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Statistics of extremes: Estimation and optimality

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(1)Tilburg University. Statistics of extremes Ferreira, A.. Publication date: 2002 Document Version Publisher's PDF, also known as Version of record Link to publication in Tilburg University Research Portal. Citation for published version (APA): Ferreira, A. (2002). Statistics of extremes: Estimation and optimality. CentER, Center for Economic Research.. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.. Download date: 18. okt. 2021.

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(277) w‰7dl {Bi lJ|5moeDcCi‹s z7x‹moei c‚¡rgx x rnŠv

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(279) i cC‰i l5pqd +% !-{ !0* šŽ #  {>mol5pHŠ

(280) i cC‰7rgzqcTx rvCvro‚ kgdl7de+mox i cU‡)ŠTd#€m0‡}mgvCvDz7€HdQcC‰5mnc # R ¬@e > G! V R mol5p– % V R Aƒ$&z7e % . . . . . . . "!  &  &  mgvCvDz7€H|qcCi rgl5v#i €H|7x ‡–cC‰5mnc Ugƒ gƒ  Q‰7rgx‹p7v#x rqsmox x ‡]z7l7i ‚¡rgeC€Hx ‡–rgl šŽ # _ ƒ2w‰7dy7i fnmoeCi‹mncCd . . l7rgeC€mox pqi‹vUcCeCi y7zqcCi rglJvCmncCi‹vU5dvTcC‰7dvDd=s rgl5pqi cCi rgl5vƒ ul5pqde Ugƒ gƒ  TŠTd#|7eCrg|BrvDd=ml7dŠ dvUcCi €mncCrge

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(290) i cC‰`cC‰7d2x i cCde+mncCz7eCdJi‹v)pqi‹vCs z5vCvDdpWƒ w‰>z5vi l`cC‰7d ‚¡rgx x rnŠ

(291) i l7kHŠTdTjUz5vUc

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(294) vCm0‡g{7ro‚z7|7|Bderge+pqdevUc+mncCi‹vUcCi‹sv cCrz5vDdQi l}cC‰7d=dvUcCi €mncCi rgl šmgv

(295) y7eCi d 5‡d +q|7x‹moi l7dp)i l Ugƒ gƒ ‘  tƒ 0 l}cC‰7i‹vs+‰5mo|qcCde

(296) ŠTd=mox‹vDrpqdmox Š

(297) i cC‰}cC‰7d=dvUcCi €mncCi rgl%ro‚  tƒ –d=|7eCrg|BrvDd=mol%mgp7mo|  cCi fgd!yBr>roc+vUcCe+mo|2|7eCrqs dpqz7eCdg{5‚¡eCrg€RŠ

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(311) x i cCde+mncCz7eCdi‹v9cCr=z5vDdTcC‰7d

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(357) x 36i € 5 6

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(359) ^  * šŽ # _ rge

(360) ^ Ž ^ _ tƒ 0 l yBrocC‰ZsmgvDdvQ|7z7eCdx ‡2l7rgl  |5moe+mo€Hd cCeCi‹s€Hd cC‰7rqp7vQpqr}l7roc=ŠTrgeCh--ƒ &l7x ‡Ji ‚

(361) ^ _ l7rgl  |5moe+mo€Hd cCeCi‹s=€Hd cC‰7rqp7v=smol yBd#vDz5ss dvCvU‚¡z7x @i l7€mo‰7x­{WªgªgŽ tƒw‰7dQz5vDdQro‚F€Hrqpqdx‹vT‚¡rgeTcC‰7dQc+moi x vDz7kgkgdvUcCdpy>‡d +>cCeCd€Hd fnmox z7dcC‰7drgeC‡kgi fgdv z5v

(362) mHvDdl5vDi y7x dQŠm0‡Hro‚9d +>cCe+mo|Brgx‹mncCi l7k‚¡eCrg€ mol}i lcCdeC€Hdpqi‹mncCd z5molcCi x d cCr=rgl7d

(363) rgzqc+vDi‹pqdTcC‰7dvCmo€H|7x dz7l7x dvCvrgl7dz5vDdvrgl7dro‚BcC‰7d

(364) kgdl7de+mox i‹vDdp!¬moeCd cCrpqi‹vUcCeCi y7zqcCi rgl5v Q¬ p . G8 [ UK 9 ?8  L ‚¡rgecC‰7rvDWd 8%‚¡rge

(365) Š

(366) ‰7i‹s+‰ K 9 ?8N/`Ž # ­Œqƒ gƒ  -*N, ‚¡rge

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(368) ‰7i‹s+‰–Ž2V. G8 V !Š

(369) ‰7deCdJ. R i‹vQm}vDz7i c+moy7x d!|BrvDi cCi fgd‚¡z7l5stcCi rgl ƒw‰7i‹v €Hdmol5v‚¡rgecC‰7d  z5molcCi x d‚¡z7l5stcCi rgl}cC‰5mnc

(370) ‚¡rge 8H/ Ž. xQ 36i € 5 U. &  Q P . & U &  . „7rge

(371) rgz7e

(372) |7eCrgy7x d€ cC‰7i‹v

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(374)  Y. R. 8 . G&. .

(375)  & # U &  9 . # G . . . '.  #. i­ƒ dgƒ9mol=d +>cCeCd€Hdz5molcCi x di‹vWx i l7hgdpcCrmolQi lcCdeC€Hdpqi‹mncCdz5molcCi x d ¡Š

(376) ‰7i‹s+‰=smolQyBd@dvUcCi €mncCdp f>i‹m–cC‰7dJd€H|7i eCi‹smoxpWƒ ‚Uƒ :y>‡£z5vDi l7k cC‰7d Q¬ p mo|7|7eCr +qi €mncCi rgl ƒ w‰7d%d +>cCeCd€Hd z5molcCi x d dvUcCi €mncCrge

(377) y5mgvDdp)rgl}cC‰7i‹veCdx‹mncCi rgl}i‹v

(378)        8

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(380)  

(381) 9 .  ­Œqƒ gƒ ,

(382)   Š

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(384) B !&%

(385) B B !

(386) 

(387) 2moeCdcC‰7dHrge+pqdeQvUc+mncCi‹vUcCi‹svQmol5p .  mol5p

(388)   vDz7i c+moy7x d}dvUcCi €mncCrge+v=‚¡rg"e . mol5p –di‹vCvD€mol {ª   ˆ>€Hi cC‰ {ª  M *Tr>rv{ª  M gr>dg{Fª  !mol5p}€mol>‡}rocC‰7de+v tƒ

(389) [yBrgz7l5p7moeC‡}smgvDd=i‹v <VŽmol5p Ž7ƒ

(390) w‰7dl%cC‰7d!vCmo€Hd d +q|7eCdvCvDi rgl ¡Š

(391) i cC‰

(392) ^ Ž smolJyBdz5vDdp2mgvmol2dvUcCi €mncCrgero‚cC‰7deCi kg‰cdl5pq|Brgi lcro‚cC‰7d |7eCrgy5moy7i x i cU‡)pqi‹vUcCeCi y7zqcCi rgl {7i l}cC‰7d#vCmo€Hd Q¬ p}vDd c  z7| ƒ w‰7d)s+‰7rgi‹s dHro‚ ¡rge#e+mncC‰7de -{FcC‰7di l5pqd + ro‚TcC‰7drge+pqde#vUc+mncCi‹vUcCi‹sv=‚¡eCrg€ Š

(393) ‰7deCd rglcC‰7d Q¬ p)mo|7|7eCr +qi €mncCi rgli‹vTyBdx i dfgdpcCr!yBdQfnmox i‹p @i‹vs eCz5s i‹mox5‚¡rgeTcC‰7d=mgss z7e+mgs ‡!ro‚ cC‰7d |7eCrqs dpqz7eCdgƒw‰7drg|qcCi €moxqfnmox z7dpqd|Bdl5p7vrglcC‰7d

(394) z7l5pqdeCx ‡>i l7kQpqi‹vUcCeCi y7zqcCi rglmol5pi‹v@meCdvDz7x c ro‚y5mox‹mol5s i l7k}fnmoeCi‹mol5s dHmol5p]y7i‹mgv=s rg€H|Brgl7dlc+vƒ 0 l]cC‰7i‹v=|5mo|Bde=ŠTd!|7eCdvDdlc#m}yBr>roc+vUcCe+mo| |7eCrqs dpqz7eCdcCr}rgyqc+moi l2cC‰7i‹vrg|qcCi €moxfnmox z7dmgp7mo|qcCi fgdx ‡gƒ=w‰7dH€Hd cC‰7rqp]i‹vQmol]d +>cCdl5vDi rgl2ro‚. . # ! 3G  # &

(395) . . . # G . . . . NG . . &. # =!

(396). .

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(398) .

(399) ).  Š

(400) ‰5mncQŠTdz5vDdp%‚¡rgergyqc+moi l7i l7kcC‰7d!rg|qcCi €mox9l>z7€yBdero‚@rge+pqdeQvUc+mncCi‹vUcCi‹svi l2dvUcCi €mncCi l7k G š†mol7i dx‹vCvDrgl2d c=mox­ƒ { ŒoŽgŽ7mol5p2†e+moi‹vD€md c=mox­ƒ {ªgªgª tƒw‰7d|5mo|Bde'mox xmol5p –di‹vCvD€mol Uªgª H|7eCdvDdlc+v)m švDi €Hi x‹moe)y7zqcJpqi -deCdlc yBr>roc+vUcCe+mo| €Hd cC‰7rqp ‚¡rge}vDrgx f>i l7kZcC‰7d–vCmo€Hd rg|qcCi €mox i cU‡|7eCrgy7x d€%{ql7rocT‚¡rgeTcC‰7dz5molcCi x dy7zqc‚¡rgeTcC‰7dQd +7s ddpqi l7k|7eCrgy5moy7i x i cU‡ro‚9m‰7i kg‰. “-œ˜n9¨ oœ™šB“. x dfgdxFŠ

