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On the Convergence of the ET Method for Extreme Upper Quantile Estimation

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(1)On the Convergence of the ET Method for Extreme Upper Quantile Estimation Stéphane Girard, Jean Diebolt. To cite this version: Stéphane Girard, Jean Diebolt. On the Convergence of the ET Method for Extreme Upper Quantile Estimation. RR-3389, INRIA. 1998. �inria-00073300�. HAL Id: inria-00073300 https://hal.inria.fr/inria-00073300 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. On the convergence of the ET method for extreme upper quantile estimation St´ephane Girard, Jean Diebolt. N ˚ 3389 Mars 1998 ` THEME 4. ISSN 0249-6399. apport de recherche.

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(151) ¨ 2 K  43 “ x `jgh`qkeŠg U X T fEtKdqi x X3vmlmfPkmoh`jiak i:”  0 t™ 1   ¡ x ong Di x r X x gnX x Yo¤f:ke`bo U f:k r œHX¦Š:X1g dqk t –h %@i ˜ ˜ o ¤ ¤   0 t™ 1   ! *{ o  &– dq> k  o ¤ ?  g  0     1 ) dbk pWdqk i ) dqk>  )  ˜ t ! `– ¥g*{ o ¥ ¤ b –h`– ¡ ¥ b  ¡. ¤ ˜ dbtWk db k ˜  o ¤ ? ¤   "p  ¥ ¤ b ¡ ¤ ) ¥£b ¡  ) ; o ¤  g 0    1 ) ¡ ˜  –ho  – dbk ˜n˜  p ¥ b ¥£b  t ! {† E ¥#P¥ ¤ b  ¡  ¡ ¤ F o o$–n`– ¥–b  db k˜  dq ¤ ¡¤) ¡ + )  ¡ k  ˜ †:Ÿ  oh`qkmŠ #" o$; œ

(152) `qg U ( Z!  dqk Af:k r * #! –`¥b ¡ ˜ dbk2¡mœHX¦iPžegpf"`bk¢g U f"g [ k  ˜    0 t™ 1   E`– ¥£Z¥ ¤ b ¡  ¡ ¤ ˜ t vo ; – dqk ˜  ˜ )  `– ¥8Z¥ ¤ b ¡  ¡ ¤ ˜ t vo  –  db –h†-[ ”™i x ohiPYX‡#¥ 9   /9jbXd Ÿ –Ì` ˜ R¶”2#¥ 9 x eXdq`qYyzf `bkK”  4– ( ˜  x eXdj`bYyzf opcml  h– ( ˜ 9jbXd ¡eg U X1k 0  ¤ `– ¥¥ ¤ b ¡  ¡ ¤ ˜ t    t_™1     ¤ – ¥ ¥ ¤ b ¡ z ¡ ¤ ˜ t=”™i x ohiPYXR T˜

(153) U X x X3”™i x X:¡K`bkN˜ g U `bo/u-fPU onX–  0 t_ 1    a 0 ¥€f:o  a ¥ ˜ . ˜ `– ¥"U !¢¥ U ¤ !©¥ † 9 ˜ Ÿ  ¤  ¤ 9:bXdŸ –Ì`q` R¶”x eXdj`bYyzf  –h( ! ¥g XZk Udqe8`qY f  t™ 1   !¥

(154) –`¥ ) ¥ ¤ opcmu g f"g/¥9$¥ ¤  † 6¥0Ÿ {eŸ R¶” `qo ' i:Šakei x Yf"dD¡ œHX%YfP›"X cLonX%i:”

(155) g U X T fEtKdqi x ”™i x Ycmdqf(œ

(156) `qg U `bkagnX1Š x f"d x X1Yf"`bk r X x Ÿ “ X iPžegpf"`bk [  yž™ ¥£b    0 t™ 1   ! † dbk R {    yž   ¡ X-vml  {dbk2–    ¥£b ¡ ˜ QT + –h†E\ ˜ Y opf rmr djXZlLiP`qkag/gÕtel?XYX-g U i r u-f:kg U X1k žLXcLonX r gni¢o U iœ g U f:g  0 t™ 1   `qo£žLiacmk r X r žLX1djiœ `j” ? ¥ Ÿ ¥ !§ ¤ mŸ Y opohcLY `bkeŠ &" o$ ¡ 0% o$; f:k r 0%  fPk r ¥"!¥ ¤ ! †P¡W–n†-[ ˜Rx X1œ x `qgnXZo  0 t™ 1   L † – ¡  ¡ ¤ ˜ t  o  –Ìdb> k  ˜  f:o  a bXd + { [LŸ Y lmlmdjtK`bkeŠ%fohf rmr dqX1l?i:`bkPg/YX1g U i r f:o`bkN{ghi–nE† \ ˜ Š:`q¤:X1og U X x X1opcedjg1Ÿ. . ‘Ö. h;i 4h]j.

(157) †P† ". ¨ ¨

(158) ¨ 2 K    f:ohX¦œ U X1k. 43]. fPk r #" o$; U iPd r Ÿ ( ! o  âv– o b bd kn ˜ –4( ˜  ! o   o 4– (. ˜ b  o   ¤  o –h(. ˜ b@ HdqknJ–4(T ˜ ZdqknJ–4(T ˜ ) B p  ¥ ) †  +  iPkmohX1šKceXZkagndqt:¡ dbk n?h–˜ (. ˜ (.  (. g! o  o k nJ4– ( ˜n˜  ) ¤  v– o 4– ( b db f:k r `qg”™i:dqdjiœ/oHg U f"g dbk nJ4– (. ˜ ˜n˜ + T h– (. ˜ ! ( o ¤  o  – o –4(  œ

(159) `jg U  #!©†“b@ Hdbk n?h– U (. x &˜ % o h– (. ˜ Ÿ ^a`bkmu3X! e8dq`qY yzf  #!©†:¡eg X X¦X3ve`bong ! ¥ ) ! ¥ohcLu U g U f"g  "p â>†   ) † b   ”™i x H  Ÿ Y lmlmdjtK`bkeŠ #" o f:k r &" o$ gni ( g!jo 4– (T ˜ f:k r * #! o 4– (T ˜ `j”7”™iPdjdqiœ/og U f:g T h– (. ˜ !N”–Ìdb k n?¤ h– (. ˜&% 4– (T o ¤ 4– (T ˜n˜h˜ ¡:fPk r œ

(160) `jg U#' X1YYf‡{eŸÃ{ –Ì` ˜ œRXiažeghf:`qk eXdj`bY yf T h– (. ˜ !¥mŸ  ' iPŠPkei x Yf"dWu1f:ohX:Ÿ Y k f:ohteYlegniPgn`bu¢X-vmlmf:kmoh`qiPk i: ” o¢ tK`jX1d r o T 4– (T ˜ ! dbk n?h– (. ˜ –nX† b VK–h† ˜h<˜ % dbk (. W¡=fPk r g U X u3iakmu3dbcmoh`jiak%”™iPdjdqiœ/o

(161) f:o

(162) fPžLi¤PX:Ÿ. " ¨ ¨

(163) ¨  2 K  43D –Ì` ˜ “ X U fE¤:X † %@i ˜  0 <™1   ! E GT  GT E “† b E † ˜ 

(164)  F o ¤ 4– G. o ¤ – G ˜ F dbk¦h– . F + T

(165) U X x X¦X3ve`bongpo ž?X-g՜HX-XZ"k G X-

(166) g G opcmu U g U f"g o ¤ ¤ – ˜ dbk¦h–  %@i ˜  0 &™1   ! E G\  G\ E > † 

(167)  F F o ¤t – ˜ ˜ ! T h– GT ˜

(168) G\  E > †  o o ¤ t¤ – –. ˜ dqk¦v–  %*i ˜ F + ¤ T

(169) U X1k G\ "

(170) %`bYledq`jXZo  0 &™1   !Ÿ” E T h– G ˜ E > †  o o ¤ t¤ – –. ˜ ˜ dbk¦h–  %@i ˜ FF + –n†Z‰ ˜ ¤ 0 –—`j` ˜ 3t r X ?ke`qgn`qiPkW¡   ?™ 1   ! T h–

(171)  ˜ Ÿ ' XZYYfmŸÃ{ Š:`q¤:XZog U X x X1opcedqg-Ÿ Ö7ÖÓO¨7©ˆ©ˆªˆ«. #" o ¤.

(172) †E{ –Ì`q`j` ˜"' X1g/cmo

(173) œ x j` ghX dqk¦v–  %*i ˜. !. – ¥#¥ ¤ b. ˜ dqk. ¡  ¡ ¤  ¥#P¥ ¤ b ¡  ¡ ¤ o ho$; ¥ ¤ b ¡ ¤ ¥#P¥ ¤ b ¡  ¡ ¤ o G . ¥ ¤ b ¡ ¤ o. * !. ! G\ . v %*i !N” E o ¤ ¤ o. – E–n– ¥ ¤ b ˜¡ ¤ ˜ dqk> ˜h˜ o – ˜ o –. ˜ – ˜ o – ˜ ! + – Y lmlmdjtK`bkeŠ – " o œ

(174) `qg U ( AfPk r W¡7œHX%ŠPX-gdbk¦–h %@i ˜ !M”%–ho – ˜n˜ Ÿ T

(175) U cmo o ¤ ¤ – ˜ dqk¦– – ˜o – ˜ + ˜ –n†+Z ˜ ˜ t¤ – ˜ F o ¤t – T œRi€u1f:ohX1o

(176) f:lLlLXZf x [ 0  R¶”#x e8dq`qY yzf  o ¤ ¤ –4( ˜  o –4( ˜ o ¤ \t –h( ˜ 9:bXd ¡Eg U X1k   <_1   !s”%– T –4G\ ˜n˜ fPk r g U X x XZohcedqg ”™iPdjdqiœ/oH” x iPY ' X1YYfLŸ {mŸ  ' i:Šakei x Yf"d r `qohg x `bžmcegn`qiPk [ x e8dq`bY yzf  o ¤ ¤ –4( ˜  o 4– ( ˜ o ¤ \t h– ( ˜ !ŸbXd Ÿ T

(177) U X1kW¡LcLon`bkeŠ–n†+Z ˜ ¡W–h†1‰ ˜Hx X1œ x `qgnXZo [  0 <™1   !Ÿ” E T –4G ˜ o ¤ ¤ – ˜ o – ˜ o ¤t – ˜ F ) œ U X x X T –4G ˜ `bo

(178) Š:`q¤:XZk žat ' X1YYfLŸ {mŸRÕg x XZohcmdjgpog U f:g k n?h– GT ˜ ˜ o ¤ ¤ – ˜ o – ˜ +  0 &™1   !Ÿ” E db G o ¤ –4G. o ¤t – ˜ F Y ”™gnX x on`bYledq` ?u1f"gh`jiakmo-¡eœHX¦Š:X1g  0 &™1   !Ÿ”%–Ìdbk % dbk ? GT ˜ + ' X1YYfmŸÃ{ o U iœ/og U f"

(179) g G !jG –h“† bVK–h† ˜h˜ Ÿ T

(180) U X x X-”™i x X:¡moh`qkLu3X p  G ) G ¶¡ œHX U fE¤:X¦dbk Ldbk G ¡mfPk r  0 &™1   !s”%–Ìdbk Tt GT ˜ –n†Z ˜ + X1kmu-X:¡#dq`qY  0 &™1   ! ¥mŸ. UeXyzf !. ‘Ö. h;i 4h]j.

(181) †Z ". ¨ ¨

(182) ¨ 2 K  43 †:Ÿ Y u1u3i x r `bkeŠgni–n†1[ ˜ ¡  0 t_™1   `qožLiacmk r X r fPoH”™i:dqdjiœ/o [ 0  `– ¥8Z¥ ¤ b ¡  ¡ ¤ ˜ t vo  – dbk ˜    t™ 1   E–`¥£Z¥ ¤ b ¡  ¡ ¤ ˜ t vo ; – dqk ˜  ) œ

(183) U `jg U 6¥ 9U   "U 9 bXd_ fP0 k r  p  ¥ T ¤ b ¡ ˜h˜ ¤ ) ¥Zbr ¡ 0 ¶Ÿz3X1oh` T r X1o1¡ g ˜ U X%Y l x iKi"”i"” ' #" X1YY f(mŸ r o#" iœ/o¦g f:g‡`qUkAg `qo‡u1f:x ohX &™1   ! ”€– –hG * f:k ?™ 1   ! r –h

(184)   Ÿ lmlmdjr tK`bkeŠ o ¤ f:k o$; œ

(185) `jg ¡ `qk f ohggn`bY1 X ( !  ¡ !j¥ ¤ b ¡ ¤ fPk `bk f¢onXZu3iPk gn`bYX ( !  ¡ * !§¥£b `qg”™i:dqdjiœ/oHg U f"g ¡ k nJ4– G ˜ ˜ fPk r  0 ?™ 1   !s” E dbk n?v–

(186)   ˜ ˜ )  0 &™1   !s” E db –h†1‚ ˜ (. o ¤ h– (. F (. o ¤ h– (. F œ

(187) `jg U r( x !NU o$;0 E–  dqk  ˜ Ÿ  iPkmohX1šKceXZkPghdjtP¡Pg U X¦l x iKi:”0`qooh`qY`qdqf x ”™i x  0 &™1   fPk r  0 ?™ 1   Ÿ ' X-gcmo u3iakmon` X g X &™1   u1f:ohX:Ÿ –Ì` ˜ R¶” ¥"!¢¥ ¤ ! †:¡eg U X1k dbk

(188) dbk  t  h– ( ˜   0 t_™1    Eh–

(189) /6

(190) ¤ ˜ t E dbkdqk  t  4– ( ˜ ) –D{:¥ ˜  h–

(191) $

(192) ¤ ˜ t E dqk GF dbk  F œ

(193) | `qg U  x pU  “† b ¡ ¤ ) “† b ¡ 7&˜f:% k r ( r !jo$–  dqk  ˜ Ÿ ˜ x x U Ug X1X Yx X¦f X3›€ve`bgong/f"¥ g

(194) 9 db>k  ~¤ !so ¤ h– 9:( bXd  opcmfPuk U g ohU cmf:žLg on`qghceghX‡`jg`qkA–D{:¥ Ÿ mLi  dbf Š:X¦X1kmiPceŠ ¡ ¤ o t"h– (. ˜ † o t:4– (T ˜  ¤   0 –v

(195) $

(196) ¤ ˜ t  –4(T ˜ dbk>o t –h(. ˜   t™ 1   –h

(197) 6

(198) ¤ ˜ t  –4(. ˜ dqk7o t –4(T ˜ + –D{m† ˜ m`bkmf"dqdqt:¡ –h†1‚ ˜ fPk r –¶{e† ˜ l x i¤K` r X  0 <_1   %@ 0 t_ 1   !N”%–Ìdb> k nJ4– G\ ˜ h– (. ˜h˜ Ÿ –Ì`q` ˜ R¶” ¥ !¢? ¥ ¤ ¡eg U XZk –¶{e† ˜ ghf:›:X1of r ` JX x XZkPg”™i x Y [ † † ¤ †  ¤ `– ¥#P¥ ¤ ˜ t  h– ( ˜ o t h– ( ˜   0 t_™1    – ¥ZP¥ ¤ ˜ t  4– ( ˜ o t 4– ( ˜ + Y o/l x X-¤K`qiPcmohdjtP¡KœRX‡u3iakmu3dbc r X‡œ

(199) `qg U –h†1‚ ˜ [  0 <_1   %@ 0 t_ 1   !N”%–Ìdb> k nJ4– G\ ˜  4– (T ˜n˜ Ÿ {eŸ R¶”  `bo ' i:Šakei x Yf"dWf:k r ¥"!¥ ¤ !©†:¡eg U X1kA–D‰ ˜ o U iœ/oHg U f:g k  ˜t  0 t_ 1   L v–

(200) 6

(201) ¤ ˜ t –Ìdbk

(202) db [ dbk  + °iœ‡¡W–h†1 ˜ u1f:kNžLX x X-œ x `qgngnXZk¢fPo  0 <_1   !N” E † dbk F ). Ö7ÖÓO¨7©ˆ©ˆªˆ«.

(203) †1[ f:k r ' 1X YYfLŸ {l x i¤K` r X1o.  0 ™?  1   !s”. Õg”™i:dqdjiœ/og U f"g Udqe8`qY f  0 <™1   %@ 0 t_™1   „ ! Udqe8`qY f " ¨ ¨

(204) ¨ +H,  ¨ K   ) “ X x ong

(205) l x i¤:X\g U f"g. †. dqk ?ˆ’&0 t . F.  0 ™?  1   %B  ™t  1   !. R. œ U XxX. E. +. ¥mŸ. † ˜ Q _’<t  ) –4G  | x x | ˜ ¤ Ÿ 3t 1X katK` o XZl X1ohX1kaghf:gn`qiPk(– X-`boho1¡ †Z‚::‚ ¡. –¶{:{ ˜. G\ G\ #!. !j. . 0 &1 ),++,+,)  0 1  k!. .  ,) ++,+)  ). y   y  F   0 <1  +,++   0 1 

(206) X(U¥ V . E. œ U X x Xsg U X  o\f x Xs`DŸ `DŸ r ŸJcLke` ”™i x Yi¤PX x  ¥ ) † ¶¡ `qo£g U X u3i xhx X1opl?iPk r `bkeŠ€i x r X x X r opf:YledqX:¡7f:k r ! V_] ^ V ¡¯†~ A2 Pb †:¡0g U X o ž?X-`bkeŠ `DŸ `DŸ r Ÿ  –h† ˜ Ÿ  oh`qkmŠ g U X¢šacLf:kagn`qdjX g x fPkmon”™i x Yf:gn`qiPkW¡KE"0   y <1 ! –h†>6 ˜ – 0  1 ˜ ¡ †  “2Ÿ T

(207) U X x X3”™i x X:¡ . . . . . . G  6G. !. E"0 \S  1 6G. >6    0 S *y <1   >6 k! k!      0 S *y &1  QP  ) >$  4Q %  !:G. S   Q. ] V  0  ! S *y <1   V_^  b  _  < ’ t ! Q   £b V

(208). œ U XxX. ˜. † ˜ – ˜ "– ˜ Ÿ T

(209) U X1kW¡ † – –. œ U X x X q` ou3dqiPohX£ghi(–n†. –h†. . S  ]. !jQ _’<t V  V_^ . –. žat%g U X  1X kag x f:d ' `bY `qg

(210) T U 1X i x X1YNŸe°iœ‡¡ h– G ˜ !No ¤ –4G ˜ Q  E †. † ˜ \ka . –h†. ˜ .  QP  . † ˜ S  y  –n† ˜n˜ ) –—¥ ) † ˜ ).   L . –˜ ˜ ˜ ?– – F n ¤ hG. n hGT o ¤ 4G\. +. ‘Ö. h;i 4h]j.

(211) †9\ ^K`qkmu-Xx e8dq`qY yzf ?n –h( ˜ !©†‡f:k r až t   ) ¡ n ¤ –hGT ˜ !Ÿ”€–h† % GT ˜ ¡?`jg”™iPdjdqiœ/oHg U f:g n ¤ 4– G\ ˜ † ˜ a ¥f:o a bXd ) E N ! ” ˜ ˜ nJ4– G o ¤ 4– G. G o ¤ –hG F `bkN¤K`jX1œ i"” ' XZYYf€{mŸ {¢–Ì` ˜ Ÿ X1kLu3X:¡ 4– G ˜ L QP o ¤ –hG ˜ ) –¶{" ˜ fPk r G\ G\ !  Q o † h– G ˜  + ¤. " ¨ ¨ 

(212) ¨  +H,  ¨ K  7 Rzk g U `bo‡l x iKi"”z¡ œRXYf:›"Xg U X ”™iPdjdqiœ

(213) `bkeŠ¢fPlml x iEvK`bYf"gh`jiakWŸ T

(214) U XX-veu-X1opo x f:k r iPY¤f x `qfPžedqX o C   0 1 ! E"0 TS *y  1  G  ¡a†  JQ f x X=u3iPkLon` r X x X r fPo `DŸ `DŸ r Ÿ-”™iPdjdqiœ

(215) `bkeŠf:k

(216) 8(¥ –h† % o ¤ – G  ˜h˜ r `bohg x `qžmcmgn`qiPkWŸ T

(217) U X x f:k r iPY ¤f x `bf:žedqX1o  0 1 !jo ¤ – G  ˜ C  0 1 f x X\g U XZk¢`DŸ `—Ÿ r Ÿ

(218) X(U¥ –h† ˜ Ÿ T

(219) U X x X3”™i x XP¡ † ]S  C  0 1 ? ! Q ^  S  0. † † ] ! o – G  ˜ †“b Q. –  1  † ˜ ^   ¤ † † –¶{[ ˜ ! o – GT ˜ b o – G\ ˜  QP  ¤ ) ¤ ¤ œ U XxX S   !:Q _’&t ] –  0 1  † ˜ Tka  L–—¥ † ˜ ¤ ) + ^  3XZon` r X1o1¡ †  ˜  † ˜ !  o ¤ – GT  ˜ ˜6o ¤ 4– G\˜ ˜ o ¤–G. o ¤ 4– G. o ¤ – G o ¤ 4– G.  6G\ ˜ o –  ˜ – \ G ¤¤ ! o ¤ – G  ˜ o ¤ h– G ˜ ) œ U X x X  `qo¦ž?X-g՜HX-XZ~k GT (fPk r G\ TŸ 3Ht &" o ¤ ¤ ¡ o ¤ ¤ –  ˜ ! ” v– o ¤ ¤ h– GT ˜h˜ Ÿ T

(220) U XZkW¡Wcmon`bkeŠ –D{:{ ˜ ¡JœRX iažeghf:`qk †  ˜  † ˜ ! ” E o  ¤ ¤ ˜ h– GT ˜ ˜ † ˜  Q. o ¤ – G\. o ¤ 4– G\. o ¤ – G\ o ¤ 4– G\ h– GT  F . . . . . . . .

(221). . Ö7ÖÓO¨7©ˆ©ˆªˆ«.

(222).

(223) †Z‰ !. ” !. V

(224). E. – ˜  † †  ˜ žat –D{: ˜ – ˜ QP o ¤ – G\ F †  ˜ žat  67 +. o ¤–G F. o ¤ ¤ hGT. o ¤ t hGT. E.

(225). . Q. `bkmf"dqdqt:¡ x X1ledbf:u-`qkeŠ`bkA–D{[ ˜ ¡eœHX¦iPžegpf"`bk¢g U f"g † ˜ E “† b  ¤ bV –n† ˜ ? ! o ¤ 4– G. Q F  b V –n† ˜ † ! ? 8b o 4– G\ ˜ ¤  Q ) ¤ rfPk g U Xsu-iPkmu-dqcmoh`qiPk ”™i:dqdjiœ/o1Ÿ m.

(226).

(227). ‘Ö. h;i 4h]j.

(228) Unit´e de recherche INRIA Lorraine, Technopˆole de Nancy-Brabois, Campus scientifique, ` NANCY 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rhˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN Unit´e de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex Unit´e de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex. ´ Editeur INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France)   .

(229)     ISSN 0249-6399.

(230)

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