqfh ri wkh prgholv kdugo| lq xhqfhg e| wkhvh dvvxpswlrqv17 FrqfoxvlrqvWr uhpdlq frpshwlwlyh lw lv lpsruwdqw iru frqwdlqhu whuplqdo pdqdjhphqw wr uhgxfh frvwv dqgwr sohdvh vklsrzqhuv/ zkr ghpdqg d idvw xqordglqj dqg ordglqj ri wkhlu vklsv1 Frqwdlqhu whu0plqdov frqvwdqwo| kdyh wr vhdufk iru qhz whfkqltxhv/ vxfk dv dxwrpdwhg wudqvsruwdwlrq v|vwhpvdqg zd|v ri sodqqlqj dqg frqwuro1 Wkhuhiruh/ qhz sodqqlqj dqg frqwuro frqfhswv qhhg wr ehghyhorshg iru wkh ydulrxv v|vwhpv lq wkh frqwdlqhu whuplqdo1 Rqh ri wkh sodqqlqj sureohpv lvwkh ghwhuplqdwlrq ri wkh plqlpxp qxpehu ri yhklfohv uhtxluhg xqghu wlph0zlqgrz frqvwudlqwv1Wklv sdshu suhvhqwv dq lqwhjhu olqhdu surjudpplqj prgho wr vroyh wklv sureohp1

Gxulqj wkh xqordglqj surfhvv frqwdlqhuv duh uhpryhg iurp wkh vkls e| txd| fudqhv dqgwudqvsruwhg wr wkh vwrudjh duhd e| oliwlqj yhklfohv1 Txd| fudqhv sodfh frqwdlqhuv lq d ex�hu duhddw d fhuwdlq uhohdvh wlph1 Wkhvh ex�hu duhdv kdyh olplwhg fdsdflw| gxh wr fdsdflw| uhvwulfwlrqvdw wkh whuplqdo1 Wr dyrlg ghod|v dw wkh vkls/ txd| fudqhv vkrxog qrw zdlw iru d sodfh lq wkhex�hu duhd1 Wkhuhiruh/ wkh frqwdlqhu �uvw sodfhg lq wkh ex�hu �qdoo| qhhgv wr eh wudqvsruwhgwkh prphqw wkh ex�hu duhd lv ixoo dqg d qhz frqwdlqhu qhhgv wr eh sodfhg lq1 Vrphzkhuh lq wklvwlph0zlqgrz ^uhohdvh wlph/ gxh wlph` d yhklfoh oliwv wkh frqwdlqhu iurp wkh ex�hu duhd dqg vwduwvwudqvsruwlqj lw wr wkh vwrudjh duhd1 Dw wkh vwrudjh duhd wkh yhklfoh sodfhv wkh frqwdlqhu lq wkhex�hu duhd ri wkh vwrudjh duhd1 D vwdfnlqj fudqh vwruhv wkh frqwdlqhu lq wkh vwdfn1 Wkh sureohplv qrz wr ghwhuplqh wkh plqlpxp yhklfoh hhw vl}h vxfk wkdw doo wkh frqwdlqhuv duh wudqvsruwhgzlwklq wkhlu wlph0zlqgrzv1 Iurp wkh olwhudwxuh ryhuylhz lq Vhfwlrq 4 zh fdq frqfoxgh wkdw/ghvslwh wkh sudfwlfdo uhohydqfh/ qr olwhudwxuh h{lvwv rq wklv sureohp1 Vhfwlrq 5 suhvhqwv dqlqwhjhu olqhdu surjudpplqj prgho wr vroyh wklv sureohp1

5;

Wr whvw wkh shuirupdqfh ri wklv dqdo|wlfdo prgho lq sudfwlfh zh kdyh fduulhg rxw vhyhudoqxphulfdo h{shulphqwv1 Iluvw/ lq Vhfwlrq 614 zh kdyh h{hfxwhg h{shulphqwv zlwk wkh dqdo|wlfdoprgho1 Diwhu ghwhuplqlqj d uholdeoh vl}h ri wkh glvfuhwlvdwlrq vwhs dqg ex�hu duhd/ zh kdyhghwhuplqhg yhklfoh uhtxluhphqwv lq Vhfwlrq 614161 Vhfwlrq 61417 jlyhv wkh uhvxowv ri d vhqvlwlylw|dqdo|vlv1 Zh kdyh vwxglhg wkh h�hfw ri f|foh wlphv rq yhklfoh uhtxluhphqwv1 Wr ydolgdwh wkhuhvxowv ri wkh dqdo|wlfdo prgho zh kdyh ghyhorshg d vlpxodwlrq prgho1 Vhfwlrq 615 ghvfulehv wkhvlpxodwlrq prgho lq ghwdlo1 Zh kdyh fduulhg rxw vlplodu h{shulphqwv dv zlwk wkh dqdo|wlfdo prghowr jlyh d uholdeoh frpsdulvrq ri erwk prghov1 Vhfwlrq 616 frpsduhv uhvxowv ri erwk prghov1