(401) ‰7i‹s+‰2i‹vvDi €Hi x‹moeƒul7x i hgd#cC‰5mnc|5mo|Bde{BŠTdpqrl7rocQmgvCvDz7€Hdmol>‡}ro‚cC‰7d|5moe+mo€Hd cCde+v h>l7rnŠ

(402) l ƒ[ x‹vDr£rgz7e)s rgl5pqi cCi rgl5v)mox x rnŠ ‚¡rge)€z5s+‰vD€mox x defnmox z7dvro‚

(403) -ƒ[w‰7d z5molcCi x d |7eCrgy7x d€ i‹v#€HrgeCd)s rg€H€HrglZi l mo|7|7x i‹smncCi rgl5v=cC‰5molZcC‰7d)i l>fgde+vDd|7eCrgy7x d€ ro‚d +7s ddp7mol5s d |7eCrgy5moy7i x i cCi dvro‚m!‰7i kg‰}x dfgdx­ƒ –d}eCdvUcCeCi‹stcrgz7e+vDdx fgdv=cCr2cC‰7d}e+mol7kgd / % ƒ w‰7i‹ve+mol7kgdi‹v!€HrvUc!i €H|BrgeDc+molc!i l mo|7|7x i‹smncCi rgl5v!mol5p£i l£cC‰7i‹v!e+mol7kgdi cHi‹v!€HrvUc!d

(404) s i dlc!cCr–s+‰7r>rvDd}m]vDd z7dl5s d : i l ­Œqƒ gƒ , cC‰5mnc#kgr>dvcCrJi lq5l7i cU‡]Š

(405) i cC‰ ƒ) x‹vDr5{9vDi l5s dŠTds rgl5vDi‹pqdeQc+moi x@|7eCrg|BdeDcCi dv{FŠTd ‰5m0fgd)cCr x i €Hi crgz7e+vDdx fgdv#cCrZvDd z7dl5s d 8 # ^ _ ƒ dl5s d%ŠTdJmoeCd%pqdmox i l7k Š

(406) i cC‰ i lcCdeC€Hdpqi‹mncCd%vDd z7dl5s dv ¡i­ƒ dgƒ {@cC‰7dJs rgeCeCdvD|Brgl5pqi l7k]rge+pqdeHvUc+mncCi‹vUcCi‹sJv !

(407)  

(408) moeCdQi lcCdeC€Hdpqi‹mncCd ^`_ # ^  Ž ^`_ ' ­Œqƒ gƒ w‰7d=€moi l}i‹pqdm!i‹vcC‰7dQ‚¡rgx x rnŠ

(409) i l7k5ƒ –d#vDddh % ­Œqƒ gƒ   moeCki lq  ‚ mgvƒ  8

(410)   S8

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(412). . #. . &. . #. Š

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(414) pqi‹vUcCeCi y7zqcCi rgl { st‚Uƒ%w‰7drgeCd€ Œqƒ Œqƒ mol5p]cC‰7d)pqi‹vCs z5vCvDi rgl–cC‰7deCdmn‚ cCde Qmol5p ]e+mol7kgdv‚¡eCrg€%{9vCm0‡g{Fx rgk cCr  ¡x rgk  cC‰7i‹vd +q|7eCdvCvDdv cC‰7d#eCdvUcCeCi‹stcCi rgl)cCri lcCdeC€Hdpqi‹mncCd#vDd z7dl5s dv

(415) mol5p}i l5s x z5pqdv

(416) cC‰7d rg|qcCi €moxTrgl7d tƒ–ˆ>i l5s d)ŠTd}moeCdx r>rgh>i l7kJ‚¡rge!mol mgp7mo|qcCi fgd€Hd cC‰7rqp ‚¡rgerg|qcCi €Hi‹vCmncCi rgl£mol5p vDi l5s 6d 8

(417) %mol5p)cC‰7dm0fgde+mokgi l7k!|7eCrgy5moy7i x i cU‡€HdmgvDz7eCd=i l ­Œqƒ gƒ  moeCd=z7l7h>l7rnŠ

(418) l {7ŠTd=eCd|7x‹mgs d cC‰7d€ Š

(419) i cC‰2vCmo€H|7x d=mol5mox rgkgz7dvƒˆ>rHŠTd=s rgl5vDi‹pqde  

(420) 8

(421)   8

(422)  % %  P     P         # / = oŒ ­Œqƒ gƒ ‘   Š

(423) ‰7deCd 8

(424)  

(425) i‹vmgvyBd ‚¡rgeCdg

(426) { 

(427) 2pqdl7rocCdvm0fgde+mokgi l7kHŠ

(428) i cC‰2eCdvD|BdstccCrcC‰7dd€H|7i eCi‹smoxpWƒ ‚Uƒ mol5p       8

(429)  %  !

(430)  

(431) 69 . %

(432)   ­Œqƒ gƒ

(433)  % Š

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