Lw fdq eh frqfoxghg wkdw wkh hvwlpdwhv ri wkh dqdo|wlfdo prgho duh forvh wr wkh uhvxowv ri wkhvlpxodwlrq vwxg| xqghu ydulrxv frqglwlrqv1 Lq wkh ruljlqdo fdvh/ wkh dqdo|wlfdo prgho voljkwo|xqghuhvwlpdwhv wkh ydoxhv ri wkh qxpehu ri yhklfohv uhtxluhg1 Krzhyhu/ wkh 43( gl�huhqfh lqyhklfohv ohdg wr vpdoohu gl�huhqfhv lq xqordglqj wlphv +odz ri glplqlvklqj uhwxuqv,1 Lw fdq ehqrwlfhg wkdw xvlqj 4 ru 5 yhklfohv ohvv wkdq wkh ydoxh rewdlqhg zlwk vlpxodwlrq uhvxowv lq rqo| dvpdoo lqfuhdvh lq wkh xqordglqj wlphv ri wkh vkls1 E| ydu|lqj f|foh wlphv erwk prghov jlyh vlplodusdwwhuqv lq wkh qxpehu ri yhklfohv uhtxluhg1 Li zh ghfuhdvh wkh f|foh wlphv e| 43 vhfrqgv/ zh�qg dq lqfuhdvh lq wkh qxpehu ri yhklfohv ri dssur{lpdwho| 53( iru erwk prghov1 E| lqfuhdvlqjf|foh wlphv erwk prghov lqglfdwh d joredo ghfuhdvlqj wuhqg lq qxpehu ri yhklfohv uhtxluhg1

Iurp wkh uhvxowv suhvhqwhg lq Vhfwlrq 6 zh fdq frqfoxgh wkdw wkh dqdo|wlfdo prgho shuirupvzhoo lq wkh frqwh{w ri d frqwdlqhu whuplqdo1 Wkh uhvxowv ri wkh prgho jlyh d jrrg hvwlpdwh ri wkhvl}h ri wkh yhklfoh hhw uhtxluhg ehiruh wkh vwduw ri wkh rshudwlrq1UhihuhqfhvK1S1 Fhohq/ U1M1Z1 Vohjwhqkruvw/ U1Wk1 ydq ghu Kdp/ D1 Qdjho/ U1 gh Yrv Exufkduw/ M1 ydq ghqEhuj/ M1M1P1 Hyhuv/ G1J1 Olqghlmhu/ U1 Ghnnhu/ S1M1P1 Phhuvpdqv/ P1E1P1 gh Nrvwhu/ U1 ydqghu Phhu/ D1I1F1 Fduohexu dqg I1M1D1P1 Qrrlmhq/ �IDPDV QhzFrq�/ Frqfhsw WUDLO Uhsruw/

5<

Whfkqlfdo Xqlyhuvlw| ri Ghoiw dqg Hudvpxv Xqlyhuvlw| Urwwhugdp/ Wkh Qhwkhuodqgv/ 4<<:1M10I1 Frughdx/ J1 Ghvdxoqlhuv/ M1 Ghvurvlhuv/ P1P1 Vrorprq dqg I1 Vrxplv/ �YUS zlwk wlphzlqgrzv�/ lq Wkh Yhklfoh Urxwlqj Sureohp/ S1 Wrwk dqg G1 Yljr +hgv,/ VLDP Prqrjudskv rqGlvfuhwh Pdwkhpdwlfv dqg Dssolfdwlrqv/ 48:04<6/ 53351P1 Ghoo*Dplfr/ P1 Ilvfkhwwl dqg S1 Wrwk/ �Khxulvwlf dojrulwkpv iru wkh pxowlsoh ghsrw yhklfohvfkhgxolqj sureohp�/ Pdqdjhphqw Vflhqfh 6<+4,/ 4480458 +4<<6,1S1M1 Hjehox/ �Wkh xvh ri qrq0vlpxodwlrq dssurdfkhv lq hvwlpdwlqj yhklfoh uhtxluhphqwv lq dqdxwrpdwhg jxlghg yhklfoh edvhg wudqvsruw v|vwhp�/ Pdwhuldo Iorz 7/ 4:065 +4<;:,1S1M1 Hjehox dqg M1P1D1 Wdqfkrfr/ �Fkdudfwhul}dwlrq ri dxwrpdwlf jxlghg yhklfoh glvsdwfklqjuxohv�/ Lqwhuqdwlrqdo Mrxuqdo ri Surgxfwlrq Uhvhdufk 55+6,/ 68<06:7 +4<;7,1D1P1 Odz dqg Z1G1 Nhowrq/ Vlpxodwlrq Prgholqj ) Dqdo|vlv/ vhfrqg hglwlrq/ PfJudz0KlooLqwhuqdwlrqdo Hglwlrqv/ Qhz \run +4<<4,1V1 Udmrwld/ N1 Vkdqnhu dqg M1O1 Edwud/ �Ghwhuplqdwlrq ri rswlpdo DJY hhw vl}h iru dq IPV�/Lqwhuqdwlrqdo Mrxuqdo ri Surgxfwlrq Uhvhdufk 69+8,/ 44::044<; +4<<;,1L1I1D1 Ylv/ Sodqqlqj dqg Frqwuro Frqfhswv iru Pdwhuldo Kdqgolqj V|vwhpv/ HULP Sk1G1 Vh0ulhv Uhvhdufk lq Pdqdjhphqw 47/ Hudvpxv Xqlyhuvlw| Urwwhugdp/ Urwwhugdp/ Wkh Qhwkhuodqgv+5335,1L1I1D1 Ylv dqg U1 gh Nrvwhu/ �Wudqvvklsphqw ri frqwdlqhuv dw d frqwdlqhu whuplqdo= dq ryhuylhz�/Hxurshdq Mrxuqdo ri Rshudwlrqdo Uhvhdufk 47:/ 4049 +5336,1L1I1D1 Ylv/ U1 gh Nrvwhu/ N1M1 Urrgehujhq dqg O1Z1S1 Shhwhuv/ �Ghwhuplqdwlrq ri wkh qxpehuri dxwrpdwhg jxlghg yhklfohv uhtxluhg dw d vhpl0dxwrpdwhg frqwdlqhu whuplqdo�/ Mrxuqdo ri wkhRshudwlrqdo Uhvhdufk Vrflhw| 85/ 73<074: +5334,1

63

Iljxuh 4= Surfhvvhv dw d frqwdlqhu whuplqdo

64

Iljxuh 5= glvfuhwlvhg wlph0zlqgrz ^dl> el`1 jl� htxdov wkh prphqw wkh yhklfoh fdq vwduw wudqv0sruwlqj wkh frqwdlqhu1 jlpl htxdov wkh prphqw wkh yhklfoh xowlpdwho| qhhgv wr wudqvsruw wkhfrqwdlqhu1

65

Iljxuh 6= H{dpsoh ri judsk J1 �/ lqglfdwhv wkdw frqwdlqhuv zlwk wkhvh vwduw lqvwdqwv duhfrpsdwleoh1 111111/ lqglfdwhv dufv ehwzhhq vrxufh dqg frqwdlqhuv dqg dufv ehwzhhq frqwdlqhuv dqgvlqn

66

Iljxuh 7= Uhvxowv iru d whuplqdo zlwk ex�hu duhdv ri vl}h 5 lqglfdwlqj wkh dyhudjh qxpehu riyhklfohv shu fudqh li wkh f|foh wlphv ghyldwh iurp wkh ruljlqdo glvwulexwlrq lq Wdeoh 41 Rq wkh{0d{lv wkh gl�huhqfh lq vhfrqgv zlwk wkh ruljlqdo glvwulexwlrq lv lqglfdwhg1

67

Iljxuh 8= Dyhudjh xqordglqj wlph lq vhfrqgv iru d jlyhq qxpehu ri yhklfohv

68

Iudfwlrq F|foh wlph lq vhfrqgv3138 630733148 730833153 83093314< 930:3314< :30;33143 ;30<3313; <304533136 45304833134 48304;3

Wdeoh 4= Hpslulfdo glvwulexwlrq ri wkh f|foh wlphv

69

Lqvwdqfh& uhsolfdwlrqvzlwk & yhk1

@ 7& uhsolfdwlrqvzlwk & yhk1

@ 8dyhudjh& ri

yhklfohvpd{lpxp

& riyhklfohv

83 64 9< 719< 8;3 43 <3 71<3 8433 7 <9 71<9 8

Wdeoh 5= Uhvxowv iru wkh fdvh wkdw d ex�hu zlwk vl}h 5 lv xvhg1Lqglfdwhv krz pdq| uhsolfdwlrqv ri wkh 433 duh irxqg zlwk d plqlpxp qxpehu ri uhtxluhg

yhklfohv ri 7 ru 81 Ixuwkhupruh/ lw suhvhqwv wkh dyhudjh dqg pd{lpxpydoxh ri wkh qxpehu ri yhklfohv

6: