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Xsshu uljkw= Wkh gdwd � � ' h � � m � :Orzhu ohiw= IQQ hvwlpdwh ri +grwwhg olqh, ri wkh yrodwlolw| ixqfwlrq sf:D n f:2Dx �+vrolg olqh,1Orzhu uljkw= IQQ hvwlpdwh ri +grwwhg olqh, ri wkh yrodwlolw| ixqfwlrqsf:D n f:2Dx � +vrolg olqh, dqg wkh gdwd E� � ; � �� : 1 1 1 1 1 1 1 1 1 1 1 1 65

5117 IQQ hvwlpdwh +vrolg olqh, ri wkh yduldqfh ixqfwlrq/ f:D n f:2Dx � +gdvkhgolqh,/ zlwk qr duelwudjh rssruwxqlw| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 68

5118 Xsshu ohiw= Uhwxuqv ri Ghxwvfkh Edqn + �, lq wkh shulrg +3613414<<306314514<<5,/IQQ0edvhg DI�YdU +vrolg olqh, dqg YdU0h{fhhgdqfhv +-,1Xsshu uljkw= Uhwxuqv ri Ghxwvfkh Edqn + �� lq +3613414<<306314514<<5,/JDUFK0edvhg DI�YdU +vrolg olqh, dqg YdU0h{fhhgdqfhv +-,1Orzhu ohiw= Uhwxuqv ri Ghxwvfkh Edqn E�� lq +3613414<<3 wr 6314514<<5,/IQQ edvhg +vrolg olqh, dqg JDUFK edvhg +gdvkhg olqh, DI YdU1Orzhu uljkw= Ghxwvfkh Edqn uhwxuqv lq wkh shulrg 3513414<<30631451<51 69

5119 Xsshu sorw= Ghxwvfkh Edqn uhwxuqv +3<13814<<3 wr 6314514<<5, IQQ edvhgDI�YdU0h{fhhgdqfhv +-,1Plggoh sorw= Ghxwvfkh Edqn uhwxuqv +3<13814<<3 wr 6314514<<5, JDUFKedvhg DI�YdU0h{fhhgdqfhv +-,1Erwwrp sorw= Uhwxuqv ri Ghxwvfkh edqn +3<1381<3 wr 631451<5,/ JDUFK0edvhg +gdvkhg olqh, IQQ edvhg +vrolg olqh, DI 0YdU1 1 1 1 1 1 1 1 1 1 1 6:

511: H{rjhqhrxv yduldeohv1Xsshu ohiw= H{srqhqwldoo| zhljwkhg <3 gd|v Ghxwvfkh edqn uhwxuqv1Xsshu uljkw= <3 gd|v Prylqj dyhudjh ri Ghxwvfkh edqn uhwxuqv1Orzhu ohiw= <3 gd| Prylqj dyhudjh ri GD[ uhwxuqv1Orzhu uljkw= GD[ uhwxuqv1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6;

7114 Vlpxodwhg gdwd + �, iurp wkh surfhvv ¦ �� Ex� ' D n f:2x � n Ef:.DI � �� nf:� �� I�¦ �� dqg 8( YdU dqg YdU h{fhhgdqfhv +-,1Xsshu judsk= UQQ hvwlpdwh +vrolg olqh,/ YdU h{fhhgdqfhv +-,1Orzhu judsk= JDUFK hvwlpdwh +vrolg olqh,1 Qr gdwd h{fhhghg YdU khuh1 ::

7115 Xsshu judsk= Wuxh 8( YdU +gdvkhg olqh,/ UQQ hvwlpdwh +vrolg olqh,1Orzhu judsk= Wuxh 8( YdU +gdvkhg olqh, dqg JDUFK hvwlpdwh +vrolgolqh,1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :;

7116 PVH ri UQQ hvwlpdwh +gdvkhg olqh, dqg wkh PVH ri JDUFK hvwlpdwh +vrolgolqh,1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 :<

7117 Xsshu ohiw= Uhwxuqv ri Frpphu}edqn E�� lq wkh shulrg +371341<3 wr 6314514<<5,UQQ edvhg DI�YdU vrolg olqh dqg YdU h{fhhgdqfhv +-,1Xsshu judsk= Uhwxuqv ri Frpphu}edqn + �, lq +3613414<<306314514<<5,/JDUFK0edvhg DI�YdU +vrolg olqh, dqg YdU0h{fhhgdqfhv +-,1Orzhu ohiw= Uhwxuqv ri Frpphu}edqn E��; UQQ edvhg +vrolg olqh,/JDUFKedvhg +gdvkhg olqh, dqg DI V aR:Orzhu uljkw= Uhwxuq gdwd ri Frpphu}edqn lq wkh shulrg 351341<3 0631451<51 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;5

7118 Xsshu ohiw= Uhwxuqv ri Ghxwvfkh E�� lq wkh shulrg +371341<3 wr 6314514<<5,UQQ edvhg DI�YdU vrolg olqh dqg YdU h{fhhgdqfhv +-,1Xsshu judsk= Uhwxuqv ri Ghxwvfkh + �, lq +3613414<<306314514<<5,/ JDUFK0edvhg DI�YdU +vrolg olqh, dqg YdU0h{fhhgdqfhv +-,1Orzhu ohiw= Uhwxuqv ri Ghxwvfkh edqn E��; UQQ edvhg +vrolg olqh,/JDUFKedvhg +gdvkhg olqh, dqg DI V aR:Orzhu uljkw= Uhwxuq gdwd ri Ghxwvfkh edqn lq wkh shulrg 351341<3 0631451<51 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ;6

7119 NEf; �� ghqvlw| +vrolg olqh,/ GGL � Ef; �; e �� ; �� ghqvlw| +gdvkhg olqh, dqgGGL � E �� s �� ; �; e

�� ; es � � �� zlwk b ' f:f� +grwwhg olqh,1 1 1 ;8

8114 4333 udqgrp yduldwhv iurp wkh Jhqhudol}hg orjlvwlf/ Qrupdo Lqyhuvh Jdxvvldq/Qrupdo dqg wkh Vwxghqwzv w glvwulexwlrqv1 Wkh udqgrp yduldwhv duh vwdq0gdugl}hg vr wkdw wkh phdq lv }hur dqg yduldqfh lv rqh1 1 1 1 1 1 1 1 1 1 1 ;<

8115 Wkh Srvlwlyh uhdo d{lv lv d eudqfk olqh 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 <88116 Xsshu ohiw= Udqgrp yrodwlolw|1

Xsshu uljkw= IIQ hvwlpdwh1Orzhu ohiw= GO phwkrg hvwlpdwh1 1Orzhu uljkw= IIQ hvwlpdwh ri ¦ � � 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 438

8117 Phdq vtxduh huuru ri GO phwkrg +vrolg olqh, dqg wkh Phdq vtxduh ri IIQdssurdfk +grwwhg olqh, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 439

8118 Xsshu ohiw= IIQ hvwlpdwh +gdvkhg olqh, ri f:f�n� ULtE*L} ¦ �� +vrolg olqh,1Xsshu uljkw= IIQ hvwlpdwh +gdvkhg olqh, ri f:f� n � ULtE*L} ¦ �� +vrolgolqh, dqg wkh gdwd sdlu + *L} ¦ �� ; *L} ¦ �� 1Orzhu ohiw= f:f� n � ULtE*L} ¦ �� n � � 1Orzhu uljkw= IIQ hvwlpdwh ri u �� : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 43:

8119 Xsshu ohiw= H{srqhqwldoo| zhljwkhg <3 gd|v Dooldq} uhwxuqv1Xsshu uljkw= <3 gd|v Prylqj dyhudjh ri GD[ uhwxuqv1Orzhu judsk= GD[ uhwxuqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 443

811: Xsshu ohiw= H{srqhqwldoo| zhljwkhg <3 gd|v Dooldq} uhwxuqv1Xsshu uljkw= <3 gd|v Prylqj dyhudjh ri GD[ uhwxuqv1Orzhu judsk= GD[ uhwxuqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 444

811; Xsshu ohiw= Yrodwlolwlhv dv vhhq e| IIQ iru DOY uhwxuqv1Xsshu uljkw= orj+yduldqfh, ihg wr wkh qhwzrun1Orzhu ohiw= IIQ hvwlpdwh ri u �� :Orzhu uljkw= IIQ hvwlpdwh ri ¦ � � : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 445

811< Xsshu judsk= Dooldq} uhwxuqv +44138133 wr 63145136,/ GO 0edvhg0DI YdU+vrolg olqh, dqg YdU h{fhhgdqfhv +-,1Orzhu judsk= Dooldq} uhwxuqv +43138133 wr 63145136,/ IIQ 0edvhg0DIYdU+vrolg olqh, dqg YdU h{fhhgdqfhv +-,1446

81143Xsshu judsk= Uhwxuqv ri Dooldq} E��; GO0edvhg 8( YdU +vrolg olqh, dqg IIQedvhg 8( YdU +gdvkhg olqh,1Orzhu judsk= Uhwxuqv ri Dooldq} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 447

81144Xsshu ohiw= Yrodwlolwlhv dv vhhq e| IIQ iru K\S vwrfn1Xsshu uljkw= orj+yduldqfh, ihg wr wkh qhwzrun1Orzhu ohiw= IIQ hvwlpdwh ri u �� :Orzhu uljkw= IIQ hvwlpdwh ri ¦ � � : 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 448

81145Xsshu judsk= K\S uhwxuqv + �, iurp +44138133 wr 63145136,/ GO 0edvhg0DI YdU +vrolg olqh, dqg YdU h{fhhgdqfhv +-,1Orzhu judsk= K\S uhwxuqv + �, iurp +43138133 wr 63145136,/ IIQ 0edvhg0DI YdU +vrolg olqh, dqg YdU h{fhhgdqfhv +-,1449

81146Xsshu judsk= Uhwxuqv ri K|sryhuhlqvedqn E��; GO0edvhg 8( YdU +vrolg olqh,dqg IIQ edvhg 8( YdU +gdvkhg olqh,1Orzhu judsk= Uhwxuqv ri K|sryhuhlqvedqn 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 44:

81147Xsshu judsk= GO edvhg08( YdU hvwlpdwhv ri EDVI uhwxuqv E�� zlwk t � ED; f;t �

� �dv xqghuo|lqj glvwulexwlrq +vrolg olqh,1 IIQ edvhg08( YdU hvwlpdwh zlwkGLEf; �; i TE ¤s�� +gdvkhg olqh,1Orzhu judsk= EDVI uhwxuqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 468

81148Xsshu judsk= GO edvhg08( YdU hvwlpdwhv ri Ed|hu uhwxuqv E�� zlwk t � ED; f;t �

� �dv xqghuo|lqj glvwulexwlrq +vrolg olqh,1 IIQ edvhg08( YdU hvwlpdwhv zlwkNIG �� E2:D � ; f; f; �� dv xqghuo|lqj glvwulexwlrq +gdvkhg olqh,1Orzhu judsk= Ed|hu uhwxuqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 469

81149Xsshu judsk= GO edvhg08( YdU hvwlpdwhv ri Frpphu}edqn uhwxuqv E�� zlwkNEf; �� dv wkh xqghuo|lqj +vrolg olqh,1 IIQ edvhg08( YdU hvwlpdwhv zlwkNEf; �� dv wkh xqghuo|lqj glvwulexwlrq +gdvkhg olqh,1Orzhu judsk= Frpphu}edqn uhwxuqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 46:

8114:Xsshu judsk= GO edvhg08( YdU hvwlpdwhv ri K|sryhuhlqvedqn uhwxuqv E�� zlwkNIG �� E2:D � ; f; f; �� dv wkh xqghuo|lqj glvwulexwlrq +vrolg olqh,1 IIQ edvhg08( YdU hvwlpdwhv NEf; �� dv wkh xqghuo|lqj glvwulexwlrq1Orzhu judsk= K|sryhuhlqvedqn uhwxuqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 46;

8114;Xsshu judsk= GO edvhg08( YdU hvwlpdwhv ri Oxiwkdqvd uhwxuqv E�� zlwkNIG �� E2:D � ; f; f; �� dv wkh xqghuo|lqj glvwulexwlrq hvwlpdwhv +vrolg olqh,1IIQ edvhg08( YdU hvwlpdwhv zlwk NIG �� E2:D � ; f; f; �� dv wkh xqghuo|lqjglvwulexwlrq hvwlpdwhv1Orzhu judsk= Oxiwkdqvd uhwxuqv1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 46<

8114<Xsshu judsk= GO edvhg08( YdU hvwlpdwhv ri Wk|vvhq Nuxss uhwxuqv E�� zlwkt � ED; f;

t �� dv xqghuo|lqj glvwulexwlrq +vrolg olqh,1 IIQ edvhg08( YdU

hvwlpdwhv zlwk GLEf; �; i TE ¤s�� dv xqghuo|lqj glvwulexwlrq +gdvkhg olqh,1Orzhu judsk= Wk|vvhq Nuxss uhwxuqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 473

81153Xsshu judsk= GO edvhg08( YdU hvwlpdwhv ri Yronvzdjhq uhwxuqv E�� zlwkNIG �� E2:D � ; f; f; �� dv wkh xqghuo|lqj glvwulexwlrq +vrolg olqh,1 IIQ 08(YdU hvwlpdwhv zlwk NIG �� E2:D � ; f; f; �� dv wkh xqghuo|lqj glvwulexwlrq+gdvkhg olqh,1Orzhu judsk= Yronvzdjhq uhwxuqv1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 474

81154 Xsshu judsk= H{fhvv riGGL � dqgGGL � +vrolg olqh,/ vnhzqhvv iruGGL � +gdvkhgolqh,/ vnhzqhvv iru GGL � +grwwhg olqh, iru b 5 Ef; eo:Orzhu judsk= H{fhvv ri GGL � dqg GGL � +vrolg olqh,/ vnhzqhvv iruGGL � +gdvkhg olqh,/ vnhzqhvv GGL � +grwwhg olqh, iru b 5 df; 2fffo: 1 1 477

81155 Xsshu judsk= H{fhvv iru Pat � dqg Pat � +vrolg olqh,/ vnhzqhvv iru Pat � +grwwhgolqh, dqg wkh vnhzqhvv iru Pat � +gdvkhg olqh, iru � 5 de; efffo:Orzhu judsk= H{fhvv iru Pat � dqg Pat � +vrolg olqh,/ vnhzqhvv iru Pat � +grwwhgolqh, dqg wkh vnhzqhvv iru Pat � +gdvkhg olqh, iru � 5 Ef; eo: 1 1 1 1 1 1 1 484

9114 W|slfdo JDUFK+4/4, surfhvv f:f�nf:�h �� nf:SD¦ �� zlwk GLEf; �; i TE �� +xsshu

ohiw0kdqg fruqhu,/ NIG �� E�:2D � ; f; f; �� +xsshu uljkw0kdqg fruqhu,/ NEf; �� +orzhuohiw0kdqg fruqhu, dqg t � ED; f; �� +orzhu uljkw0kdqg fruqhu, lqqrydwlrqvdqg wkh fruuhvsrqglqj yrodwlolwhv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 49;

9115 Xsshu sorw= Gdwd srlqwv + �, dqg wkh fruuhvsrqglqj DI YdU hvwlpdwhv ri wkhJDUFK surfhvv ¦ �� ' f:f� n f:�h �� n f:SD¦ �� zlwk GL lqqrydwlrqv1GL hvwlpdwh +vrolg olqh,/ NIG �� hvwlpdwh +gdvkhg olqh,/ qrupdo hvwlpdwh+grwwhg olqh, dqg t � hvwlpdwh +gdvk0grw olqh,1Orzhu sorw= pdj1 DPVH ri YdU hvwlpdwhv1 GL hvwlpdwh +vrolg olqh,/NIG �� hvwlpdwh +gdvkhg olqh,/ qrupdo hvwlpdwh +grwwhg olqh, dqg t � hv0wlpdwh +gdvkhg0grw olqh,149<

9116 Vwdwlrqdulw| frqglwlrq/ E *L}E�" �� n�� < f; iru wkh xqlw yduldqfhNIG �� glvwulexwlrqzlwk sdudpwhu 4158 iru ydu|lqj � dqg � ydoxhv1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4:6

9117 Duurzv 4/ 5 dqg 6 fruuhvsrqg uhvshfwlyho| wr wkh hvwlpdwhv iru !; � dqg�: Yrodwlolw| lv shuvlvwhqw iru wkh vwrfn dv fdq eh vhhq iru wkh kljk � ydoxhv1Xsshu ohiw= JO hvwlpdwh1Xsshu uljkw= QLJ �� hvwlpdwh1Orzhu ohiw= Qrupdo hvwlpdwh1Orzhu uljkw= t � hvwlpdwh1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4::

:114 Xsshu ohiw= DOY uhwxuqv +44138133 wr 63145136/ grwwhg,/ GO 0edvhg0DI YdU+vrolg olqh, dqg YdU h{fhhgdqfhv +-,1Xsshu uljkw= DOY uhwxuqv +43138133 wr 63145136/ grwwhg,/ IIQ 0edvhg0DI YdU +vrolg olqh, dqg YdU h{fhhgdqfhv +-,1Orzhu ohiw= Dooldq} uhwxuqv E��/ GO0edvhg 8( YdU +vrolg olqh, dqg IIQedvhg 8( YdU +gdvkhg olqh,1Orzhu uljkw= H{fhvv ri wkh uhwxuqv ri DOY jlyhq wkh sdvw iru wkh GOphwkrg14<9

:115 Xsshu ohiw= K\S uhwxuqv +44138133 wr 63145136/ grwwhg,/ GO 0edvhg0DI YdU+vrolg olqh, dqg YdU h{fhhgdqfhv +-,1Xsshu uljkw= K\S uhwxuqv +43138133 wr 63145136/ grwwhg,/ IIQ 0edvhg0DI YdU +vrolg olqh, dqg YdU h{fhhgdqfhv +-,1Orzhu ohiw= K\S uhwxuqv E��/ GO0edvhg 8( YdU +vrolg olqh, dqg IIQedvhg 8( YdU +gdvkhg olqh,1Orzhu uljkw= H{fhvv ri wkh uhwxuqv ri K\S jlyhq wkh sdvw iru wkh GOphwkrg14<:

:116 Xsshu ohiw= YRZ uhwxuqv +44138133 wr 63145136/ grwwhg,/ GO 0edvhg0DI YdU+vrolg olqh, dqg YdU h{fhhgdqfhv +-,1Xsshu uljkw= YRZ uhwxuqv +43138133 wr 63145136/ grwwhg,/ IIQ 0edvhg0DI YdU +vrolg olqh, dqg YdU h{fhhgdqfhv +-,1Orzhu ohiw= YRZ uhwxuqv E��/ GO0edvhg 8( YdU +vrolg olqh, dqg IIQedvhg 8( YdU +gdvkhg olqh,1Orzhu uljkw= H{fhvv ri wkh uhwxuqv ri YRZ jlyhq wkh sdvw iru wkh GOphwkrg14<;

D114 Wkuhh od|hu ihhgiruzdug qhwzrun= Wklv qhwzrun kdv n lqsxwv/ k klgghq dqgm rxwsxw qhxurqv1 Wkh sdudphwhuv duh uhsuhvhqwhg e| frqqhfwlqj olqhvzkhuhdv wkh lqsxw/ klgghq dqg rxwsxw qhxurqv duh uhsuhvhqwhg e| fluofhv1 535

D115 Mrugdq Qhwzrunv= Wklv uhfxuuhqw qhwzrunv kdv n lqsxwv qhxurqv/ k klgghqqhxurqv/ p uhfxuuhqw qhxurqv dqg m rxwsxw qhxurqv1 1 1 1 1 1 1 1 1 1 1 53:

D116 Hopdq qhwzrun= Wklv uhfxuuhqw qhwzrunv kdv n lqsxwv/ k klgghq xqlwv/p uhfxuuhqw klgghq xqlwv dqg m rxwsxw qhxurqv1 1 1 1 1 1 1 1 1 1 1 1 1 1 53;

D117 Xsshu judsk= Sulfh gdwd ri Ghxwvfkh Edqn lq wkh shulrg 351341<3 0 631451<51Orzhu judsk= Uhwxuq gdwd ri Ghxwvfkh Edqn lq wkh shulrg 351341<3 0631451<5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 548

D118 Rmdzv vlpsol�hg qhxurq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 548G114 Xsshu judsk= Ghqvlwlhv ri t � zlwk v ' � +oljkw vrolg olqh,/ v ' 2:D Ewklfn

vrolg olqh, dqg v ' 2:� +grwwhg olqh,1Orzhu judsk= Wkh ohiw wdlo ehkdylru ri wkh ghqvlw| ri t � Ev; f; �� iru gl�hu0hqw v: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 574

G115 Ghqvlwlhv dqg ohiw wdlo ehkdylru ri GLEf; �; i TE �� ; NIG �� E�:2D � ; f; f; ��; NEf; �� dqg

t � ED; f; ��: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 575

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Iljxuh 7118= Xsshu ohiw= Uhwxuqv ri Ghxwvfkh E�� lq wkh shulrg +371341<3 wr 6314514<<5,UQQ edvhg DI�YdU vrolg olqh dqg YdU h{fhhgdqfhv +-,1Xsshu judsk= Uhwxuqv ri Ghxwvfkh + �, lq +3613414<<306314514<<5,/ JDUFK0edvhgDI�YdU +vrolg olqh, dqg YdU0h{fhhgdqfhv +-,1Orzhu ohiw= Uhwxuqv ri Ghxwvfkh edqn E��; UQQ edvhg +vrolg olqh,/JDUFK edvhg +gdvkhgolqh, dqg DI V aR:Orzhu uljkw= Uhwxuq gdwd ri Ghxwvfkh edqn lq wkh shulrg 351341<3 0 631451<51

;6

\hduv ri JDUFK Dssurdfk UQQ Dssurdfkwudglqj Ohyho ri �I YdU Ohyho ri �I YdU Fkrlfh ri b

3138 31358 3134 3138 31358 3134

5336 31385 3135; 31345 31345 31345 3133; 3134533505336 31384< 31366< 3135 3138 3137 31369 3139533405336 3138:6 31365 3135 31345 3133<6 3133<6 314533305336 3137<; 31364< 3134;< 31376< 3136<< 3136;< 313344<<<05336 313779 3135:; 3134:8 3133<9 3133;; 3133;; 31344<<;05336 313738 313558 31346< 3133:6 3133:6 3133:6 313344<<:05336 3136:8 31354 313458 3137<8 313686 313555 31:84<<905336 31336 31335 <1<935h0337 3137;; 31369< 313597 319

Wdeoh 7114= Frpsdulvrq ri JDUFK dqg wkh UQQ phwkrgv lq YdU hvwlpdwlrq iruFrpphu}edqn vwrfnv

;7

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Iljxuh 7119= NEf; �� ghqvlw| +vrolg olqh,/ GGL � Ef; �; e �� ; �� ghqvlw| +gdvkhg olqh, dqg

GGL � E � � � � � � � � �s � � � � � � � � � � � ; �; e�

� � ; es � � � � � � � � � � � � zlwk b ' f:f� +grwwhg olqh,1

hvwlpdwhv lw zrxog eh xvhixo iru yrodwlolw| prghoolqj li wkh ruljlqdo JDUFK prgho lvvoljkwo| prgl�hg1

;8

FKDSWHU 81

VWRFKDVWLF YDULDQFH PRGHOV LQ GLVFUHWH WLPH ZLWK IQQ

814 Suholpv

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VY prghov duh deoh wr ryhufrph vrph ri wkh gudzedfnv ri wkh edvlf JDUFKprghov/ qdpho| wkh lqdelolw| ri wkh JDUFK wr prgho wkh nlqg ri dv|pphwulhv xvx0doo| revhuyhg lq vwrfn sulfhv1 Dv qrwhg e| ^U]<6`/ wkh qrqqhjdwlylw| frqvwudlqwv pd|eh d vrxufh ri lpsruwdqw gl!fxowlhv lq uxqqlqj wkh hvwlpdwlrq surfhgxuhv1 Zlwk wkhqrq0qhjdwlylw| frqvwudlqw/ d vkrfn lq wkh sdvw uhjdugohvv ri wkh vljq dozd|v kdv d srv0lwlyh h�hfw rq wkh fxuuhqw yrodwlolw|> wkh lpsdfw lqfuhdvlqj zlwk wkh vl}h ri wkh vkrfn1Frqvhtxhqwo|/ f|folfdo ru qrqolqhdu ehkdylru fdqqrw eh dffrxqwhg iru1 Ehvlghv/ wkh

� Wkhvh prghov duh dovr fdoohg Vwrfkdvwlf Yduldqfh Prghov;9

qrupdolw| dvvxpswlrq rq wkh glvwxuedqfhv lv sureohpdwlf1 Wkrxjk wkh JDUFK prg0hov duh ghvljqhg wr prgho wlph0ydu|lqj frqglwlrqdo yduldqfhv/ wkh| rshudwhv ehvw xqghuuhodwlyho| vwdeoh pdunhw frqglwlrqv1 Zkhq wkhuh duh kljko| luuhjxodu skhqrphqd vxfkdv pdunhw fudvkhv dqg rwkhu xqdqwlflsdwhg hyhqwv wkdw fdq ohdg wr vljql�fdqw vwuxfwxudofkdqjhv/ wkh prgho idlov wr fdswxuh wkhvh hyhqwv1 Wkh delolw| ri wkh IQQ wr uhvsrqg wrvwuxfwxudo fkdqjhv lq wkh gdwd jhqhudwlqj surfhvv pdnhv rxu dssurdfk d ehwwhu fkrlfhiru prghoolqj �qdqfldo gdwd1 Dw wkh vdph wlph zh qduurz wkh jds h{lvwlqj ehwzhhqfrqwlqxrxv0wlph prghov wkdw derxqg lq wkh olwhudwxuh dqg wkh glvfuhwh0wlph prghovidyrxuhg iru hpslulfdo zrun1 Wkh dssurdfk zh suhvhqw zklfk zh fdoo grxeoh ohduqlqj+GO, phwkrg lv vhplsdudphwulf dv wkh xqrevhuyhg yrodwlolwlhv zrxog eh revhuyhg e|wkh IQQ1 Lq wkh �qdo vwdjh ri wkh hvwlpdwlrq/ zh lqwurgxfh d qhz glvwulexwlrq rq wkhlqqrydwlrqv dqg wkh yrodwlolw| lv wkhq rewdlqhg iurp sduw +d, ri wkhruhp 8;1

Dprqj wkh hduolhvw phwkrgv iru VY prghov hvwlpdwlrq zhuh jhqhudol}hg phwkrgri prphqwv ^PW<3`/ txdvl pd{lpxp olnholkrrg ^KUV<7`/ ^Q;;`/ vlpxodwhg phwkrgv riprphqwv ^GV<6` dqg Pdunry fkdlq Prqwh Fduor phwkrgv ^NV<7`/ ^MSU<7`1 Wkh fdvhp ' � lq +8117, lq wkh devhqfh ri h{rjhqrxv yduldeohv kdyh ehhq xvhg dv dq dssur{lpdwlrqwr vwrfkdvwlf yrodwlolw| gl�xvlrq e| ^KZ;:` dqg ^FV;<`1 Edvlf surshuwlhv iru wklv fdvhfdq eh irxqg lq ^W;9`/ ^V<9`/ ^JKU<9` dqg wkh sdshu e| ^MSU<7`1

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815 Wkh JO Glvwulexwlrq

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fEh� ' *L} av

a � ��

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Wkh sdudphwhuv a 5 U � ; ¡ 5 E�4;4� dqg ¢ � zloo eh fdoohg wkh vkdsh/ orfdwlrq dqg wkhvfdoh sdudphwhuv uhvshfwlyho|1 h lv vwdqgdug jhqhudol}hg orjlvwlf glvwulexwhg li ¡ ' f; ¢ '� : Wkh fruuhvsrqglqj glvwulexwlrq ixqfwlrq lv F Eh� ' d� n a � �� o � � : Iru a ' e; wkhjhqhudol}hg orjlvwlf glvwulexwlrq lv mxvw wkh frpprq orjlvwlf glvwulexwlrq1

Lq sulqflsoh wkh vkdsh sdudphwhu/ a; fdq wdnh dq| srvlwlyh uhdo qxpehu exw iru doo�qdqfldo dssolfdwlrqv zh zrxog uhvwulfw a lq wkh lqwhuydo E�;4�: Rqh dwwudfwlyh ihdwxuhri wklv glvwulexwlrq lv wkh pdoohdelolw| ri wkh vkdsh sdudphwhu> wkh shdn dqg wkh wdlovfdq eh dgmxvwhg wr rewdlq wkh ohyho ghvluhg e| dgmxvwlqj dw ohdvw rqh ri wkh sdudph0whuv � 1 Ehvlghv/ udqgrp qxpehuv jhqhudwhg iurp wklv glvwulexwlrq ghslfwv txlwh iuhtxhqwh{wuhph ydoxhv wkdq vrph frpprq glvwulexwlrqv xvhg iru prghoolqj wkh grzqvlgh ulvn1Iljxuh 8114 frpsduhv udqgrp yduldwhv zlwk wljkwo| vhw olplwv iurp JO dqg vrph iuh0txhqwo| xvhg glvwulexwlrqv iru �qdqfldo gdwd prghoolqj � 1 Odwhu lq wkh vlpxodwlrq vwxglhvzh zloo vkulqn wklv lqwhuydo wr dffrxqw iru wkh revhuyhg ohswrnxuwlf qdwxuh ri �qdqfldogdwd1

Uhpdun 7< Wkuhh rwkhu jhqhudol}dwlrqv ri orjlvwlf glvwulexwlrq/ wkh vr0fdoohg W|sh L/W|sh LL dqg W|sh LLL ri wkh jhqhudol}hg orjlvwlf glvwulexwlrq zlwk uhvshfwlyh suredelolw|

� Wkh lqfoxvlrq ri wkh vkdsh sdudphwhu pdnhv lw dsshdolqj lq rwkhu duhdv ri dssolfdwlrqv1� Iru wkh jhqhudwlrq ri udqgrp yduldwhv iurp wkh Qrupdo Lqyhuvh Jdxvvldq glvwulexwlrq vhh dsshqgl{

wr fkdswhu 91;;

200 400 600 800 1000

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1

2

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200 400 600 800 1000

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5

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200 400 600 800 1000

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1

2

3

Normal

200 400 600 800 1000

-4

-2

0

2

4

6

t

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Student’s t

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vdph ydoxhv xvhg lq jhqhudwlqj wkh vhulhv1 D IQQ hvwlpdwh zlwk �yh qhxurqv lq wkhklgghq od|hu zlwk v|pphwul}hg orjlvwlf ixqfwlrq dv dfwlydwlrq ixqfwlrq jdyh wkh ehvwshuirupdqfh1 Lq wkh hvwlpdwlrq ri ¡ �� dqg ¦ �� zh xvhg irxu qhxurqv hdfk lq wkh klgghqod|hu zlwk wkh vdph dfwlydwlrq ixqfwlrq1 Dv fdq eh vhhq iurp �jxuh 8116/ erwk phwkrgfrxog qrw h�hfwlyho| uhvsrqg wr wkh kljk shdnv lq wkh gdwd dqg dsshdu wr vhw dq xsshuerxqg iru wkh udqgrp yrodwlolw|1 Rq wkh rwkhu kdqg/ wkh IQQ jdyh kljkhu yrodwlolw|hvwlpdwhv wkdq wkh GO phwkrg1 Wkh orzhu uljkw ri wkh vdph �jxuh vkrzv wkh IQQhvwlpdwh ri ¦ �� ' sV arE*L} ¦ �� =x � � � � dqg �jxuh 8117 vkrzv wkh phdq vtxduh huuruv riwkh wzr phwkrgv1 Iljxuh 8118 +xsshu ohiw, vkrzv wkh IQQ hvwlpdwh +gdvkhg olqh, rif:f�n� ULtE*L} ¦ �� (' � Evrolg0olqh, zkhuhdv wkh xsshu uljkw fxuyh vkrzv wkhvh wzr judskvsorwwhg wrjhwkhu zlwk wkh gdwd sdlu E*L} ¦ �� ; *L} ¦ �� ' �:

Wkh IQQ hvwlpdwh yhu| zhoo lq wkh uhjlrq zkhuh wkh gdwd lv ghqvh dqg jdyh dsrruhu hvwlpdwh lq wkh vsduvh uhjlrq1 Wklv lv qr vxusulvh/ w|slfdo ri dq| qrqsdudphwulfhvwlpdwlrq surfhgxuh1 Iljxuh 8118 +orzhu ohiw, lv wkh jhqhudwhg gdwd iurp *L} ¦ �� zkhuhdvwkh orzhu uljkw lv wkh IQQ hvwlpdwh ri ¡ �� :

Lq wkh qh{w vwxg| ri wkh vlpxodwlrq/ zh h{dplqh hljkw gl�huhqw Jhupdq vwrfnv=Dooldq} DJ +DOY,/ EDVI DJ +EDV,/ Ed|hu DJ +ED\,/ Frpphu}edqn DJ +FEN,/K|sryhuhlqvedqn +K\S,/ Ghxwvfkh Oxiwkdqvd DJ +OXG,/ Wk|vvhq Nuxss +WK\, dqgYronvzdjhq DJ +YRZ, iru wkh shulrg 35134133 wr 63145136 dqg frpsduh lwv edfnwhvwlqjshuirupdqfh zlwk wkh IQQ hvwlpdwh1 Zh kdyh fkrvhq lq hdfk dvvhw 60glphqvlrqdoh{rjhqhrxv yduldeohv frqvlvwlqj ri h{srqhqwldoo| zhljkwhg bf�gd|v klvwrulfdo yduldqfh

437

1000 2000 3000 4000 5000

2

4

6

8

1000 2000 3000 40000.5

1

1.5

2

2.5

1000 2000 3000 4000

1.7

1.8

1.9

2

2.1

1000 2000 3000 4000

0.5

0.6

0.7

0.8

0.9

Iljxuh 8116= Xsshu ohiw= Udqgrp yrodwlolw|1Xsshu uljkw= IIQ hvwlpdwh1Orzhu ohiw= GO phwkrg hvwlpdwh1 1Orzhu uljkw= IIQ hvwlpdwh ri ¦ ��

438

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.002

0.004

0.006

0.008

0.01

0.012DL errorFFN error

Iljxuh 8117= Phdq vtxduh huuru ri GO phwkrg +vrolg olqh, dqg wkh Phdq vtxduh ri IIQdssurdfk +grwwhg olqh,

439

-4 -2 0 2-4

-2

0

2

4

-10 -5 0 5 10-10

-5

0

5

10

15

20FFN estimateTrue functionData points

1000 2000 3000 4000 5000

-4

-2

0

2

4

1000 2000 3000 4000 5000-4

-2

0

2

4

Iljxuh 8118= Xsshu ohiw= IIQ hvwlpdwh +gdvkhg olqh, ri f:f�n� ULtE*L} ¦ �� (' � +vrolg olqh,1Xsshu uljkw= IIQ hvwlpdwh +gdvkhg olqh, ri f:f� n � ULtE*L} ¦ �� (' � +vrolg olqh, dqg wkhgdwd sdlu + *L} ¦ �� (' ; *L} ¦ �� 1Orzhu ohiw= f:f� n � ULtE*L} ¦ �� (' � n � � 1Orzhu uljkw= IIQ hvwlpdwh ri u �� :

43:

ri h � ; h{srqhqwldoo| zhljkwhg bf �gd|v klvwrulfdo yduldqfh ri h � ;

EW � �(' '� )[

� 0 ' �

G Eh � � � � 7h � ) �; 7h � ) ' �bf

� )[

� 0 'h � � �

wkh uhwxuqv d � �(' ri wkh pdunhw lqgh{ dqg bf gd|v prylqj dyhudjh/ PD � �(' ' '� ) S � )7 0 ' d � � 7

+Vhh �jxuh 8119, dv dq lqglfdwru ri orqj whup pdunhw pryhphqw1 Zh kdyh fkrvhq ' f:bD; G dv wkh vxp ri wkh jhrphwulf vhulhv n � n : : :n � ) : Lq wkh GO dssurdfkzh orrvh <4 wudglqj gd|v iru lqlwldol}dwlrq zkhuhdv lq wkh IQQ edvhg phwkrg wkh gdwdvwduwv iurp Zhgqhvgd| Pd| 43/ 5333 dv zh orrvh <3 wudglqj gd|v iru lqlwldol}dwlrq1 Wkhuhvxowv duh wdexodwhg lq wdeoh 81141 Wkh � � E� > f� lv vlpso| wr dyrlg wkh lqfoxvlrq ri dfrqvwdqw whup lq *L} ¦ �� surfhvv1 Iljxuh 811: vkrzv wkh IQQ hvwlpdwlrq dw hyhu| vwdjhri wkh ohduqlqj surfhvv iru wkh DOY uhwxuqv1 Iljxuh 811; vkrzv wkh 8( YdU hvwlpdwhviru wkh GO dqg wkh IQQ dssurdfkhv1 Lq �jxuh 811< +xsshu judsk,/ wkh wzr phwkrgvduh sorwwhg wrjhwkhu dqg wkh orzhu sorw vkrzv wkh DOY uhwxuqv1 Rq wkh zkroh wkh GOdssurdfk lv surplvlqj frpshwlqj yhu| zhoo zlwk wkh IQQ phwkrg1 Ri frxuvh/ li rqhlv zloolqj wr glvvlsdwh pruh wlph lq wkh ohduqlqj surfhvv/ d pxfk ehwwhu uhvxowv duh wreh h{shfwhg1 Iljxuh 81143 wkurxjk 81145 vkrz vlplodu uhvxowv iru K\S vwrfn1 Revhuyhwkh xqxvxdo kljk hvwlpdwh dw odj 633 iru wkh K\S vwrfn xqghu uhodwlyho| vwdeoh pdunhwfrqglwlrqv e| wkh GO phwkrg +�jxuh 81145/ xsshu judsk,1 Wklv fdq eh h{sodlqhg e| wkhidfw wkdw wkh IQQ hvwlpdwh ¦ � dw wklv odj ylrodwhv wkh frqvwudlqwv ¦ � < �:H��H dqg vrrqh fdq gr glvfulplqdqw frqglwlrqdo yrodwlolw| hvwlpdwlrq +vhh uhpdun 8<,1

Zh vxppdul}h wkh hvwlpdwlrq surfhvv lq wkh iroorzlqj dojrulwkp=

Dojrulwkp 96 +Grxeoh ohduqlqj,Vwhs 4= Pdnh d �uvw dssur{lpdwlrq ri ¦ � e| wkh phwkrg ghvfulehg lq fkdswhu 5 ru dv

43;

Vwrfn IQQ edvhg GO edvhgOhyho ri �I YdU Ohyho ri �I YdU � �f:fD f:f2D f:f� f:fD f:f2D f:f�

DOY 31377< 313585 3133<; 313837 3135:7 3133<< 31<EDVI 3137: 3135;7 313464 3137<6 3135:7 3133<< 31;9ED\ 3137: 31365; 31356 313738 313596 313475 31:FEN 313847 31354< 31343< 3137<6 31354< 313388 31<;K\S 313858 313574 31343< 3137<6 313585 31344 31<48OXG 3137: 31356 313464 313837 31354< 3133:: 31<4;WK\ 3137: 3135;7 313486 3137<6 313585 31344 4YRZ 31376; 313574 313453 313837 31353; 3133;; 31;98

Wdeoh 8114= Frpsdulvrq ri IIQ dqg GO hvwlpdwhv ri YdU iru gl�huhqw Jhupdq vwrfnvlq wkh shurlg 35134133 wr 631451361 JO lv wkh xqghuo|lqj glvwulexwlrq

43<

200 400 600 800

0.5

1

1.5

2

2.5

3

3.5

x 10-3

200 400 600 800

-6

-4

-2

0

2

4x 10

-3

100 200 300 400 500 600 700 800 900 1000-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Iljxuh 8119= Xsshu ohiw= H{srqhqwldoo| zhljwkhg <3 gd|v Dooldq} uhwxuqv1Xsshu uljkw= <3 gd|v Prylqj dyhudjh ri GD[ uhwxuqv1Orzhu judsk= GD[ uhwxuqv

443

200 400 600 800

0.5

1

1.5

2

2.5

3

3.5

x 10-3

200 400 600 800

-6

-4

-2

0

2

4x 10

-3

100 200 300 400 500 600 700 800 900 1000-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Iljxuh 811:= Xsshu ohiw= H{srqhqwldoo| zhljwkhg <3 gd|v Dooldq} uhwxuqv1Xsshu uljkw= <3 gd|v Prylqj dyhudjh ri GD[ uhwxuqv1Orzhu judsk= GD[ uhwxuqv

444

200 400 600 800

0.02

0.03

0.04

0.05

0.06

200 400 600 800

-8

-7.5

-7

-6.5

-6

-5.5

200 400 600 800

-8

-7.5

-7

-6.5

-6

200 400 600 800

0.2

0.4

0.6

0.8

1

1.2

Iljxuh 811;= Xsshu ohiw= Yrodwlolwlhv dv vhhq e| IIQ iru DOY uhwxuqv1Xsshu uljkw= orj+yduldqfh, ihg wr wkh qhwzrun1Orzhu ohiw= IIQ hvwlpdwh ri u �� :Orzhu uljkw= IIQ hvwlpdwh ri ¦ �� :

445

0 100 200 300 400 500 600 700 800 900

-0.1

-0.05

0

0.05

0.1

0 100 200 300 400 500 600 700 800 900

-0.1

-0.05

0

0.05

0.1

Iljxuh 811<= Xsshu judsk= Dooldq} uhwxuqv +44138133 wr 63145136,/ GO 0edvhg0DI YdU+vrolg olqh, dqg YdU h{fhhgdqfhv +-,1Orzhu judsk= Dooldq} uhwxuqv +43138133 wr 63145136,/ IIQ 0edvhg0DI YdU +vrolg olqh,dqg YdU h{fhhgdqfhv +-,1

446

0 100 200 300 400 500 600 700 800 900

-0.1

-0.05

0

0.05

0.1

0.15ALV returnsDL estimateFFN estimate

100 200 300 400 500 600 700 800 900 1000

-0.1

-0.05

0

0.05

0.1 ALV returns

Iljxuh 81143= Xsshu judsk= Uhwxuqv ri Dooldq} E��; GO0edvhg 8( YdU +vrolg olqh, dqgIIQ edvhg 8( YdU +gdvkhg olqh,1Orzhu judsk= Uhwxuqv ri Dooldq}

447

200 400 600 8000.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

200 400 600 800

-9

-8

-7

-6

-5

200 400 600 800

-8

-7.5

-7

-6.5

-6

200 400 600 800

0.5

1

1.5

Iljxuh 81144= Xsshu ohiw= Yrodwlolwlhv dv vhhq e| IIQ iru K\S vwrfn1Xsshu uljkw= orj+yduldqfh, ihg wr wkh qhwzrun1Orzhu ohiw= IIQ hvwlpdwh ri u �� :Orzhu uljkw= IIQ hvwlpdwh ri ¦ �� :

448

0 100 200 300 400 500 600 700 800 900

-0.2

-0.1

0

0.1

0.2

0 100 200 300 400 500 600 700 800 900-0.2

-0.1

0

0.1

0.2

Iljxuh 81145= Xsshu judsk= K\S uhwxuqv + �, iurp +44138133 wr 63145136,/ GO 0edvhg0DI YdU +vrolg olqh, dqg YdU h{fhhgdqfhv +-,1Orzhu judsk= K\S uhwxuqv + �, iurp +43138133 wr 63145136,/ IIQ 0edvhg0DI YdU +vrolgolqh, dqg YdU h{fhhgdqfhv +-,1

449

0 100 200 300 400 500 600 700 800 900

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1 HYP returnsDL estimateFFN estimate

100 200 300 400 500 600 700 800 900 1000-0.15

-0.1

-0.05

0

0.05

0.1HYP returns

Iljxuh 81146= Xsshu judsk= Uhwxuqv ri K|sryhuhlqvedqn E��; GO0edvhg 8( YdU +vrolgolqh, dqg IIQ edvhg 8( YdU +gdvkhg olqh,1Orzhu judsk= Uhwxuqv ri K|sryhuhlqvedqn

44:

ghvfulehg lq ^GI37`1Vwhs 5= Hvwlpdwh ¡ �� e| plqlpl}lqj wkh rswlpl}dwlrq sureohp

�n� �

�[� 0 � � '

d*L} ¦ �� fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( ��o �

e| wkh Ohyhqehuj0Pdutxdugw dojrulwkp1Vwhs 6= Hvwlpdwh ¦ �� e| plqlpl}lqj wkh rswlpl}dwlrq sureohp

�n� �

�[� 0 � � '

dE*L} ¦ �� ¡ �� fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( ��o �

djdlq e| wkh Ohyhqehuj0Pdutxdugw dojrulwkp1 Dowhuqdwlyho|/ hvwlpdwh © �� ' EEE*L} ¦ �� =x � � � � e|plqlpl}lqj

�n� �

�[� 0 � � '

dE*L} ¦ �� fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( ��o �

e| wkh Ohyhqehuj0Pdutxdugw dojrulwkp dqg jhw dq hvwlpdwh ¦ �� ' © �� ¡ �� ri ¦ �� :Vwhs 7= Li ¦ �� Ei� < �:H��H iru doo � � i � n; vxevwlwxwh ¡ �� dqg ¦ �� lq wkhruhp 8; wr jhwdq hvwlpdwh _V arEh � =x � �(' � ri V arEh � =x � �(' �:Vwhs 8= ¦ �� Ei� � �:H��H iru dw ohdvw rqh � � i � n; xvh wkh phwkrg ghvfulehg lq fkdswhu.:

Pd{lpxp olnholkrrg Hvwlpdwlrq Wr ghdo zlwk wkh txhvwlrq ri wkh shuirupdqfh ri IQQlq frpelqdwlrq zlwk pd{lpxp olnholkrrg hvwlpdwlrq zh vxssrvh wkdw

*L} ¦ �� ' � *L} ¦ �� (' n � � ; � � �GLEf; ¦ �� ; a� +81143,

Vwdwlrqdulw| ri wkh *L} ¦ �� surfhvv lpsolhv wkdw EE*L} ¦ �� ' f; V arE*L} ¦ �� =x � � � � '¦ ��

�� dqg khqfh *L} ¦ �� GLEf; ¦ �� ; a�: Ohw Ex � � �� 0 ' eh wkh *L} ¦ �� dv vhhq e| wkh

44;

IQQ lq dffrugdqfh zlwk +8117,1 Wkhq iru d �{hg a 5 E�;4�; wkh ydoxhv ri � dqg ¦ �� fdqeh rewdlqhg e| vroylqj wkh rswlpl}dwlrq sureohp

4�?� * � �� dn *L}E¦ �s�� n

s�� *L} a¦ �

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x 7 n 2�[7 0 '

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dv wkh frqvwdqwv ri wkh olnholkrrg ixqfwlrq sod|v qr uroh lq wkh plqlpl}dwlrq1 Qrwh wkdwwklv sureohp kdv wr eh vroyhg qxphulfdoo|1 Xvlqj wkh Dooldq} vwrfnv xqghu wkh vdphfrqglwlrqv glvfxvvhg suhylrxvo|/ zh vroyhg wkh rswlpl}dwlrq sureohp xvlqj PDWODEyhuvlrq 9141 Ri wkh 48 gl�huhqw lqlwldo jxhvv hvwlpdwhv iru d�; ¦ �� o; iru wkh rswlpl}dwlrqurxwlqh/ 43 jdyh wkh vdph plqlpdo remhfwlyh ydoxh1 Lq rwkhu zrugv/ wkh remhfwlyhixqfwlrq dsshdu �dw dw wkh rswlpxp dqg vr wkhuh zdv wkh gl!fxow| ri zklfk ydoxhwr vhohfw iru d�; ¦ �� o: Iru rqh ri wkhvh 43 plqlpdo remhfwlyh ydoxhv xvhg lq fdofxodwlqjV arEh � =x � �(' � lq frqvxowdwlrq zlwk wkhruhp 93/ zh rewdlq wkh iroorzlqj uhvxowv= dwDI�txdqwloh riGLEf; �; i TE 6

� � ��; e:b2I ri wkh gdwd srlqwv h{fhhghg YdUzlwk d fkrlfhri � � ' e:2D � �f �� : Dw 2:DI dqg �I �txdqwloh ri GLEf; �; i TE 6� � �� wkh shuirupdqfhzhuh uhvshfwlyho| 2:bDI dqg �:e2I ri wkh gdwd h{fhhglqj YdU1 Lqfuhdvlqj ru ghfuhdvlqj� � eh|rqg wklv ydoxh glg qrw lpsuryh wkh shuirupdqfh ri YdU h{fhhgdqfhv dw wkhvh ohyhovdowrjhwkhu1 Zh dovr wulhg hvwlpdwlqj wkh YdU lq wkh devhqfh ri h{rjhqrxv yduldeohv dqgrewdlqhg wkh iroorzlqj uhvxow= dw DI�txdqwloh ri GLEf; �; i TE 6� � ��; e:bbI ri wkh gdwdsrlqwv h{fhhghg YdU zlwk d fkrlfh ri � � ' S:D � �f �� : Dw 2:DI dqg �I �txdqwlohri GLEf; �; i TE 6

� � �� wkh shuirupdqfh zhuh uhvshfwlyho| �:2bI dqg �:DI ri wkh gdwdh{fhhglqj YdU1 Wkhuh duh qxpehu ri rshq txhvwlrqv khuh= frxog zh kdyh rewdlqhg dehwwhu uhvxow li wkh xqrevhuydeohv zhuh revhuyhg zlwk UQQ lqvwhdg dqg surfhhg zlwk wkhpd{lpxp olnholkrrgB Ri frxuvh/ khuh wrr zh vwloo kdyh wr erwkhu derxw zklfk ydoxhv wr

44<

fkrrvh iru d�; ¦ �� o dv wkh remhfwlyh ixqfwlrq lv �dw dw wkh rswlpxp1 Frxog wkh hvwlpdwhvlpsuryh li zh lqfuhdvh wkh qxpehu ri qxpehu ri odjv lq wkh *L} ¦ �� surfhvvB Revhuyhwkdw dv wkh qxpehu ri odjv lqfuhdvhv wkh remhfwlyh ixqfwlrq ehfrphv pruh gl!fxow wrplqlpl}h1 Iru lqvwdqfh li zh dvvxph

*L} ¦ �� ' � *L} ¦ �� ' n � *L} ¦ �� n � � ; � � �GLEf; ¦ �� ; a�

wkhq zh zrxog kdyh wr vroyh wkh sureohp

4�?� * � * � �� dn *L}E¦� n *L} a¦

�[7 0 '

x 7 n 2�[7 0 '

*L}E� n a � $� � � �o

xqghu wkh frqvwudlqwv ¦ � ' ¦ ��2¤

U dwm��S�� 0 ' � � e 7 �� m � > f: Wklv rswlpl}dwlrq sureohp

pd| eh vroyhg e| vroylqj wkh xqfrqvwudlqhg sureohp

4�?� * � * � �� dn *L}E¦� n *L} a¦

�[7 0 '

x 7 n 2�[7 0 '

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� 0 'y� E¦ ��¦ � n

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zkhuh S �

� 0 ' y� E¦ ��¦ � n�¦ �� lv d eduulhu ixqfwlrq dqg wkh Ey� � �� 0 ' duh d vhtxhqfh ri ghfuhdv0

lqj ydoxhv ^E34`/ sj 4831 Xvlqj wkh iplqfrq frppdqg lq wkh rswlpl}dwlrq Wrroer{xqghu wkh frqvwudlqwv ¦ � > f/ zh wulhg zlwk pdq| lqlwldo hvwlpdwhv ehiruh wkh rs0wlpdo vroxwlrq zdv orfdwhg1 Xqiruwxqdwho|/ wkh uhvxowv zhuh txlwh vlplodu wr wkdw ri+81143,1 Dw DI�txdqwloh ri GLEf; �; i TE 6

� � ��; eD gdwd srlqwv h{fhhghg YdU uhsuhvhqw0lqj e:b2I zlwk � � ' D:��f �� : Dw 2:DI dqg �I�txdqwlohv wkh hvwlpdwhv zhuh qrw lp0suhvvlyh zlwk 2H dqg �� gdwd srlqwv h{fhhglqj V aR uhsuhvhqwlqj uhvshfwlyho| �:fSI dqg�:e2I ri wkh gdwd srlqwv1 Lq hdfk ri wkhvh wzr surfhgxuhv zh orrvh <3 wudglqj gd|v irulqlwldol}dwlrq1 Rq wkh zkroh/ wkh GO shuirupv ehwwhu wkdq wkh IQQ.pd{lpxp olnhol0krrg edvhg phwkrgv1 Wkhuh duh/ krzhyhu/ vrph rshq txhvwlrqv= frxog zh kdyh rewdlqhg

453

d ehwwhu hvwlpdwh iru wkh GO phwkrg li d gl�huhqw xqghuo|lqj glvwulexwlrq zhuh xvhgBZkdw derxw UQQ iroorzhg UQQ ru IQQ iroorzhg e| UQQB Wkhvh duh rshq txhvwlrqvwkdw qhhgv ixuwkhu lqyhvwljdwlrqv1

818 Uhvxowv iru vrph frpprq glvwulexwlrqv

Lq wklv vhfwlrq zh lqyhvwljdwh wkh h�hfw rq wkh YdU hvwlpdwh dv wkh xqghuo|lqjglvwulexwlrq ydulhv iru wkh GO phwkrg1 Lq frqvhtxhqfh zh jlyh uhvxowv iru wkhruhp 93iru rwkhu frpprq glvwulexwlrqv +l1h1 NIG � � ; N dqg wkh t � � xvhg iru prghoolqj wkhgrzqvlgh ulvn1

Wkh surfhhglqjv vr idu prwlydwhv wkh iroorzlqj jhqhudo uhvxow ydolg iru frqwlqxrxvglvwulexwlrqv vxssruwhg rq U zlwk glvwulexwlrq sdudphwhuv qrw ghshqg rq wkh vxssruw=Ohw X eh d sdudphwhu vsdfh dqg ifEh( X�j � � � ; suredelolw| ghqvlw| dvvrfldwhg zlwk wkhudqgrp yduldeoh/ h; zlwk glvwulexwlrq F Eh( X� zlwk uhvshfw wr wkh Ohehvjxh phdvxuh:Zhzulwh h�F EX�: Ohw M � Et�; t 5 U eh wkh prphqw jhqhudwlqj ixqfwlrq ri h:

Gh�qlwlrq 97 Y � *L}F EX� li *L} Y �F EX�:

Wkhruhp 98 Ohw Y � *L}F EX�: Wkhq EY � ' M � En�; zkhuh M � Et�; t 5 U lv wkh pr0

phqw jhqhudwlqj ixqfwlrq ri wkh glvwulexwlrq F EX� zlwk ghqvlw| fEh( X� vxssruwhg rqU; qrw ghshqglqj rq X:

Surri1 Li Z ' *L} Y �F EX� wkhq Y kdv ghqvlw|

�yfE*L} y; ��; y > f:

454

Wr vhh wklv/ qrwh wkdw ehfdxvh Z�F EX�; f � E*L} y� ' fE*L} y( X�: Ohw f � Ey� eh wkh ghqvlw|ri Y zlwk fruuhvsrqglqj glvwulexwlrq ixqfwlrq F � Ey�: Wkhq

F � Ey� ' EE� �� ' EE� "�# �� *L} y�~}��

� ' F � Ez�; y > f

E| wkh gl�huhqwldelolw| ri F � Ey� zh kdyh

f � Ey� ' dF � Ey�dy ' dF � Ez�

dzdzdy ' �

yf � E*L} y�:

KhqfhEY � '

]

�� y� �yfE*L} y; ��dy

Wkh vxevwlwxwlrq z ' *L} y lpso| wkdw

EY � ']

� e /� fEz; ��dz ' Ee � / ' M � En�:

Wkh wkhruhp lpso| wkdw rqfh wkh prphqw jhqhudwlqj ixqfwlrq/ M � Et�; t 5 U riF EX� lv nqrzq/ wkh nth prphqwv derxw f ri *L}F EX� lv rewdlqhg e| vlpso| uhsodflqjt zlwk n: Rq wkh rwkhu kdqg/ lq wkh fdvh zkhuh wkh M � Et�; t 5 U ri F EX� lv whglrxv wrghulyh/ zh fdq flufxpyhqw wkh sureohp yld wkh prphqwv derxw f ri *L}F EX� wkdw pd|eh hdvlhu wr rewdlq lq d forvhg irup1

H{dpsoh 99 Frqvlghu wkh glvwulexwlrq zlwk ghqvlw|

fEh� ' e �� ¢KE�� i TE�e ��

� �; h 5 E�4;4�; � > f; � > f +81144,

Wkh prphqw jhqhudwlqj ixqfwlrq fdq eh rewdlqhg e| frqvlghulqj wkh nth prphqwv derxwf ri wkh ghqvlw|

gEh� ' e � � � � h � � � '¢KE�� i TE� �

�h$� e � � � �; h 5 Ef;4� +81145,

455

zklfk fdq hdvlo| eh vhhq dv �

� �� e ��KE�� KE� n n¢�:

Khqfh e| wkhruhp 98/ M � Et� ri +81144, lv vlpso|

M � Et� ' � � � e � �KE�� KE� n t¢�

iurp zklfk lqwhuhvwlqj surshuwlhv iru fEh� fdq eh ghgxfhg1

Wkh qhz ghqvlw| +81144, fdq eh xvhg iru prghoolqj wkh ulvn ri dq dvvhw � irudssursuldwho| fkrvhq � dqg �: Wkh vnhzqhvv lv hqwluho| frqwuroohg e| � dqg fdq ehiuhho| dgmxvwhg wr rewdlq wkh ohyho ghvluhg1 Zkhq � ' � dqg � ' �; zh jhw wkh Jxpehoglvwulexwlrq1 Revhuyh dovr wkdw wkh ghqvlw| +81145,/ frqwdlqv wkh jdppd glvwulexwlrqe| fkrrvlqj ¡ ' f dqg ¢ ' � dqg iru wkdw pdwwhu h{srqhqwldo dqg wkh fkl0vtxduhglvwulexwlrqv dv vshfldo fdvhv1

Frqyhuvho|/ zh kdyh wkh iroorzlqj jhqhudo uhvxow ydolg iru frqwlqxrxv glvwulexwlrqvvxssruwhg rq U � zlwk glvwulexwlrq sdudphwhuv qrw ghshqg rq wkh vxssruw1 Ohw X eh dsdudphwhu vsdfh dqg fEh( X�; h 5 U � suredelolw| ghqvlw| dvvrfldwhg zlwk wkh udqgrpyduldeoh/ h; zlwk glvwulexwlrq F Eh( X� zlwk uhvshfw wr wkh Ohehvjxh phdvxuh: Zh zulwhh�F EX�: Ohw M � Et�; t 5 U eh wkh prphqw jhqhudwlqj ixqfwlrq ri h:

Gh�qlwlrq 9: Zh vd| wkdw wkh udqgrp yduldeoh Y � i TF EX� li i TY �F EX�:

Wkhruhp 9; Ohw i TY �F EX�: Wkhq M � Et� ' Eh � ; zkhuh Eh � ; t 5 U lv wkh twkprphqw derxw f ri wkh glvwulexwlrq F EX� zlwk ghqvlw| fEh( X�; h 5 U � :

� Xvh wkh vxevwlwxwlrq y @ �� x �� e � :

� Vhh vhfwlrq 819/ iru vrph vwdwlvwlfdo surshuwlhv456

Surri1 Ohw Z ' i TY; wkhq f � Ee � � ' fEe � ( X�: Ohw f � Ey� eh wkh ghqvlw| riY zlwk fruuhvsrqglqj glvwulexwlrq ixqfwlrq F � Ey�: Wkhq

F � Ey� ' EE� � �� ' EE� 143 5 �� i T y� ~} ��

� ' F � Ez�; y 5 U

E| wkh gl�huhqwldelolw| ri F � Ey� zh kdyh

f � Ey� ' dF � Ey�dy ' dF � Ez�

dzdzdy ' i TEy�f � Ei TEy��:

QrzM � Et� '

]

� e � � i TEy�f � Ei TEy�( ��dy

Xvlqj wkh vxevwlwxwlrq h ' i TEy�; zh jhw

M � Et� ']

�� h � fEh; ��dh ' Eh �

Wkhruhp 9< Iru hyhu| qrqghjhqhudwh glvwulexwlrq zlwk ghqvlw| vxssruwhg rq U � uhvshfwlyho|U; wkhuh h{lvwv d qhz ghqvlw| vxssruwhg rq U uhvshfwlyho| U � e| dsso|lqj d vprrwk wudqv0irupdwlrq +vxssruw gr qrw ghshqg rq wkh ghqvlw| sdudphwhuv,1

Surri1 +l, Iluvw zh vkrz wkdw li fEh( �� lv d qrqghjhqhudwh ghqvlw| vxssruwhgrq U � zlwk glvwulexwlrq ixqfwlrq F � Eh( ��; wkhq wkhuh h{lvwv d ghqvlw| vxssruwhg rq U:

Fodlp= e � fEe � ( �� lv d qrqghjhqhudwh ghqvlw| vxssruwhg rq U:Ohw h ' e � dqg vxssrvh wkdw y kdv ghqvlw| fEy( �� zlwk glvwulexwlrq F � Ey( ��: Wkhq

F � Ey( �� ' P EY � y� ' P Ee � � e � � ' F � Eh( ��; y 5 U

QrzfEy( �� ' dF � Ey( ��

dy ' dF � Eh( ��dh

dhdy ' e � fEe � ( ��:

457

Wklv lv d ghqvlw| vxssruwhg rq U dv e � fEe � ( �� f dqg]

� e � fEe � ( ��dy ']

�� fEm( ��dm ' �:

+ll, Qh{w/ zh vkrz wkdw li fEh( �� lv d qrqghjhqhudwh ghqvlw| vxssruwhg rq U zlwk glv0wulexwlrq ixqfwlrq F � Eh( ��; wkhq wkhuh h{lvwv d ghqvlw| vxssruwhg rq U � :Fodlp= �yfE*L} y( �� lv d qrqghjhqhudwh ghqvlw| vxssruwhg rq U � :

Ohw h ' *L} y dqg vxssrvh wkdw y kdv ghqvlw| fEy( �� zlwk glvwulexwlrq F � Ey( ��: Wkhq

F � Ey( �� ' P EY � y� ' P E*L} Y � *L} y� ' F � Eh( ��; y 5 U �

Qrz

fEy( �� ' dF � Ey( ��dy ' dF � Eh( ��

dhdhdy ' �

yfE*L} y( ��; y 5 U �

Wklv lv d ghqvlw| vxssruwhg rq U dv �yfE*L} y( �� f dqg

]

��yfE*L} y( ��dy '

]

� fEm( ��dm ' �:

Pdun wkdw wkh uhvxow lv ydolg iru glvwulexwlrqv zkrvh ghqvlw| gr qrw ghshqg rqwkh ghqvlw| sdudphwhuv +h1j1 wkh uhvxow lv qrw ydolg iru wkh Sduhwr glvwulexwlrq,1 Zhzloo odwhu xvh wklv uhvxow wr pdnh dqhz glvwulexwlrqv wkdw kdyh qrw ehhq h{sorlwhg dwwkh �qdqfldo pdunhw1

Xqghu NIG � � dvvxpswlrq iru wkh lqqrydwlrqv zh kdyh wkh iroorzlqj uhvxow=

458

Wkhruhp :3 Ohw Dvvxpswlrq 89 krog dqg vxssrvh wkdw

*L} ¦ �� ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� n � � ; � � �NIG � � E��; f; f; ¦ �� :

Li " � �NIG � � E7�; f; f; �� lq +8117,; wkhq iru 7� > ¦ ��+l, V arEh � =x � �(' � ' i Td¡ �� s7� � � 7�¦ �� o:+ll, h{Eh � =x � �(' � > fzkhuh ¡ �� ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� lv wkh phdq ri *L} ¦ �� jlyhq wkh sdvw x � � � 1

Surri1 +l, Pdun wkdw wkh vwdwlrqdulw| ri *L} ¦ �� lpso| wkh vwdwlrqdulw| ri h � : Ohw¡ �� dqg ¦ �� eh uhvshfwlyho| wkh h{shfwdwlrq dqg wkh yduldqfh ri *L} ¦ �� jlyhq wkh sdvw/x � � � : Wkhq

¡ �� ' EE*L} ¦ �� =x � � � � ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( ��

dv E� � ' EE� � =x � � � � ' f; dqg


dv fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� lvx � � � �phdvxudeoh1 Khqfh *L} ¦ �� NIG � � E7�; f; ¡ �� ; ¦ �� dqgvr iru 5 U; *L} ¦ �� NIG � � E7�; f; ¡ �� ; � ¦ �� jlyhq x � � � :Lw iroorzv iurp wkhruhp 98/ +vhh dovr dsshqgl{ wr fkdswhu 9, wkdw

EEe �� "�# �� =x � � � � ' i Td ¡ �� t7� � � 7� � ¦ �� n 7�o +81146,

Qrz iru ' �; V arEh � =x � �(' � ' E" �� EEe "�# � �� ehfrphv

V arEh � =x � �(' � ' i Td¡ �� t7� � � 7�¦ �� n 7�o

+ll, Zlwk ' 2; lw iroorzv iurp +81146, wkdwEEEh � �¡ � � � =x � �(' � ' E" �� EEi TE*L} ¦ �� ehfrphv

E" �� i Td2¡ �� t7� � � e7�¦ �� n 7�o

459

Wkhuhiruh

exEh � =x � � ' � ' EEEh � � ¡ � � � =x � � ' �EV arEh � =x � �(' ��

' E" ��EE" �� ~} �

��

i Td2¡ �� s7� � � e7�¦ �� n 7�oEi Td¡ �� s7� � � 7�¦ �� n 7�o� �� ~} �

�

��;

zkhuh � � ' � n �7� lv wkh nxuwrvlv ri " � dqg 7� > e¦ �� : Lw vx!fhv wr vkrz wkdw wkh

h{suhvvlrq J > �: Qrwh wkdw J ' i Td2s7� � � 7�¦ �� s7� � � e7�¦ �� 7�o: Li 2s7� � � 7�¦ �� s7� � � e7�¦ �� 7� � f; lw iroorzv wkdw ¦ �� f zklfk lv d frqwudglfwlrq wr wkh srvlwlylw|¦ �� 1 Khqfh wkh rqo| dgplvvleoh ydoxhv iru wkh h{srqhqw ri wkh h{suhvvlrq J lv wkh fdvhzkhuh 2s7� � � 7�¦ �� s7� � � e7�¦ �� 7� > f: Khqfh J 5 E�;4� dqg vr wkh uhvxow iroorzv1

Iru wkh qrupdo glvwulexwlrq zh kdyh wkh iroorzlqj=


*L} ¦ �� ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� n � � ; � � �NEf; ¦ �� :

Li " � �NEf; �� lq +8117,; wkhq+l, V arEh � =x � �(' � ' e � � � $� � ��

+ll, h{Eh � =x � �(' � > fzkhuh ¡ �� ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� lv wkh phdq ri *L} ¦ �� jlyhq wkh sdvw x � � � 1

Surri1 +l, Ohw ¡ �� dqg ¦ �� eh uhvshfwlyho| wkh h{shfwdwlrq dqg wkh yduldqfh ri*L} ¦ �� jlyhq wkh sdvw/ x � � � : Wkhq

¡ �� ' EE*L} ¦ �� =x � � � � ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( ��45:

dv E� � ' EE� � =x � � � � ' f; dqg


dv fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� lv x � � � �phdvxudeoh1 Khqfh *L} ¦ �� NE¡ �� ; ¦ �� dqg vrb *L} ¦ �� NEb¡ �� ; b � ¦ �� jlyhq x � � � iru b 5 U: Iurp wkhruhp 98/

EEe � "�# �� =x � � � � ' i TEb¡ �� n �2b � ¦ �� : +81147,

QrzV arEh � =x � �(' � ' E" �� EEe "�# � ��

dqg zlwk b ' � lq +81147,/ zh rewdlq

V arEh � =x � �(' � ' i TE¡ �� n �2¦ �� :

+ll, Zlwk b ' 2 lq +81147,; EEEh � � ¡ � � � =x � �(' � ' E" �� EEi TE*L} ¦ �� ehfrphv

E" �� i TE2¡ �� n 2¦ �� :

Khqfh

exEh � =x � �(' � ' EEEh � � ¡ � � � =x � �(' �EV arEh � =x � �(' �� ' � � i TE¦ ��

' �Ei TE¦ �� > f

dv ¦ �� > f dqg vr i TE¦ �� 5 E�;4� �� :D vlplodu uhvxow iru wkh fdvh zkhuh *L} ¦ �� iroorz d vlpsoh ARE�� surfhvv fdq eh

irxqg lq ^JKU<9` dqg ^V<;`/ sjv 470481 Wkh odwwhu uhihuhqfh glvfxvvhv vhyhudo dgydqwdjhvdqg ydulrxv jhqhudol}dwlrq ri wklv prgho1

Iru wkh v|pphwulf vwxghqwzv t glvwulexwlrq zh kdyh wkh iroorzlqj=�� lv wkh nxuwrvlv ri " � zklfk lv 6 iru wkh qrupdo glvwulexwlrq1

45;

Ohppd :5 Ohw h� *L} t � Ev; ¡; ¦ � �: Wkhq

Eh � ' 2 i TEn¡�BE �

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� )+* - �ULt�Exn¦sv � 2�

E� n x � � � $�

dx

Surri1 Revhuyh wkdw li h� *L} t � Ev; ¡; ¦ � �; wkhq h kdv ghqvlw|

fEh� ' E� n "�# � � � � �� $�

hBE �

� ; '� �¦sv � 2 ; h > f

Khqfh/ wkh prphqwv derxw f; fdq eh zulwwhq

Eh � ' 2 i TEn¡�BE �

� ; '� �]

� )+* - �ULt�Exn¦sv � 2�

E� n x � � � $�

dx


*L} ¦ �� ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� n � � ; � � �t � Ev; f; ¦ �� :

Li " � �t � Ev; f; �� lq +8117,; wkhq+l, V arEh � =x � �(' � ' 2 i TE¡ ��

BE �

� ; '� �U

� ) * - �ULt�Ex¦ �

sv � 2�E� n x � � � $

�dx

+ll, h{Eh � =x � �(' � > f iru dssursuldwho| fkrvhq v;zkhuh ¡ �� ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� lv wkh phdq ri *L} ¦ �� jlyhq wkh sdvw x � � � 1


¡ �� ' EE*L} ¦ �� =x � � � � ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( ��

dv E� � ' EE� � =x � � � � ' f; dqg

¦ �� ' V arE*L} ¦ �� =x � � � � ' V arE� � =x � � � � ' V arE� � � ' ¦ ��45<

dv fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� lv x � � � �phdvxudeoh1 Wkhq *L} ¦ �� t � Ev; ¡ �� ; ¦ �� dqg vriru dq| vhohfwhg lq U; *L} ¦ �� t � Ev; ¡ �� ; � ¦ �� jlyhq x � � � : Xvlqj ohppd :5 zh rewdlq

EEe �� "�# �� =x � � � � '2 i TE ¡ �� BE �

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sv � 2�E� n x � � � $

�dx +81148,

Khqfh zlwk ' �; zh jhw

V arEh � =x � �(' � ' E" �� EEe "�# ��

' 2 i TE¡ �� BE �

� ; '� �]

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sv � 2�E� n x � � � $

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+ll, Zlwk ' 2 lq +81148,/ EEEh � � ¡ � � � =x � �(' � ' E" �� EEi TE*L} ¦ �� ehfrphv

EEEh � � ¡ � � � =x � �(' � ' E" �� 2 i TE2¡ �� BE �

� ; '� �]

� ) * - �ULt�E2x¦ �

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dqg vrexEh � =x � � ' � ' BEv2 ;

�2��

U� ) * - �

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dx2EU � )+* - �

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dx� � � �

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BEv2 ;�2��

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sv � 2�E� n x � � � $

�dx > SE

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�dx� � +81149,

zkhuh � � '� � � � � � � � � lv wkh nxuwrvlv ri " � : Dv e � � � n x; ULt�E2x¦ �

sv � 2� < � nx¦ �

sv � 2 dqg ULt�Ex¦ �sv � 2� < � n x¦ �

sv � 22 : Khqfh wkh lqhtxdolw| +81149, fdq

eh zulwwhq � �

�v � e

� BE '� ;�

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sv � 2Ev � e�

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Dv � $�* ��

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fkrvhq dssursuldwho|1� � Qrwh wkdw U �

u +4 . u � , � du @ +c �� , � B+ � �

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463

Uhpdun :7 Wkh lqwhjudo]

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sv � 2�E� n x � � � $

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fdq eh hydoxdwhg iru hyhu| v > f: Ri frxuvh/ iru �qdqfldo gdwd prghoolqj v ' D vkrxogeh vx!flhqw1 Lwv ydoxh fdq eh rewdlq e| wkh uhvlgxh wkhruhp �� dv

]

� ) * - �ULt�Ex¦ �

sv � 2�E� n x � � � dx ' ¤

e d�¦ �s� t�?E¦ �

s�� n � ULtE¦ �s�� ¦ �� ULtE¦ �

s��o

Uhpdun :8 Dv wkh ydolglw| ri wkh lqhtxdolw| +8114:, ghshqgv rq v dqg ¦ � ; qrw rqo|vkrxog v eh dssursuldwho| fkrvhq exw wkdw ¦ � vkrxog qrw eh odujh1 Li wkh IQQ hvwlpdwhri ¦ � lv wrr odujh wkhq v kdv wr eh fkrvhq elj hqrxjk wr dffrxqw iru wkh odujh ydoxh ri¦ � : Wklv zloo hqvxuh wkdw +8114:, lv dozd|v wuxh1 Dowhuqdwlyho|/ rqh fdq gr glvfulplqdqwfrqglwlrqdo hvwlpdwhv ri ¦ � +vhh fkdswhu :,1

Gdwd dqdo|vlv zlwk frpprq glvwulexwlrqv Zh lqyhvwljdwh wkh shuirupdqfh ri wkhvhuhvxowv xqghu uhdo olih gdwd1 Lq sduwlfxodu/ zh zlvk wr lqyhvwljdwh krz zhoo wkh GOphwkrg uhvsrqg wr h{wuhph hyhqwv lq wkh gdwd dv wkh xqghuo|lqj glvwulexwlrq ydulhvdqg frpsduh lwv edfnwhvwlqj shuirupdqfh zlwk wkh JO dqg wkh IQQ fdvh1 Zkdw gr zhkrsh wr dfklhyhB Wr vhohfw wkh ehvw glvwulexwlrq wr lpsrvh rq " � vr dv wr rewdlq wkhehvw hvwlpdwh ri wkh yrodwlolw|1 Zh h{dplqh wkh hljkw Jhupdq vwrfnv glvfxvvhg suhylrxvo|xqghu wkh vdph frqglwlrqv zlwk wkh vdph h{rjhqhrxv yduldeohv1 Lq wkh IQQ fdvh wrr/ zhydu| wkh xqghuo|lqj glvwulexwlrq1 Wdeoh 8115 vxppdul}hv wkh uhvxowv rewdlqhg= wkh YdUh{fhhgdqfhv iru hdfk phwkrg zlwk ydu|lqj xqghuo|lqj glvwulexwlrq1 Wkh GO phwkrgzdv ehwwhu lq dw ohdvw �yh ri wkh vwrfnv h{fhsw wkh DOY dqg EDVI1 Lq wkh ED\ vwrfn/wkh GO xqghuhvwlpdwhg dw 8( dqg ryhuhvwlpdwhg dw 4( exw ehwwhu wkdq wkh IQQ fdvhdw 518( dqg 4(1 Zh revhuyh iurp wkh wdeoh wkdw t � qhyhu jdyh d ehvw hvwlpdwh iru dq| ri

�� Vhh dq| frpsoh{ dqdo|vlv wh{werrn1 Vhh iru h{dpsoh ^V97`1464

wkh vwrfn lq wkh IQQ fdvh1 Rq wkh rwkhu kdqg/ lq wkh GO dssurdfk/ wkh t � glvwulexwlrqzrxog eh idyrxuhg lq wkuhh ri wkh hljkw vwrfnv dqg vr lv wkh NIG � � glvwulexwlrq1 Ixuwkhulqyhvwljdwlrqv/ krzhyhu/ kdv wr eh pdgh rq wkh h�hfw ri YdU dv ¢ ulvhv dqg 7� fkdqjhv lqwkh fdvh ri wkh odwwhu glvwulexwlrq1 Lq wkh FEN vwrfn/ N dv wkh xqghuo|lqj glvwulexwlrqlv frq�uphg e| wkh GO phwkrg zkhuhdv NIG � � dv xqghuo|lqj lv frq�uphg lq OXGdqg YRZ1 Iljxuh 81146 +xsshu sorw, vkrzv wkh GO edvhg08( YdU hvwlpdwhv ri EDVIuhwxuqv E�� zlwk t � ED; f;

t �

�� dv xqghuo|lqj glvwulexwlrq +vrolg olqh, wrjhwkhu zlwk IQQ

edvhg08( YdU hvwlpdwh zlwk GLEf; �; i TE ¤s�� +gdvkhg olqh,1Wkh orzhu sorw ri �jxuh 81146 vkrzv wkh EDVI uhwxuqv1 Vlplodu judskv duh sorwwhg iruhdfk vwrfn1 Lq hdfk ri wkh judskv/ wkh glvwulexwlrq wkdw jdyh wkh ehvw YdU hvwlpdwh lqwkh IQQ phwkrg lv sorwwhg wrjhwkhu zlwk wkh GO hvwlpdwh1

465

Vwrfn IQQ edvhg GO edvhgXqghuo|lqj Ohyho ri �I YdU Xqghuo|lqj Ohyho ri �Iglvwulexwlrq f:fD f:f2D f:f� glvwulexwlrq f:fD f:f

DOY N 31377< 313585 3133<; GL 313837 313EDVI GL 3137;4 313585 31343< t � 3137:4 313ED\ NIG � � 3137;4 31364: 31354< t � 313749 313FEN N 313847 31354< 31343< N 3137:4 313K\S N 313858 313574 31343< NIG � � 3137;6 313OXG NIG � � 3137: 31353; 31343< NIG � � 313837 313WK\ GL 313836 313596 31343< t � 3137<6 313YRZ NIG � � 31377< 313574 3133<; NIG � � 3137<6 313

Wdeoh 8115= Frpsdulvrq ri IIQ dqg GO hvwlpdwhv ri YdU iru gl�huhqw Jhupdq vwrfnv

466

Lq hdfk ri wkhvh wzr dssurdfkhv wkh qxpehu ri qhxurqv xvhg lq wkh klgghq od|huglg qrw h{fhhg vl{ zlwk wkh wdq0vljprlg wudqvihu ixqfwlrq dv dfwlydwlrqv1 Iru hyhu|hvwlpdwh e| wkh IQQ dssurdfk zh fdq rewdlq d ehwwhu hvwlpdwh iru GO phwkrg irudw ohdvw rqh ri wkh xqghuo|lqj glvwulexwlrqv1 Rq wkh rwkhu kdqg/ wkh GO dssurdfkwdnhv dw ohdvw wzlfh dv pxfk wlph dv wkh IQQ phwkrg1 Ri frxuvh/ wkhuh duh rwkhu rshqtxhvwlrqv= frxog wkh hvwlpdwhv lpsuryh li zh shuplw vnhzqhvv lq wkh glvwulexwlrqdodvvxpswlrqvB Zkdw derxw pl{wxuhv ri wkh glvwulexwlrqvB Wkdw lv vxssrvh " � lq +8117, lvNIG � � glvwulexwhg dqg � � lq +8118, lv vd|GL glvwulexwhg1 Gr wkhvh pl{wxuh glvwulexwlrqvohdg wr lpsuryhg yrodwlolw| hvwlpdwhvB Frxog zh kdyh rewdlqhg d ehwwhu hvwlpdwh lq wkhfdvh ri wkh NIG � � li wkh sdudphwhu 7� lv lqfuhdvhg ru ghfuhdvhg voljkwo|B Zkdw derxwwkh glvwulexwlrqdo dvvxpswlrq +81144,B Dovr ixuwkhu lqyhvwljdwlrqv kdv wr eh fduulhg rxwwr vhh wkh h�hfw rq wkh YdU hvwlpdwhv dv wkh qxpehu ri ghjuhhv ri iuhhgrp lqfuhdvhviru wkh t � glvwulexwlrq1 Wkh shuirupdqfh ri GO dssurdfk zlwk lqfuhdvlqj ru ghfuhdvlqjgdwd vl}h dovr qhhgv vrph lqyhvwljdwlrqv1

Zh frqfoxgh wkdw iru erwk phwkrgv/ dqg dq| jlyhq �qdqfldo gdwd lw lpsruwdqw wrhvwlpdwh wkh yrodwlolwlhv xqghu gl�huhqw glvwulexwlrqdo dvvxpswlrqv dqg wkh glvwulexwlrqwkdw jlyhv wkh ehvw hvwlpdwh ri wkh YdU eh xvhg lq wkh dqdo|vlv1

819 Uhvxowv iru qhz glvwulexwlrqv zlwk srvvleoh dv|pphwu| lq wkh lqqrydwlrq

Lq wklv vhfwlrq zh lqyhvwljdwh wkh yrodwlolw| xqghu vrph qhz glvwulexwlrqv fdsdeohri prghoolqj wkh uhwxuqv ri dq| ulvn| dvvhw1

467

0 100 200 300 400 500 600 700 800 900

-0.15

-0.1

-0.05

0

0.05

0.1

0.15 BASF returnsDL estimateFFN estimate

100 200 300 400 500 600 700 800 900 1000

-0.1

-0.05

0

0.05

0.1Returns of BASF

Iljxuh 81147= Xsshu judsk= GO edvhg08( YdU hvwlpdwhv ri EDVI uhwxuqv E�� zlwkt � ED; f;

t �

�� dv xqghuo|lqj glvwulexwlrq +vrolg olqh,1 IIQ edvhg08( YdU hvwlpdwh zlwk

GLEf; �; i TE ¤s�� +gdvkhg olqh,1Orzhu judsk= EDVI uhwxuqv

468

0 100 200 300 400 500 600 700 800 900

-0.1

0

0.1

0.2

0.3

0.4BAY returnsDL estimateFFN estimate

100 200 300 400 500 600 700 800 900 1000

-0.1

0

0.1

0.2

0.3 Returns of BAY

Iljxuh 81148= Xsshu judsk= GO edvhg08( YdU hvwlpdwhv ri Ed|hu uhwxuqv E�� zlwkt � ED; f;

t �

�� dv xqghuo|lqj glvwulexwlrq +vrolg olqh,1 IIQ edvhg08( YdU hvwlpdwhv zlwk

NIG � � E2:D � ; f; f; �� dv xqghuo|lqj glvwulexwlrq +gdvkhg olqh,1Orzhu judsk= Ed|hu uhwxuqv

469

0 100 200 300 400 500 600 700 800 900

-0.2

-0.1

0

0.1

CBK returnsDL estimateFFN estimate

100 200 300 400 500 600 700 800 900 1000

-0.1

-0.05

0

0.05

0.1

0.15CBK returns

Iljxuh 81149= Xsshu judsk= GO edvhg08( YdU hvwlpdwhv ri Frpphu}edqn uhwxuqvE�� zlwk NEf; �� dv wkh xqghuo|lqj +vrolg olqh,1 IIQ edvhg08( YdU hvwlpdwhv zlwkNEf; �� dv wkh xqghuo|lqj glvwulexwlrq +gdvkhg olqh,1Orzhu judsk= Frpphu}edqn uhwxuqv

46:

0 100 200 300 400 500 600 700 800 900

-0.15

-0.1

-0.05

0

0.05

0.1

0.15 HYP returnsDL estimateFFN estimate

100 200 300 400 500 600 700 800 900 1000-0.15

-0.1

-0.05

0

0.05

0.1HYP returns

Iljxuh 8114:= Xsshu judsk= GO edvhg08( YdU hvwlpdwhv ri K|sryhuhlqvedqn uhwxuqvE�� zlwk NIG � � E2:D � ; f; f; �� dv wkh xqghuo|lqj glvwulexwlrq +vrolg olqh,1 IIQ edvhg08(YdU hvwlpdwhv NEf; �� dv wkh xqghuo|lqj glvwulexwlrq1Orzhu judsk= K|sryhuhlqvedqn uhwxuqv

46;

0 100 200 300 400 500 600 700 800 900

-0.15

-0.1

-0.05

0

0.05

0.1

0.15 LUD returnsDL estimateFFN estimate

100 200 300 400 500 600 700 800 900 1000-0.15

-0.1

-0.05

0

0.05

0.1

0.15Returns of LUD

Iljxuh 8114;= Xsshu judsk= GO edvhg08( YdU hvwlpdwhv ri Oxiwkdqvd uhwxuqv E�� zlwkNIG � � E2:D � ; f; f; �� dv wkh xqghuo|lqj glvwulexwlrq hvwlpdwhv +vrolg olqh,1 IIQ edvhg08(YdU hvwlpdwhv zlwk NIG � � E2:D � ; f; f; �� dv wkh xqghuo|lqj glvwulexwlrq hvwlpdwhv1Orzhu judsk= Oxiwkdqvd uhwxuqv1

46<

0 100 200 300 400 500 600 700 800 900-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15 THY returnsDL estimateFFN estimate

100 200 300 400 500 600 700 800 900 1000

-0.15

-0.1

-0.05

0

0.05

0.1

Returns of THY

Iljxuh 8114<= Xsshu judsk= GO edvhg08( YdU hvwlpdwhv ri Wk|vvhq Nuxss uhwxuqvE�� zlwkt � ED; f;

t �

�� dv xqghuo|lqj glvwulexwlrq +vrolg olqh,1 IIQ edvhg08( YdU hvwlpdwhv

zlwk GLEf; �; i TE ¤s�� dv xqghuo|lqj glvwulexwlrq +gdvkhg olqh,1Orzhu judsk= Wk|vvhq Nuxss uhwxuqv

473

0 100 200 300 400 500 600 700 800 900-0.1

-0.05

0

0.05

0.1

0.15VOW returnsDL estimateFFN estimate

100 200 300 400 500 600 700 800 900 1000

-0.05

0

0.05

0.1Returns of VOW

Iljxuh 81153= Xsshu judsk= GO edvhg08( YdU hvwlpdwhv ri Yronvzdjhq uhwxuqv E�� zlwkNIG � � E2:D � ; f; f; �� dv wkh xqghuo|lqj glvwulexwlrq +vrolg olqh,1 IIQ 08( YdU hvwlpdwhvzlwk NIG � � E2:D � ; f; f; �� dv wkh xqghuo|lqj glvwulexwlrq +gdvkhg olqh,1Orzhu judsk= Yronvzdjhq uhwxuqv1

474

Wkh JJO � dqg wkh JJO � glvwulexwlrqv Rqh dqrpdo| ri wkh GL glvwulexwlrq +dovr irudoo v|pphwulf glvwulexwlrq zlwk qr vnhzqhvv sdudphwhu, lv wkdw wkh glvwulexwlrq gr qrwjlyh surylvlrq iru srvvleoh dv|pphwu| lq wkh xqfrqglwlrqdo glvwulexwlrq ri uhwxuqv1 Zhfruuhfw wklv dqrpdo| e| lqwurgxflqj wkh e�sdudphwhu idplo| ri glvwulexwlrqv gh�qhgehorz=

Gh�qlwlrq :9 D udqgrp yduldeoh h kdv wkh GGL � glvwulexwlrq +lq vkruw h�GGL � E¡; ¢ � ; a; b�,li lwv ghqvlw| lv jlyhq e|

fEh� ' b *L} a¢a � � ��

d� n a � �� o � � ' ; �4 < h < 4; b > f

Lw kdv wkh GGL � glvwulexwlrq +lq vkruw h�GGL � E¡; ¢ � ; a; b�, li lwv ghqvlw| lv jlyhq e|

fEh� ' b *L} a¢a ��

d� n a �� o � � ' ; �4 < h < 4; b > f

Wkh sdudphwhuv ¡ 5 E�4;4�; ¢ � > f; a 5 U � dqg b > f duh uhvshfwlyho| wkh orfdwlrq/vfdoh/ vkdsh dqg wkh vnhzqhvv1

Wkh glvwulexwlrq ixqfwlrqv iru GGL � dqg GGL � duh jlyhq uhvshfwlyho| e|

F � Eh� ' d� n a � �� o � � ; F � Eh� ' �� d� n a

�� o � �

Revhuyh wkdw wkh GGL � glvwulexwlrq jhqhudol}hv wkh W|sh L jhqhudol}hg orjlvwlf glvwul0exwlrq ri ^E<5`/ ^EO;;` dqg iru b ' �; wkh glvwulexwlrq lv vlpso| wkh GL glvwulexwlrq1

Wkh GGL � glvwulexwlrq lv srvlwlyho| vnhzhg iru b 5 E�;4�; qhjdwlyho| vnhzhgiru b 5 Ef; �� dqg v|pphwulf zkhq b ' �: Rq wkh kdqg/ GGL � glvwulexwlrq lv qhjdwlyho|vnhzhg iru b 5 E�;4�; srvlwlyho| vnhzhg iru b 5 Ef; �� dqg v|pphwulf zkhq b ' � +vhh

475

�jxuh 81153,1 Udqgrp qxpehuv iurp wkh GGL � dqg wkh GGL � glvwulexwlrqv fdq ehrewdlqhg uhvshfwlyho| yld wkh uhodwlrq

¡� ¢*L} a *L}E �

�sUEf; ��

¡n ¢*L} a *L}E �

�s�� UEf; ��

zkhuh UEf; �� lv wkh xqlirup glvwulexwhg udqgrp yduldeohv rq wkh xqlw glvn1

Vrph vwdwlvwlfdo surshuwlhv

Sursrvlwlrq :: Ohw h�GGL � E¡; ¢ � ; a; b�: Wkhq iru n 5 Q; Eh � fdq eh zulwwhq dv�[� 0 )

� �r�¡ � � � E�� E ¢

*L} a�� J �

zkhuh J � ' U ') E*L}E '�� dm:

Wkh surri fdq eh ghgxfhg dv lq wkh GL fdvh1 D vlplodu uhvxow fdq eh ghgxfhgiru wkh GGL � :

Sursrvlwlrq :; Ohw h�GGL � E¡; ¢ � ; a; b�: Wkhq+l, Eh ' ¡ ¢

*L} a d«Eb�� «E��o+ll, V arEh� ' E ¢

*L} a� � d«� Eb� n « � E��o

+lll, �Eh� ' d« � � Eb�� « � � E��od« � Eb� n « � E��o �

�

+ly, h{Eh� ' « � � � Eb� n « � � � E��d« � Eb� n « � E��o � > f iru doo b

zkhuh «Eu�; « � Eu�; « � � Eu� dqg « � � � Eu� duh gljdppd/ wuljdppd/ whwudjdppd dqg shq0wdjdppd ixqfwlrqv uhvshfwlyho| �� 1

�� « �� +b, @ d � �

db � � +orj �+b,, zkhuh �+b, @ U � t � e � dt: Vhh ^DV:5`/ sjv 58805:4

476

0 200 400 600 800 1000 1200 1400 1600 1800 2000-2

-1

0

1

2

3

0 0.5 1 1.5 2 2.5 3 3.5 4-2

0

2

4

6

Excess of GGL+ and GGL-

Skewness of GGL+

Skewness of GGL-

Iljxuh 81154= Xsshu judsk= H{fhvv ri GGL � dqg GGL � +vrolg olqh,/ vnhzqhvv iruGGL � +gdvkhg olqh,/ vnhzqhvv iru GGL � +grwwhg olqh, iru b 5 Ef; eo:Orzhu judsk= H{fhvv ri GGL � dqg GGL � +vrolg olqh,/ vnhzqhvv iru GGL � +gdvkhgolqh,/ vnhzqhvv GGL � +grwwhg olqh, iru b 5 df; 2fffo:

Zh revhuyh wkdw wkh vnhzqhvv dqg wkh h{fhvv ghshqgv rqo| rq wkh vnhzqhvv sd0udphwhu b: Wkh surri zloo qrw eh vkrzq dv lw fdq hdvlo| eh ghgxfhg iurp wkhlu uhvshfwlyhfkdudfwhulvwlf ixqfwlrqv

' � Et� ' be 7 � � BEbn i ¢*L} at; �� i ¢

*L} at�; ' � Et� ' be 7 � � BEb� i ¢*L} at; � n i ¢

*L} at�

Dv fdq eh vhhq iurp wkh �jxuhv 81153 +xsshu judsk,/ wkh h{fhvv wkdw grplqdwhvwkh vnhzqhvv ghfuhdvhv vwhdglo| uhdfklqj lwv plqlpxp durxqg �:�e.H zkhq b ' �:22 dqgwkhq dwwdlqv lwv kljkhvw ydoxh dv b $ f: Zkhq b h{fhhgv �:22 Exsshu judsk,; lw lqfuhdvhv

477

vwhdglo| xqwlo durxqg 2:e Ezkhq b ' DHf � zkhuh lw grhv qrw lqfuhdvh ixuwkhu/ i:e*�4 ��

�� ) exEh� ' 2:e1 Rq wkh rwkhu kdqg/ lq wkh fdvh ri GGL � glvwulexwlrq/ vnhzqhvv

lv prqrwrqh lqfuhdvlqj dwwdlqlqj lwv rswlpxp ri �:� dw b ' �2: Iru wkh GGL � ; wkhvnhzqhvv lv prqrwrqh ghfuhdvlqj dwwdlqlqj lwv plqlpdo ri ��:� zkhq b ' �2:

Sursrvlwlrq :< Ohw h�GGL � E¡; ¢ � ; a; b� wkhq LE ' n � h� ' GGLE ' n � ¡; �� ¢ � ; a; b� iru ' ; � lq U:

Wkh sursrvlwlrq surfodlp wkdw wkh vkdsh dqg wkh vnhzqhvv uhpdlq xqfkdqjhgxqghu fkdqjhv lq orfdwlrq dqg vfdolqj1

Sursrvlwlrq ;3 +l,Ohw h�GGL � E¡; ¢ � ; a; b� dqg vxssrvh V arEh� ' ¦ � dqg Eh 'f; wkhq wkhuh h{lvwv xqltxh �a 5 U � vxfk wkdw h�GGL � E¦ � ' � � � � �s �� ' � ; ¦ � ; �a; b�:+ll,Ohw h�GGL � E¡; ¢ � ; a; b� dqg vxssrvh V arEh� ' ¦ � dqg Eh ' f; wkhq wkhuh h{lvwvxqltxh �a 5 U � vxfk wkdw h�GGL � E¦ � � � � � ' �s � � � � � � � ' � ; ¦ � ; �a; b�:

Surri1 +l, h�GGLE¡; ¢ � ; a; b� lpsolhv wkdw

V arEh� ' ¢*L} a d«

� Eb� n « � E��o ' ¦ �

e| dvvxpswlrq1 Wkxv a ' i TE �� s« � Eb� n « � E�� dqg wkh ghqvlw|/ fEh� ri h ehfrphv

b¦t« � Eb� n « � E�� i Td�s« � Eb� n « � E��E � � � �� o

d� n i TE�s« � Eb� n « � E��E � � � �� o � � 'Wkhuhiruh h�GGLE¡; ¦ � ; i TEs« � Eb� n « � E��; b�: Qrz Eh ' f lpso| wkdw ¡ ' ��E«E��«Eb�� / �� ' ¦s« � Eb� n « � E�� dqg vr h�GGLE��E«E��«Eb��; ¦ � ; �a; b�; �a ' i TEs« � Eb� n « � E��1

+ll, Vlplodu wr +l,1478

Uhpdun ;4 Wr xvh wkh GGL � glvwulexwlrqv iru wkh JDUFK idplo| prghov dvvxphwkh lqqrydwlrqv �� /

" � �SGGL � EE«Eb�� «E��s« � Eb� n « � E�� ; �; �a; b�:

Gh�qlwlrq ;5 D udqgrp yduldeoh h lv *L}GGL � +lq vkruw *L}GGL � E¡; ¢ � ; a; b�, li

*L} h�GGL � E¡; ¢ � ; a; b�:

Ohppd ;6 h� *L}GGL � E¡; ¢ � ; a; b� / wkhq Eh � ' be �� BEbn ¢*L} a; � n ¢

*L} a�;¢

*L} a 5E� b

n;�n� iru GGL � dqg ¢

*L} a 5 E��n;

bn� iru GGL �

Surri1 Iroorzv iurp wkhruhp 98 dqg wkh fkdudfwhulvwlf ixqfwlrq ri GGL � dqgGGL � glvwulexwlrqv1

Fruuhvsrqglqj wr wkhruhp 8; zh kdyh wkh iroorzlqj uhvxow iru wkh GGL � fdvh=

Wkhruhp ;7 Ohw Dvvxpswlrq 89 krog dqg vxssrvh wkdw

*L} ¦ �� ' fE*L} ¦ �� ' ; : : : ; *L} ¦ �� ( �� n � � ; � � �GGL � E¦ � E«E�� «Eb��s« � Eb� n « � E�� ; ¦ �� ; �a; b�

Li " � �SGGL � E � � ' � � � � �s �� ' � ; �; �a; b�; �a ' i TEs« � Eb� n « � E�� wkhq+l, V arEh � =x � �(' � ' b i TE¡ �� n � � ' � � � � � �s �� ' � ¦

� �BEb �� "�#�� ; � ��

"�#��

+ll, �Eh � =x � �(' � ' � � sKEb�� ¦ �

2 *L} �a �� ' �

�¦ �

2 *L} �a �� ¦ �

*L} �a �� ' �

¦ �

*L} �a ��

+lll, h{+h � =x � �(' � > fzkhuh ¡ �� ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� lv wkh phdq ri *L} ¦ �� jlyhq wkh sdvw x � � � 1

�� Wkh GGL zlwk phdq 3 dqg yduldqfh 4 zrxog eh fdoohg vwdqgdug GGL ghqrwhgSGGL + � �

��

� � �s�� ; 4; �a; b,:

479


¡ �� ' EE*L} ¦ �� =x � � � � ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( ��

dv E� � ' EE� � =x � � � � ' f; dqg


dv fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� lv x � � � �phdvxudeoh1 Khqfh

*L} ¦ �� GGL � E¡ �� n ¦ � E«E�� «Eb��s« � Eb� n « � E�� ; ¦ �� ; �a; b�

vr iru dq| 5 U

*L} ¦ �� GGL � d E¡ �� n ¦ � E«E�� «Eb��s« � Eb� n « � E�� ; � ¦ �� ; �a; bo

jlyhq x � � � dv wkh sdudphwhuv �a dqg b duh lqyduldqw xqghu vfdolqj +sursrvlwlrq :<,1 Lwiroorzv iurp ohppd ;6 wkdw

EEi TE *L} ¦ �� =x � � � � ' b i Td E¡ �� n¦ � E«E�� «Eb��s« � Eb� n « � E�� oBEb ¦ �

*L} �a; � ¦ �

*L} �a� +8114;,

Qrz zlwk ' � lq +8114;, zh kdyh

V arEh � =x � �(' � ' E" �� EEi TE*L} ¦ ��

' bE" �� e �� $�� $� � BEb ¦ �

*L} �a; � ¦ �

*L} �a�

+ll, Qrz

�Eh � =x � �(' � 'EEEh � � ¡ � �

� =x � �(' �EV arEh � =x � �(' ��

' � � EEi TE� *L} ¦ � �dEEi TE*L} ¦ �� o

��

47:

zkhuh � � lv wkh frh!flhqw ri vnhzqhvv ri " � : Fkrrvlqj ' �

� lq +8114;, zh kdyh

EEEi TE� *L} ¦ � �� ' b i Td�2E¡ �� n ¦ � E«E�� «Eb��s« � Eb� n « � E�� oBEb �¦ �

2 *L} �a; � �¦ �

2 *L} �a�

dqg vr

�Eh � =x � �(' � '� �sb

BEb � ��

� "�# � ; ��

� "�#�� dBEb ��

"�#�� ; � �� "�#�� o �� ' � � sKEb� KEb

� ��

� "�#�� KE��

� "�#�� dKEb ��

"�#�� KE� �� "�# � �o ��

+lll, EEEh � � ¡ � � � =x � �(' � ' E" �� EEi TE*L} ¦ �� dqg ' 2 lq +8114;, zh rewdlq

EEEh � � ¡ � � � =x � �(' � ' bE" �� e � �� $� � � � � �� $� � BEb 2¦ �

*L} �a; � 2¦ �

*L} �a�:

Wkh exEh � =x � �(' � 'EEEh � � ¡� � =x � �(' �EV arEh � =x � �(' ��

�� ' �b

E" ��EE" ��

BEb � � � "�#�� ; � � � � "�#�� EBEb ��

"�#�� ; � �� "�#�� : Zlwk

� � ' E" ��EE" �� ' � n « � � � Eb� n « � � � E��

d« � Eb� n « � E��o �

zh zloo �uvw vkrz wkh uhvxow iru wkh GGL � fdvh1Iru wkh GGL � zh pxvw vkrz wkdw iru � b

2 < ¦ �

*L} �a < �2

� � BEbn 2¦ �

*L} �a; �� 2¦ �

*L} �a� > �bEBEbn ¦ �

*L} �a; �� ¦ �

*L} �a�� +8114<,

iru hyhu| b < 4: Fkrrvh � > f vr vpdoo wkdw ¦ �

*L} �a � �2��: Wkhq +8114<, fdq eh zulwwhq

� � BE� n b� 2�; 2�� > �bEBE�2 n b� 2�; �2 n ��

Qrz BE� n b� 2�; 2�� * KE2�� $ n4 dv � $ f: Dovr dv � $ f; BE� n b� 2�; '� n �� $BE�2 n b; '� � zklfk d frqvwdqw iru hyhu| b: Wklv �qlvkhv wkh surri iru wkh GGL � fdvh1

Iru wkh GGL � fdvh zh pxvw vkrz wkdw iru ��2 < ¦ �

*L} �a < b2

� � BEb� 2¦ �

*L} �a; � n2¦ �

*L} �a� > �bEBEb� ¦ �

*L} �a; � n¦ �

*L} �a�� +81153,47;

iru hyhu| b < 4: Djdlq fkrrvh � > f vr vpdoo wkdw ¦ �

*L} �a � b2 � �: Wkhq wkh lqhtxdolw|

+81153, fdq eh zulwwhq

� � BE2�; � n b� 2�; � > �bEBE b2 n �; � n b2 � ��

Dv BE� n b � 2�; 2�� * KE2�� $ n4 dv � $ f dqg BE �� n �; � n �

� � �; � $ BE �� ; � n�

� �; frqvwdqw iru hyhu| b dqg vr wkh uhvxow lv wuxh iru wkh GGL � fdvh wrr1

Wkh Sdw � ; Sdw � dqg wkh Sdw glvwulexwlrqv

Gh�qlwlrq ;8 D udqgrp yduldeoh h lv Pat � 0glvwulexwhg zlwk sdudphwhuv ¡; �; v dqg� Eiru vkruw h�Pat � E¡; �; v; �� li lwv ghqvlw| lv jlyhq e| htxdwlrq +81144,1 h�Pat � E¡; �; v;�� li h kdv ghqvlw|

f � Eh� ' e � �� ¢KE�� i TE�e � ��

� �; h 5 E�4;4�; � > f; � > f

¡ dqg � duh orfdwlrq sdudphwhuv zkhuhdv v dqg � duh uhvshfwlyho| wkh vfdoh dqgwkh vnhzqhvv sdudphwhuv1 Wkh glvwulexwlrq ixqfwlrqv duh uhvshfwlyho| jlyhq e|

F � Eh� ' �� K 7 � � Ei TE� � �2��

� ; �� dqg F � Eh� ' K 7 � � Ei TE� � �� ; ��

zkhuh K 7 � � lv wkh lqfrpsohwh jdppd ixqfwlrq �� 1 Wkh glvwulexwlrqv duh xqltxho| ghwhu0plqhg e|

' � Et� ' e � 7 � �� % "�# � �KE�� KE�� itv� dqg ' � Et� '

e 7 � � � � "�# � �KE�� KE� n itv�

Sursrvlwlrq ;9 Ohw h�Pat � E¡; �; v; ��: Wkhq iru ' ; � lq U;

LE ' n � h� ' Pat � E� ' n � ¡; �; � v; ��:�� +�x; �, @

�� U ��

e � t � � dt47<

Sursrvlwlrq ;: Ohw h�Pat � E¡; �; v; ��: Wkhq iru ' ; � lq U;

LE ' n � h� ' Pat � E ' n � ¡; �; � v; ��:

Wkh sdudphwhuv � dqg � lv lqyduldqw xqghu fkdqjhv lq orfdwlrq dqg vfdolqj1Krzhyhu/ pdun wkh qhjdwlrq iru ' lq wkh Pat � fdvh$

Sursrvlwlrq ;; Ohw h�Pat � E¡; �; v; ��: Wkhq wkh iroorzlqj krog=+l, Eh ' �d¡n vE«E�� n *L} ��o+ll, V arEh� ' v � « � E��+lll, �Eh� ' � « � � E��

d« � E��o �� > f iru doo �:

+ly, h{Eh� ' « � � � E��d« � E��o � > f iru doo �

zkhuh «; « � ; « � � dqg « � � � duh uhvshfwlyho| wkh gljdppd/ wuljdppd/ whwudjdppd dqgshqwdjdppd ixqfwlrqv1

Uhpdun ;< Doo hyhq prphqw derxw R ri wkh Pat � dqg wkh Pat � glvwulexwlrqv duh wkhvdph1 Doo rgg prphqwv derxw R ri wkh Pat � duh qhjdwlyh ri wkh wkdw ri wkh Pat � : Lq frq0vhtxhqfh/ wkh deryh sursrvlwlrq dovr krogv iru wkh Pat � glvwulexwlrq h{fhsw wkh Eh dqg�Eh� gl�hu lq vljq1

Fohduo|/ wkh Pat � dqg wkh Pat � glvwulexwlrqv duh uhvshfwlyho| srvwlyho| dqgqhjdwlyho| vnhzhg dqg wkh h{fhvv lv dozd|v srvlwlyh +vhh �jxuh 81154, � , 1

Li V arEh� ' ¦ � dqg wkh Eh ' f; wkhq h�Pat � E � � � �s �� ; �;¦s« � E�� ; �� dqg lv

vwdqgdug Pat � glvwulexwhg zkhq ¦ ' �: Wkh Vwdqgdug Pat � glvwulexwlrq zloo eh ghqrwhge| SPat � E � � �s �� ; �;

�s« � E�� ; ��: Vlploduo|/ wkh li V arEh� ' ¦ � dqg wkh Eh ' f; wkhq��8 Vhh dovr kwws=22pdwkzruog1zroiudp1frp2Sro|jdppdIxqfwlrq1kwpo

483

0 500 1000 1500 2000 2500 3000 3500 4000-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6Excess for Pat+ and Pat.-

Skewness for Pat.-

Skewness for Pat.+

0 0.5 1 1.5 2 2.5 3 3.5 4-2

0

2

4

6

Excess for Pat.+ and Pat.

-

Skewness for Pat.-

Skewness for Pat.+

Iljxuh 81155= Xsshu judsk= H{fhvv iru Pat � dqg Pat � +vrolg olqh,/ vnhzqhvv iruPat � +grwwhg olqh, dqg wkh vnhzqhvv iru Pat � +gdvkhg olqh, iru � 5 de; efffo:Orzhu judsk= H{fhvv iru Pat � dqg Pat � +vrolg olqh,/ vnhzqhvv iru Pat � +grwwhg olqh,dqg wkh vnhzqhvv iru Pat � +gdvkhg olqh, iru � 5 Ef; eo:

484

h�Pat � E� � � � �s � � � � ; �;¦s« � E�� ; �� dqg wkh vwdqgdug Pat � glvwulexwlrq zloo eh ghqrwhg

e| SPat � E� � � �s �� ; �;�s« � E�� ; ��: Wr xvh wkh glvwulexwlrqv iru wkh JDUFK idplo|

prghov/ � fdq eh �{hg lq dgydqfh exw kdv wr eh fkrvhq dssursuldwho| dqg dvvxphwkh lqqrydwlrqv/ " � kdv hlwkhu wkh SPat � ru SPat � glvwulexwlrq1 Frpsduhg wr wkhvwdqgdug qrupdo glvwulexwlrq/ wkh SPat � dqg SPat � duh rqo| voljkwo| vnhzhg zkhq� lv prghudwho| odujh1

Fruuhvsrqglqj wr wkhruhp 93 zh kdyh wkh iroorzlqj uhvxow iru wkh SPat � dqgwkh SPat � glvwulexwlrqv1

Wkhruhp <3 Ohw Dvvxpswlrq 89 krog dqg vxssrvh wkdw

*L} ¦ �� ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� n � � ; � � �Pat � E ¦ � «E��s« � E�� ; �;¦ �

s« � E�� ; ��:

Li " � �SPat � E «E��s« � E�� ; �;�s« � E�� ; ��; wkhq iru � > ¦ �

s« � E�� ;+l, V arEh � =x � �(' � ' �

KE�� i TE¡ �� n� � � � �s � � � � �KE�� ¦ �

s« � E��

+ll, �Eh � =x � �(' � '� � sKE��

dKE�� ¦ �s« � E��o

��

KE�� ¦ �

2s« � E��

+lll, h{Eh � =x � �(' � > f iru dssursuldwho| fkrvhq �;zkhuh ¡ �� ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� lv wkh phdq ri *L} ¦ �� jlyhq wkh sdvw x � � � 1


¡ �� ' EE*L} ¦ �� =x � � � � ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( ��

dv E� � ' EE� � =x � � � � ' f; dqg

¦ �� ' V arE*L} ¦ �� =x � � � � ' V arE� � =x � � � � ' V arE� � � ' ¦ ��485

dv fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� lv x � � � �phdvxudeoh1 Wkhq

*L} ¦ �� SPatE�¡ �� ¦ � «E��s« � E�� ; �;¦ �

s« � E�� ; ��

dqg iru 5 U;

*L} ¦ �� SPatE E�¡ �� ¦ � «E��s« � E��; �; ¦ �

s« � E�� ; ��

jlyhq wkh sdvw x � � � dv wkh orfdwlrq sdudphwhu � dqg wkh vnhzqhvv sdudphwhu � lvlqyduldqw xqghu vfdolqj1 Khqfh

EEi TE *L} ¦ �� =x � � � � '�

KE�� i TE� E�¡ �� ¦ � «E��s« � E��KE�� ¦ �s« � E��: +81154,

Wkhuhiruh iru ' �; zh kdyh

V arEh � =x � �(' � ' EEi TE*L} ¦ ��

' �KE�� i TE¡ �� n ¦ � «E��s« � E��KE�� ¦ �

s« � E��

+ll, Vlplodu wr wkh surri wkhruhp ;7 +ll,1+lll, EEEh � �¡ � � � =x � � ' � ' E" �� EEi TE*L} ¦ �� dqg zlwk ' 2 lq +81154, zh kdyh

EEEh � � ¡ � � � =x � �(' � ' E" ��i TE2E¡ �� n

�� s � � � � ��KE�� KE�� 2¦ �

s« � E��

Khqfh

exEh � =x � �(' � ' � � KE��KE�� 2¦ �

s« � E��EKE�� s ��

� �

Lw vx!fhv wr vkrz wkdw iru � > 2¦ �s« � E��

� � KE�� 2¦ �s« � E�� > �EKE�� ¦ �

s« � E�� :486

Fkrrvh � > f vr vpdoo wkdw ¦ �s« � E��

2 � �: Wkhq zh fdq fkrrvh � vxfk wkdw deryhlqhtxdolw| fdq eh zulwwhq

� � KE2�� > �EKE�2 n ��

Ohwwlqj � $ f |lhogv wkh ghvluhg uhvxow1


*L} ¦ �� ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� n � � ; � � �Pat � E� ¦ � «E��s« � E�� ; �;¦ �

s« � E�� ; ��:

Li " � �SPat � E� «E��s« � E�� ; �;�s« � E�� ; ��; wkhq

+l, V arEh � =x � �(' � '�

KE�� i TE¡ �� s � � � � �KE�n ¦ �s« � E��

+ll, �Eh � =x � �(' � '� � sKE��

dKE� n ¦ �s« � E��o

��

KE� n �¦ �

2s« � E��



¡ �� ' EE*L} ¦ �� =x � � � � ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( ��

dv E� � ' EE� � =x � � � � ' f; dqg



*L} ¦ �� SPat � E¡ �� ¦ � «E��s« � E�� ; �;¦ �

s« � E�� ; ��487

dqg iru 5 U;

*L} ¦ �� SPat � E E¡ �� ¦ � «E��s« � E��; �; ¦ �

s« � E�� ; ��

jlyhq wkh sdvw x � � � dv wkh orfdwlrq sdudphwhu � dqg wkh vnhzqhvv sdudphwhu � lvlqyduldqw xqghu vfdolqj1 Khqfh +vhh wkhruhp 98,

EEi TE *L} ¦ �� =x � � � � '�

KE�� i TE E¡ �� ¦ � «E��s« � E��KE� n ¦ �s« � E��: +81155,

Wkhuhiruh iru ' �; zh kdyh

V arEh � =x � �(' � ' EEi TE*L} ¦ ��

' �KE�� i TE¡ �� ¦ � «E��s« � E��KE� n ¦ �

s« � E��

+ll, Vlplodu wr wkh surri wkhruhp ;7 +ll,1+lll, EEEh � �¡ � � � =x � � ' � ' E" �� EEi TE*L} ¦ �� dqg zlwk ' 2 lq +81155, zh kdyh

EEEh � � ¡ � � � =x � �(' � ' E" ��i TE2E¡ �� s ��

KE�� KE� n 2¦ �s« � E��

Khqfh

exEh � =x � �(' � ' � � KE��KE� n 2¦ �

s« � E��EKE�n � �s � � � � ��

� �

Lw vx!fhv wr vkrz wkdw iru

� � KE� n 2¦ �s« � E�� > �EKE� n ¦ �

s« � E�� :

Fkrrvh � > f vr vpdoo vxfk wkdw ¦ �s« � E��

2 n �: Wkhq zh fdq fkrrvh � vxfk wkdwwkh deryh lqhtxdolw| fdq eh zulwwhq

� � KE2�� > �EKE�2 n ��

Ohwwlqj � $ f |lhogv wkh ghvluhg uhvxow1488

Gh�qlwlrq <5 Zh vd| wkdw h�PatE¡; �; v; �� li

h��2Pat � E¡; �; v; �� n �2Pat � E¡; �; v; ��:

Wkh PatE¡; �; v; �� glvwulexwlrq lv v|pphwulf dqg kdv doo rgg prphqwv }hur1Rwkhu lqwhuhvwlqj surshuwlhv fdq hdvlo| eh ghgxfhg1 Li V arEh� ' ¦ � dqg Eh ' f; wkhqh lv Pat glvwulexwhg li

h��2Pat � E ¦«E��s« � E�� ; �;¦s« � E�� ; �� n

�2Pat � E� ¦«E��s« � E�� ; �;

¦s« � E�� ; ��

dqg lv vwdqgdug Sdw glvwulexwhg zkhq ¦ ' �: Wkh Vwdqgdug Pat glvwulexwlrq zloo ehghqrwhg e| SPatE � � �s �� ; �;

�s« � E�� ; ��:Zh kdyh wkh iroorzlqj uhvxowv iru wkh Pat glvwulexwlrq=


*L} ¦ �� ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� n � � ; � � �PatE ¦ � «E��s« � E�� ; �;¦ �

s« � E�� ; ��

Li " � �SPatE «E��s« � E�� ; �;�s« � E�� ; ��; wkhq iru � > ¦ �

s« � E�� ;+l, V arEh � =x � �(' � '

�2KE��

%i TE¡ �� n ¦ � «E��s« � E��KE�� ¦ �

s« � E�� n i TE¡ �� ¦ � «E��s« � E��KE� n ¦ �s« � E��

&

+ll, h{Eh � =x � �(' � > fzkhuh ¡ �� ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( �� lv wkh phdq ri *L} ¦ �� jlyhq wkh sdvw x � � � 1

Surri1 +l, Zh �uvw revhuyh wkdw li h�SPatE �� s �� ; �;¦ �

s« � E�� ; �� wkhq

h��2Pat � E ¦«E��s« � E�� ; �;¦s« � E�� ; �� n

�2Pat � E� ¦«E��s« � E�� ; �;

¦s« � E�� ; ��489

glvwulexwhg1 Ohw ¡ �� dqg ¦ �� eh uhvshfwlyho| wkh h{shfwdwlrq dqg wkh yduldqfh ri *L} ¦ �� jlyhqwkh sdvw/ x � � � : Wkhq

¡ �� ' EE*L} ¦ �� =x � � � � ' fE*L} ¦ �� (' ; : : : ; *L} ¦ �� ( ��

dv E� � ' EE� � =x � � � � ' f; dqg



*L} ¦ �� 2SPat � E�¡ �� ¦ � «E��s« � E�� ; �;¦ �

s« � E�� ; �� n�2Pat � E¡ �� ¦ � «E��s« � E�� ; �;

¦ �s« � E�� ; ��

dqg iru 5 U;

*L} ¦ �� 2PatE E�¡ �� ¦ � «E��s« � E��; �; ¦ �

s« � E�� ; ��n�2Pat � E E¡ �� ¦ � «E��s« � E��; �;

¦ �s« � E�� ; ��:

jlyhq wkh sdvw/ x � � � : Khqfh

EEi TE *L} ¦ �� =x � � � � ' �2KE��

%

i TE� E�¡ �� ¦ � «E��s« � E��KE�� ¦ �s« � E��

n i TE E¡ �� ¦ � «E��s« � E��KE� n ¦ �s« � E��

&

+81156,

Zlwk ' �; zh jhw

V arEh � =x � �(' � ' �2KE��

%i TE¡ �� n ¦ � «E��s« � E��KE�� ¦ �

s« � E��

n i TE¡ �� ¦ � «E��s« � E��KE� n ¦ �s« � E��

&

+lll, EEEh � � ¡ � � � =x � �(' � ' E" �� EEi TE*L} ¦ �� dqg zlwk ' 2 zh kdyh +81156,

EEEh � � ¡� � =x � �(' � ' E" ��2KE��

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^E<8`/ qrupdo ru Vwxghqwzv w lqqrydwlrqv li wkh gdwd zdv uhdoo| jhqhudwhg e| JDUFK+4/4,prgho1 Iru wklv uhdvrq zh dsso| wkhruhp << wr vlpxodwhg gdwd ri vl}h n ' 833 uhsuhvhqw0lqj dssur{lpdwho| 2 |hduv wudglqj gd|v iurp wkh JDUFK+4/4, surfhvv= h � ' ¦ � " � zkhuh¦ �� ' !n �h �� (' n �¦ �� (' zlwk gl�huhqw sdudphwhu fkrlfhv iru !; � dqg � exw shuvlvwhqwlq yrodwlolw| rqo|1 Zh �uvw dvvxph wkdw " � �GLEf; �; i TE 6� � �� dqg fdofxodwh wkh dyhudjhphdq vtxduh huuru/ +DPVH,

AMSE ' �n

�[7 0 '

dV̂ aR�

�� Ei�� V̂ aR�

�� Ei�o �

ri wkh JDUFK0ulvn iru hdfk glvwulexwlrq zkhuh V̂ aR�

�� ' ¦ �� q �� ; V̂ aR�

�� ' ¦ �

� q �� 1 Qrwhwkdw wkh q �� lv ��txdqwloh ri wkh uhvshfwlyh glvwulexwlrq/ ¦ �� lv wkh wuxh xqghuo|lqj yrodwlo0lw| dqg ¦ �

� lv wkh hvwlpdwhg yrodwlolw|1 Qh{w zh dvvxph/ " � �NIG � � E�:2D � ; f; f; �� dqg fdo0fxodwh DPVH ri wkh JDUFK0ulvn dqg wkh uhvxowv wdexodwhg1 Wkh surfhgxuh lv uhshdwhgiru wkh qrupdo dqg wkh Vwxghqwzv t glvwulexwlrqv dqg wkh uhvxowv wdexodwhg +vhh wdeoh914,1 Wkh wdeoh vkrzv wkh pdjql�hg DPVH +pdj1 DPVH, � iru wkh YdU hvwlpdwhv dw8(/ 518( dqg 4( zkhq wkh wuxh xqghuo|lqj glvwulexwlrq zhuh GL dqg wkh fruuhvsrqglqjhvwlpdwhv e| wkhNIG � � ; N dqg wkh t � : Iljxuh 9114 vkrzv d w|slfdo JDUFK+4/4, surfhvvzlwk ydu|lqj lqqrydwlrqv1 Dv fdq eh vhhq iurp wkh �jxuh 9114 +wrs uljkw, dvvxplqj dNIG � � uhvxowhg lq dq h{wuhpho| kljk ydoxhv iru wkh uhwxuqv1 Wkh fruuhvsrqglqj pdj1DPVH iru wkh YdU fdofxodwlrqv zdv htxdoo| yhu| kljk +vhh wdeoh 915,1 Uhvxowv iru wkhfdvh zkhuh wkh lqqrydwlrqv duh qrupdo dqg wkh vwxghqwzv t lv uhohjdwhg wr wkh dsshqgl{wr wkh fkdswhu +wdeohv G114 dqg G115,1

� DPVH lv pdjql�hg n wlphv49:

0 200 400 600 800 1000-3

-2

-1

0

1

2

3datavolatility

0 200 400 600 800 1000-1

-0.5

0

0.5

1

1.5

2x 10

28

datavolatility

0 200 400 600 800 1000-3

-2

-1

0

1

2

datavolatility

0 200 400 600 800 1000-3

-2

-1

0

1

2

3

4

datavolatility

Iljxuh 9114= W|slfdo JDUFK+4/4, surfhvv f:f� n f:�h �� (' n f:SD¦ �� ' zlwkGLEf; �; i TE 6

� � �� +xsshu ohiw0kdqg fruqhu,/ NIG � � E�:2D � ; f; f; �� +xsshu uljkw0kdqg fru0qhu,/ NEf; �� +orzhu ohiw0kdqg fruqhu, dqg t � ED; f; �� +orzhu uljkw0kdqg fruqhu, lqqryd0wlrqv dqg wkh fruuhvsrqglqj yrodwlolwhv

49;

0 100 200 300 400 500 600 700 800 900 1000-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

dataGLNIGσ

2

Nts

0 100 200 300 400 500 600 700 800 900 10000

0.005

0.01

0.015

0.02

0.025

0.03GLNIGσ

2

Nts

Iljxuh 9115= Xsshu sorw= Gdwd srlqwv + �, dqg wkh fruuhvsrqglqj DI YdU hvwlpdwhv riwkh JDUFK surfhvv ¦ �� ' f:f�n f:�h �� (' nf:SD¦ �� (' zlwk GL lqqrydwlrqv1 GL hvwlpdwh+vrolg olqh,/ NIG � � hvwlpdwh +gdvkhg olqh,/ qrupdo hvwlpdwh +grwwhg olqh, dqg t �

hvwlpdwh +gdvk0grw olqh,1Orzhu sorw= pdj1 DPVH ri YdU hvwlpdwhv1 GL hvwlpdwh +vrolg olqh,/ NIG � � hvwlpdwh+gdvkhg olqh,/ qrupdo hvwlpdwh +grwwhg olqh, dqg t � hvwlpdwh +gdvkhg0grw olqh,1

49<

Glvwulexwlrq Sdudphwhuv pdj1 DPVH dw ohyho ��I YdU! � � f:fD f:f2D f:f�

GLEf; �; i TE 6� � �� 3134 316 3198 513:;7 6154:: 813954

3134 314 31;8 619;:4 81:3;3 ;1<:<;

NIG � � E�:2D � ; f; f; �� 3134 316 3198 517385 61;878 9164<43134 314 31;8 61;6:; 914837 4313;63

NEf; �� 3134 316 3198 419;:; 516<95 616:933134 314 31;8 61938< 8144<4 :15459

t � ED; f; �� 3134 316 3198 61356< 71<53< ;176533134 314 31;8 7145:: 91:4:5 4418433

Wdeoh 9114= 833 vlpxodwhg gdwd iurp JDUFK+4/4, zlwk JO lqqrydwlrqv

4:3

Glvwulexwlrq Sdudphwhuv pdj1 DPVH dw ohyho ��I YdU! � � f:fD f:f2D f:f�

GLEf; �; i TE 6� � �� 3134 316 3198 519768h.363 713<58h.363 9176;7h.3633134 314 31;8 :15:94h.33; 414597h.33< 41::54h.33<

NIGE�:2D � ; f; f; �� 3134 316 3198 5163:3h.35< 619<:5h.35< 913945h.35<3134 314 31;8 51<748h.33; 71:46<h.33; :1:5;4h.33;

NEf; �� 3134 316 3198 61:47:h.35< 815:68h.35< :17634h.35<3134 314 31;8 613<77h.33; 716<63h.33; 914;<8h.33;

t � ED; f; �� 3134 316 3198 51999<h.363 716733h.363 :17699h.3633134 314 31;8 918;6;h.33; 413:47h.33< 41;68<h.33<

Wdeoh 9115= 833 vlpxodwhg gdwd iurp JDUFK+4/4, zlwk QLJ lqqrydwlrqv

4:4

Wkh frpshwlwlrq zdv uhodwlyho| nhhq dv fdq eh vhhq iurp wkh xsshu fxuyh ri�jxuh 9115 zkhuh doo wkh glvwulexwlrqv uhvsrqghg txlwh zhoo wr mxpsv lq wkh gdwd1 Dsorw ri wkh fruuhvsrqglqj pdj1 DPVH iru JDUFK0YDU fdofxodwlrq uhyhdo wkdw wkhGL grhv ehwwhu iroorzhg e| wkh qrupdo/ wkh NIG � � dqg wkh t � glvwulexwlrqv1

Vwdwlrqdulw| frqglwlrq iru JDUFK+4/4, zlwk QLJ � � lqqrydwlrqv D JDUFK+4/4, vh0txhqfh kdv d xqltxh vwdwlrqdu| vroxwlrq li/ dqg rqo| li

E *L}E�" �� n �� < f

^Q<3`1 Wr fkhfn wkh vwdwlrqdulw| frqglwlrq iru JDUFK+4/4, zlwkNIG � � lqqrydwlrqv/ zhlqyhvwljdwh wkh ehkdylru ri E *L}E�" �� n�� iru ydu|lqj �; � zkhq " � �NIG � � glvwulexwhg1Iru m odujh hqrxjk/

�m

�[

� 0 '*L}E�" �� n �� * E *L}E�" �� n ��

Dv fdq eh vhhq iurp �jxuh 9116/ vwdwlrqdulw| ri wkh JDUFK+4/4, surfhvv zlwkNIG � � lqqrydwlrqvghshqg rq wkh fkrlfh ri wkh sdudphwhuv � dqg �: Vwdwlrqdulw| lv suhvhuyhg iru ydoxhv ri� forvh wr xqlw| dqg ydoxhv ri � forvh wr }hur1 Vwdwlrqdulw| dovr ghshqg rq wkh fkrlfhri sdudphwhuv iru wkh NIG � � glvwulexwlrq1

Lq wkh qh{w vwxg|/ zh jhqhudwh h � �' ; : : : ; h � ��)�) ; m ' �; : : : ; �fff ' M dqg

fdofxodwh wkh DPVH iru hdfk sdwk1 Wkhq wkh h{shfwhg dyhudjh phdq vtxduh huuru/H+DPVH, fdq eh dssur{lpdwhg e|

�M

�

[

� 0 'AMSE � � * EEAMSE�

zkhuh AMSE � � lv wkh DPVH iru sdwk m: Pdun wkdw wkh EEAMSE� zloo eh fdofxodwhgiru hdfk glvwulexwlrq h{fhsw wkh NIG � � glvwulexwlrq zkhuh vwdwlrqdulw| lv qrw dozd|v

4:5

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

0

0.1

0.2

α = 0.3 β = 0.65

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

-0.2

-0.1

0

0.1

α = 0.1 β = 0.85

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

-0.6

-0.4

-0.2

α = 0.005 β = 0.925

Iljxuh 9116= Vwdwlrqdulw| frqglwlrq/ E *L}E�" �� n �� < f; iru wkh xqlw yduldqfhNIG � � glvwulexwlrq zlwk sdudpwhu 4158 iru ydu|lqj � dqg � ydoxhv1

4:6

jxdudqwhhg iru rxu sdudphwhu fkrlfhv1 D fduhixo dqdo|vlv ri wkh wdeohv G116 wkurxjk G118+vhh dsshqgl{ wr fkdswhu 9, uhyhdo wkdw wkh v|pphwulf vwxghqwzv t glvwulexwlrq shuiruphgzruvh ryhudoo zlwk wkh odujhvw H+DPVH, luuhvshfwlyh ri wkh xqghuo|lqj lqqrydwlrq1 WkhGL lv vxshulru ryhudoo iroorzhg e| wkh qrupdo dqg wkhNIG � � glvwulexwlrqv1 Exw gr wkhvhuhvxowv wudqvodwhv wr uhdo olih vlwxdwlrqv zkhq wkh h{dfw glvwulexwlrq ri wkh lqqrydwlrqvduh xqnqrzqB

Rxu qh{w vwxg| ri wkh vlpxodwlrq h{dplqhv wkh shuirupdqfh ri hljkw Jhupdqvwrfnv/ qdpho|/ Dooldq} DJ +DOY,/ EDVI DJ +EDV,/ Ed|hu DJ +ED\,/ Frpphu}edqnDJ +FEN,/ K|sryhuhlqvedqn +K\S,/ Ghxwvfkh Oxiwkdqvd DJ +OXG,/ Wk|vvhq Nuxss+WK\, dqg Yronvzdjhq DJ +YRZ, xqghu wkhvh glvwulexwlrqv1 Zh fdofxodwh wkh YdU dw��txdqwloh iru hdfk glvwulexwlrq iru wkh |hduv 5336/ 533505336/ 533405336/ 533305336dqg frpsduh zlwk wkh ��txdqwloh ydoxhv ri f:fD; f:f2D dqg f:f�: Wkh uhvxowv duh vkrzqlq wdeohv 9116 wkurxjk 9119 ehorz1 D fduhixo dqdo|vlv ri wkh uhvxowv lq wkh wdeoh uhyhdo wkdwwkh GL glvwulexwlrq shuiruphg ehwwhu ryhudoo d frq�updwlrq ri wkh uhvxowv irxqg lq wkhvlpxodwlrq1 Wkh v|pphwulf vwxghqwzv t zlwk �yh ghjuhhv ri iuhhgrp/ t � ED; f;

t �

�� dqg

wkh qrupdo glvwulexwlrqv/ duh zruvh ryhudoo , 1 Wkh NIG � � zdv voljkwo| ehwwhu wkdq wkhGL zlwk wkh 5336 gdwd exwGL zdv vxshulru dv zh lqfuhdvhg wkh qxpehu ri wudglqj |hduv

� 1 Dq lpsruwdqw revhuydwlrq lv wkdw dq| sduwlfxodu vwrfn kdv dq rswlpdo glvwulexwlrq lqlwv YdU hvwlpdwlrq dqg hyhq iru dq| sduwlfxodu vwrfn gl�huhqw lqqrydwlrqv zloo kdyh wr ehxvhg dw gl�huhqw ohyhov ri ��I: Iru lqvwdqfh lq wkh ED\ vwrfn/ wkrxjk wkhGL grhv ehwwhuryhudoo/ dw 2:DIYdU wkhNIG � � jdyh d ehwwhu hvwlpdwh luuhvshfwlyh ri wkh gdwd vl}h1 Wklv

8 Dq dgmxvwphqw lq wkh qxpehu ri ghjuhhv ri iuhhgrp pd| lpsuryh lwv shuirupdqfh1� Dq dgmxvwphqw lq wkh sdudphwhu/ �� ri wkh NIG � � pd| ohdg wr dq lpsuryhphqw lq wkh YdU

hvwlpdwlrq14:7

lv dovr wuxh iru wkh YRZ vwrfnv zkhuh wkh qrupdo jdyh d ehwwhu hvwlpdwh dw DI YdU +vhhwdeoh 9117, Dovr lqfuhdvlqj wkh vl}h ri wkh gdwd pd| uhtxluh d fkdqjh lq glvwulexwlrqdodvvxpswlrq iru wkh lqqrydwlrq1 Iru h{dpsoh/ wkh FEN vwrfn iru wkh shulrg 35134136 wr63145136 +wdeoh 9116, uhtxluhv wkh qrupdo glvwulexwlrq iru dq rswlpdo hvwlpdwlrq ri wkhYdU dw doo ohyhov1 Lq wkh shulrg 35134135 wr 63145136 +wdeoh 9117,/ wkh GL glvwulexwlrqjdyh d ehwwhu hvwlpdwh ri wkh YdU dw doo ohyhov wrjhwkhu zkhuhdv wkh NIG � � glvwulexwlrqjdyh d ehwwhu hvwlpdwh iru wkh shulrg 35134134 wr 63145136 +wdeoh 9118,1 Zkhq wkh vl}h riwkh gdwd lv lqfuhdvhg wr irxu |hduv wudglqj shulrg +wdeoh 9119,/ rqh pd| kdyh wr frqvlghuxvlqj wkh NIG � � ru wkh t � glvwulexwlrq1 Wklv glvwulexwlrqdo fkdqjhv zlwk lqfuhdvlqjgdwd vl}h fdq eh h{sodlqhg e| vwuxfwxudo fkdqjhv lq wkh xqghuo|lqj vwrfkdvwlf surfhvvhv1Rq wkh rwkhu kdqg/ vwrfnv vxfk dv ED\ lq zklfk wkh GL glvwulexwlrq shuiruphg ehwwhuluuhvshfwlyh ri wkh gdwd vl}h pd| eh dwwulexwhg wr uhodwlyho| vwdeoh xqghuo|lqj vwrfkdvwlfsurfhvv1

Zh frqfoxgh wkdw iru dq| jlyhq �qdqfldo gdwd dqg dq| jlyhq ��IYdU/ lwv hvvhqwldowr hvwlpdwh wkh YdU iru doo srvvleoh glvwulexwlrqdo dvvxpswlrqv dqg wkh glvwulexwlrq wkdwjlyhv d ehwwhu hvwlpdwh eh xvhg lq wkh dqdo|vlv1 Dovr dv wkh sdudphwhuv duh vhqvlwlyhwr wkh klvwrulfdo gdwd xvhg lq wkh YdU hvwlpdwlrq/ rqh kdv wr frqvlghu wkh ohqjwk riwkh klvwrulfdo gdwd xvhg1 D yhu| orqj gdwd shulrg zlwk pdq| rxwolhuv lv xqolnho| wr ehvxlwdeoh ehfdxvh h{wuhph pryhv iurp d yhu| glvwdqw sdvw frxog kdyh dq dgyhuvh h�hfwrq wkh sdudphwhu hvwlpdwhv dqg iru wkdw pdwwhu wkh JDUFK0ulvn1 Hpslulfdo dqdo|vlvfrqgxfwhg e| ^PV33` uhsruwv wkdw JDUFK prghov �wwhg wr gdlo| wlph vhulhv fryhulqjrqh ru wzr |hduv ri gdwd duh lq ehwwhu djuhhphqw zlwk wkhruhwlfdo uhvxowv1 Wkhuh lvdw ohdvw rqh rshq txhvwlrq= frxog wkh hvwlpdwhv lpsuryh li zh shuplw vnhzqhvv lq wkh

4:8

lqqrydwlrqv dv lwv riwhq revhuyhg wkdw dvvhw uhwxuqv duh vnhzhg ^GS<:`B Wkhuhiruh wkhulvn hvwlpdwhv xqghu glvwulexwlrqv wkdw shuplwv vnhzqhvv vxfk dv wkh GGL � / NIG/wkh vnhzhg qrupdo ri ^D;9` qhhgv wr eh lqyhvwljdwhg dv qhjohfwlqj vnhzqhvv frxog eldvlqihuhqfh derxw wkh ulvn ehlqj vrxjkw1

4:9

1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

11 year trading 2 years trading3 years trading4 years trading

1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

Parameter estimates for Allianz stock with increasing data size

1

2

3

1

2

3

1

2

3

1

2

3

Iljxuh 9117= Duurzv 4/ 5 dqg 6 fruuhvsrqg uhvshfwlyho| wr wkh hvwlpdwhv iru !; � dqg�: Yrodwlolw| lv shuvlvwhqw iru wkh vwrfn dv fdq eh vhhq iru wkh kljk � ydoxhv1Xsshu ohiw= JO hvwlpdwh1Xsshu uljkw= QLJ � � hvwlpdwh1Orzhu ohiw= Qrupdo hvwlpdwh1Orzhu uljkw= t � hvwlpdwh1

4::

Vwrfn GlvwulexwlrqGLEf; �; i TE 6

� � �� NIG � � E�:2D � ; f; f; �� NEf; ��Ohyho ri ��I YdU Ohyho ri ��I YdU Ohyho ri ��I YdUf:fD f:f2D f:f� f:fD f:f2D f:f� f:fD f:f2D f:f�

DOY 31389 3135; 3133; 31385 31365 31349 31389 31365 3135;EDVI 31377 3135; 31345 3137; 31369 3133; 31377 31369 31357ED\ 31385 31357 3135 31385 31357 3135 3135; 31357 31349FEN 31385 31345 3133; 3137; 31349 31337 31385 31357 1345K\S 31385 31357 31349 3137; 31357 31337 31385 3135; 3135OXG 31369 3133; 31337 31369 31349 3133; 31369 31357 31345WK\ 31385 3135 31337 3137 31349 31337 31389 31365 3133;YRZ 3137; 3135 3133; 3137; 31357 31345 31385 31365 3135

Wdeoh 9116= YdU hvwlpdwhv iru DOY/ EDV/ ED\/ FEN/ OXG/ K\S/ WK\ dqg YRZ ir

4:;



DOY 31376< 31355 31337 31378< 31355 31337 31386< 3135<< 3134EDVI 31386< 31368< 3133; 31388< 31366< 3133; 31384< 31368< 3135ED\ 31386< 31358< 3134 3138:< 31357 3134 31388< 31357 3134FEN 3137<< 3135:< 3133; 3137<< 3135:< 31339 3137<< 31364< 3134K\S 3137<< 3135 31339 3137:< 31355 31337 3137<< 31357 3134OXG 3136<< 3134 31339 3136<< 3134 31339 3136<< 31349 3133WK\ 31388< 31355 3134 3138:< 31355 3133; 31394< 31358< 3134YRZ 31374< 31349 31345 3136:< 31349 31337 3137<< 31355 3134


4:<

Vwrfn GlvwulexwlrqGLEf; �; i TE 6� � �� NIG � � E�:2D � ; f; f; �� NEf; ��Ohyho ri ��I YdU Ohyho ri ��I YdU Ohyho ri ��I YdUf:fD f:f2D f:f� f:fD f:f2D f:f� f:fD f:f2D f:f�

DOY 3137<6 31359: 31343: 31383: 31359: 313386 31387: 3135<6 3134EDVI 3137<6 3135<6 3133<6 31385 31363: 3133<6 3137; 3135<6 3134ED\ 31383: 3135<6 313466 31387: 31359: 313466 3136;: 31359: 3134FEN 31387: 3135; 313466 31387: 31359: 31345 31385 31363: 3134K\S 3137<6 3134:6 31337 3137<6 3134:6 31337 3137; 31359: 3133OXG 3137; 313546 3133<6 3137; 3134;: 313386 31379: 31355: 3134WK\ 31385 31355: 3133<6 31383: 3135 31339: 31385 3135<6 3134YRZ 31383: 31355: 31343: 3137; 3134;: 31343: 31383: 3135<6 3134


4;3



DOY 31377; 31355< 31344 31376; 3134<< 3133; 31375; 31358< 3134EDVI 31378; 31359< 3133; 3137;; 31359< 31339 31379; 3135;< 3134ED\ 3137;; 31356< 3134 31385; 31356< 3133< 31374; 31353< 3134FEN 31383; 31363< 31346< 3137<; 31359< 3134 31379; 3135<< 3134K\S 31384; 31354< 3133< 31383; 31353< 3133: 31384; 31359< 3134OXG 3137:; 31353< 3133; 3137<; 3134;< 3133; 3137:; 31358< 3134WK\ 31387; 3135:< 31345 31387; 31358< 3133< 31386; 3135;< 3134YRZ 31379; 3134<< 31344 31379; 3134:< 31344 31379; 31359< 3134


4;4

Xqghu glvwulexwlrqv zlwk srvvleoh dv|pphwu|Zh dsso| wkhruhp 433 wr d vlpxodwhg gdwdri vl}h n ' 833 dqg :83 iurp wkh JDUFK+4/4, surfhvv= h � ' s! n �h �� (' n �¦ �� (' " � zkhuh" � kdv d glvwulexwlrq gl�huhqw iurp wkh �yh glvwulexwlrqv zlwk sdudphwhu fkrlfh dv lqwkh v|pphwulf fdvh1 Zh fdofxodwh wkh DPVH ri wkh JDUFK0ulvn iru hdfk glvwulexwlrqdqg iru hdfk gdwd vl}h1 Wkh glvwulexwlrq zlwk wkh vpdoohvw DPVH +ru vpdoohvw pdj1DPVH, vkrxog eh idyrxuhg iru wkh ulvn fdofxodwlrq wkdq rqh zlwk wkh odujhvw DPVH+ru odujhvw pdj1 DPVH,1 Wkh uhvxowv duh wdexodwhg iru hdv| frpsdulvrq ri wkh pdj1DPVH +vhh wdeohv G119 wkurxjk G11<,1 Wkh GGL � glvwulexwlrqv zhuh vxshulru zkhq wkhxqghuo|lqj glvwulexwlrqv zhuh t � luuhvshfwlyh ri wkh gdwd vl}h +vhh wdeohv G11: dqg G11<,1Wkh NIG � dqg wkh NIG � glvwulexwlrqv shuiruphg zruvh ryhudoo1 Zkhq wkh wuxh xq0ghuo|lqj glvwulexwlrq zhuh qrupdo dqg wkh gdwd vl}h zhuh 833/ wkh SPat � glvwulexwlrqrxwshuiruphg wkh SPat � iroorzhg e| wkh qrupdo1 Krzhyhu/ zkhq wkh gdwd vl}h lv lq0fuhdvhg wr :83/ wkh qrupdo rxwshuiruphg wkh SPat � iroorzhg e| SPat � : Rq wkh rwkhukdqg/ zkhq wkh xqghuo|lqj glvwulexwlrq zhuh Vwxghqwzv t; wkh SPat � dqg wkh t � zhuhdssur{lpdwho| rq d sdu zkhuhdv wkh SPat � shuiruphg voljkwo| zruvh zkhq wkh gdwdvl}h zhuh 8331

Qh{w/ zh jhqhudwh h � �' ; : : : ; h � ��)�) ; m ' �; : : : ; �fff ' M dqg fdofxodwh wkh

H+DPVH, zkhq wkh lqqrydwlrqv duhGL; qrupdo dqg Vwxghqwzv t +zh xvh wkh vdph jhqhu0dwhg gdwd dv lq wkh v|pphwulf fdvh ,1 Wkh uhvxowv duh uhohjdwhg wr wdeohv G1143/ G1144 dqg

4;5

G1145 lq wkh dsshqgl{ zkhuh zh kdyh xvhg wkh deeuhyldwlrqv

SGGL � Eb� ' SGGL � E�E«Eb�� «E��s« � Eb� n « � E�� ; �; �a; b�;

SGGL � Eb� ' SGGL � E E«Eb�� «E��s« � Eb� n « � E�� ; �; �a; b�

SPat � E�� ' SPat � E «E��s« � E�� ; �;�s« � E�� ; ��

SPat � E�� ' SPat � E� «E��s« � E�� ; �;�s« � E�� ; ��

Lw lv fohdu iurp wkh wdeoh G1143 dqg G1145 wkdw zkhq wkh lqqrydwlrqv iru wkh jhqhudwhggdwd zhuh GL dqg Vwxghqwzv w wkh GGL � glvwulexwlrqv zhuh vxshulru wkdq wkh ehvwhvwlpdwh iru wkh v|pphwulf fdvh +l1h1 wkh GL dqg wkh NIG � � �1 Zkhq wkh jhqhudwhggdwd kdv wkh qrupdo lqqrydwlrqv/ wkh SPat � zdv forvhvw wr wkh qrupdo hvwlpdwh1 Wkhqrupdo jhqhudwhg gdwd zdv wkh rqo| gdwd wkdw zdv kdug wr ehdw zlwk d vlqjoh fkrlfh risdudphwhu lq wkh dv|pphwu| fdvh1 Ri frxuvh/ li zh zhuh wr ydu| wkh sdudphwhu fkrlfhiru wkh dv|pphwulf glvwulexwlrqv dfurvv hdfk sdwk d pxfk vpdoohu EEAMSE� lv wr ehh{shfwhg iru hdfk dv|pphwulf glvwulexwlrq1

4;6

Vwrfn GlvwulexwlrqSGGL � SGGL � SPat �

Ohyho ri ��I YdU Ohyho ri ��I YdU Ohyho ri ��I Yf:fD f:f2D f:f� f:fD f:f2D f:f� f:fD f

DOY 3137<< ) � � � 31357 3134 ) � �) � ) � � � $�� f:fD�b ' � � 31357

) � )�) �f:f� 3137:< ' �� 3

EDVI 3137<< ' � � 31364< 31339 3137:< ) � � 3135:< 31337 3137<< ' � ) 3

ED\ 3137<< ' � � 31355 3134 3137<< ) � � � 3135 3134 31384< ' �� 3

FEN 3137<< ' 3135:< 3133; 3137<< ' 3135:< 3133; 3137<< 3

K\S 3137<< ' 31357 ) � � 3133; 3137<< ' � ' 31355 3133; 3137<< ' � )�) 3

OXG 31376< ) � � � 31357 3133; 3136<< � � � 3134; 3133; 3136:< � )�)�) 3

WK\ 3137<< ' � � 31355 3133; 3137<< ) � �� 31355 3133; 3137<< � ) 3

YRZ 3137<< ) � � 31357 31345 3137<< ' � � 31355 31345 31378< � )�)�) 3

Wdeoh 911:= YdU hvwlpdwhv iru hljkw Jhupdq vwrfnv iru wkh shulrg 35134135 wr 63145136 zlwk s

4;7

Vwrfn GlvwulexwlrqSGGL � SGGL � SPat �

Ohyho ri ��I YdU Ohyho ri ��I YdU Ohyho rf:fD f:f2D f:f� f:fD f:f2D f:f� f:fD

DOY 3137<6 ' 31359: 31343: 3137<6 ) � �

� 31357 3133; 3137<6

EDVI 3137<6 ' 3135<6 3133<6 3137<6 ' 3135<6 3133<6 31377

ED\ 31379: ' � '�� 31359: 31345 31379: ) � � 31359: 31345 31369

FEN 31379: ' � � 313586 31343: 31379: ) � � � � 313586 3133<6 31379:

K\S 3137<6 ' 31359: ) � � � 3133; 3137<6 ' 313586 ' � � � 3133; 31379:

OXG 3137<6 ) � � � � 313546 3133<6 3137<6 ' � ' 31355: 3133<6 31379:

WK\ 31385 ' 31355: 3133<6 31385 ' � ) � � 313586 31343: 31379:

YRZ 31383: ) � �

� 31357 31343: 3137<6 ) � �

� 313546 31343: 31379:

Wdeoh 911;= YdU hvwlpdwhv iru hljkw Jhupdq vwrfnv iru wkh shulrg 35134134 wr 63145136 zlwk s

4;8

Lq wkh qh{w vwxg|/ zh uh0h{dplqh ulvn hvwlpdwhv iru wkh hljkw Jhupdq vwrfnvglvfxvvhg suhylrxvo| exw rqo| iru wkh SGGL � ; SGGL � ; SPat � dqg SPat � zlwk qru0pdo dv wkh vwdqgdug1 D fduhixo dqdo|vlv ri wkh wdeohv lq 911: dqg 911; uhyhdo wkdw wkhSGGL � zdv ehwwhu dqg wkh qrupdo zdv zruvh ryhudoo � 1 Wkh SPat � rxwshuiruphg wkhSPat � ; d vwuhqjwkhqlqj ri zkdw zdv irxqg lq wkh vlpxodwlrq vwxglhv1 Wkhuh duh vrphihz srlqwv wr qrwh lq wkh ryhudoo vwxglhv1 Iluvwo|/ iru dssursuldwho| fkrvhq sdudphwhu/�; wkh SPat � dqg wkh SPat � uhvsrqg ehwwhu wr h{wuhphv lq wkh gdwd wkdq wkh qrupdoexw gl�huhqw sdudphwhu/ �; pd| eh qhfhvvdu| iru gl�huhqw YdU ohyhov1 Wklv revhuydwlrqsurpswv d ixuwkhu duhd iru ixuwkhu uhvhdufk iru wkh JDUFK idplo| prghov wkdw zh whupyglvfulplqdqw JDUFK prghoolqjz1 Vhfrqgo|/ zkhwkhu ru qrw d srvlwlyh/ v|pphwulf ruqhjdwlyh glvwulexwlrq lv wr eh xvhg lq wkh YdU hvwlpdwlrq dovr ghshqgv rq wkh gdwdvl}h1 Wklugo|/ dv|pphwulhv lq wkh uhwxuq gdwd fdqqrw eh ljqruhg lq vr idu dv wkh YdUhvwlpdwlrq lv frqfhuqhg1

Zh frqfoxgh e| uhfrpphqglqj wkh xvh riGGL � glvwulexwlrqv iru wkhGARCH idplo|prghov exw furvvfkhfn |rxu dqdo|vlv zlwk wkh SPat � dqg wkh SPat � glvwulexwlrqv$

917 Frpsdulvrq ri uhvxowv zlwk WJDUFK Prgho

Lq wklv vhfwlrq zh frpsduh rxu uhvxowv zlwk wkh WJDUFKprgho zlwk frqglwlrqdoyrodwlolw| vshfl�fdwlrq

¦ �� ' ! n �h �� (' n �¦ �� (' n � � h �� (' � � - * ) � Eh � �(' �� Lq wdeohv 911: dqg 911;/ a phdqv wkh YdU hvwlpdwh lv a dqg wkh fkrlfh ri sdudphwhu xvhg

lq wkh glvwulexwlrq lv b: Dovr wkh fkrlfh ri sdudphwhuv iru wkh VSdw � dqg VSdw duh SPat � @^469; 469; 5333; 469; 4533; 4533; 463; 469`; SPat @ ^4533; 4533; 4533; 4533; 4533; 469; 4533; 533`iru ^DOY/EDVI/ED\/FEN/K\S/OXG/WK\/YRZ`

4;9

zlwk qrupdo lqqrydwlrqv iru 8( YdU dqg zlwk qrupdo/ Vwxghqwzv t zlwk irxu dqg vl{ghjuhhv ri iuhhgrp iru 4( YdU iru wkh hljkw Jhupdq vwrfnv1 Zlwk wzr ru wkuhh |hduvwudglqj gdwd +wdeohv 911< dqg 91143, lw fohdu frpsdulqj zlwk wkh wdeoh 911: dqg 911;wkdw wkh SGGL � zdv ehwwhu hvwlpdwh wkdq qrupdo dw erwk 8( dqg 4( YdU1 H{fhswwkh OXG vwrfn/ wkh SPat � zdv dovr ehwwhu wkdq wkh qrupdo dw erwk 8( dqg 4( YdU1Dw 4( YdU/ t � Ee; f; �� zrxog eh suhihuuhg lq 6 +FEN/ WK\ dqg YRZ, ri wkh hljkwvwrfnv zkhuhdv wkh SGGL � zrxog eh suhihuuhg lq 5 +DOY dqg OXG,1 Dw 4( YdUiru t � ES; f; ��; SGGL � zrxog reylrxvo| eh suhihuuhg wr wkh t � : Iru erwk t � Ee; f; �� dqgt � ES; f; ��; SGGL � zdv ehwwhu lq wkh ryhudoo 4( YdU iru wkh 6 |hdu wudglqj gdwd +wdeoh91143,1

4;:

Vwrfn NEf; �� t � Ee; f; �� t � ES; f; ��Ohyho ri ��I YdU Ohyho ri ��I YdU Ohyho ri ��I YdUf:fD f:f� f:f� f:f�

DOY 3138:< 31347 31335 3134EDVI 3138<< 31349 31339 31347ED\ 31386< 31347 3134 3134FEN 3137<< 31349 3134 31347K\S 31388< 31347 3133; 31345OXG 31374< 3133; 31339 3133;WK\ 31398< 31349 3134 31347YRZ 31384< 31345 3134 31345

Wdeoh 911<= YdU hvwlpdwhv iru DOY/ EDV/ ED\/ FEN/ OXG/ K\S/ WK\ dqg YRZ iruwkh shulrg 35134135 wr 63145136 xqghu WJDUFK

4;;

Vwrfn NEf; �� t � Ee; f; �� t � ES; f; ��Ohyho ri ��I YdU Ohyho ri ��I YdU Ohyho ri ��I YdUf:fD f:f� f:f� f:f�

DOY 3139 3134:6 3133; 31345EDVI 3137<6 313466 31343: 313466ED\ 31377 31349 31347: 3135FEN 31385 3134;: 31349 3134:6K\S 31385 31345 313386 3133;OXG 313786 31345 31339: 31345WK\ 313866 313466 3133<6 31345YRZ 313866 313466 31343: 31343:

Wdeoh 91143= YdU hvwlpdwhv iru DOY/ EDV/ ED\/ FEN/ OXG/ K\S/ WK\ dqg YRZiru wkh shulrg 35134134 wr 63145136 xqghu WJDUFK

4;<

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4<3

Wkhruhp 435 Ohw dvvxpswlrq 434 krog dqg vxssrvh wkdw

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Eh � ' d � M � Et�dt � m � � � ' ¡ � I � n 2¡�I � n � � I �

Eh � ' d � M � Et�dt � m � � � ' ¡ � I � n �¡ � �I � n �¡� � I � n � � I �

Eh � ' d � M � Et�dt � m � � � ' ¡ � I � n e¡ � �I � n S¡ � � � I � n e¡� � I � n � � I �

:

:

:

Eh � ' d � M � Et�dt � m � � � ' ¡ � I � n n¡ �

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nEn� ��2- ¡ �

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� I � n : : :

nnEn� �� : : :En� r n ��r- ¡ �

� E ¢*L} a�

I � n : : : n E ¢*L} a�

� I � � �

Wkh iroorzlqj uhvxow zdv xvhg lq wkh surri ri ohppd 84=

Fodlp 458 Iru dq| n 5 Q; Eh �� ' S �

�� EEh� � covEh �� ; h� n EEh� � � � ; n � i n �:

Surri1 Ohw P Ek� eh=

Eh � � � '�[

�� EEh� � covEh � �

� ; h� n EEh� � � � ; k � i n �

P E�� G Eh � ' covEh; h� nEhcovE�; h� n EEh� � ' covEh; h� n EEh� � dqg vr P E�� lv wuxh1Qrz P Ek n �� lv=

Eh � � � '� � �[��

EEh� � covEh � � � �� ; h� n EEh� � � � ; k � i

55<

Zh suryh e| lqgxfwlrq wkdw P Ek� lpsolhv P Ek n �� dqg khqfh frqfoxgh wkdw P En� lvwuxh iru doo n 5 Q: E| P Ek n ��; zh kdyh

Eh � � � '�[

�� EEh� � � � covEh � �

� ; h� n EEh� � � �

' covEh � � � ; h� n Ehd�[

�� EEh� � covEh � �

� ; h� n EEh� � � �

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o

' covEh � � � ; h� n EhEh � � � :

Wklv lv P Ek n ��-

563

DSSHQGL[ G1


714 Qrupdo Lqyhuvh Jdxvvldq glvwulexwlrq

Wkh suredelolw| ixqfwlrq iru wkh Qrupdo Lqyhuvh Jdxvvldq +QLJ, lqwurgxfhg e|^E<8` lv fkdudfwhul}hg e| irxu sdudphwhuv E�; �; ¡; �� zkhuh � lv wkh vwhhsqhvv sdudphwhu/� lv dq dv|pphwu| sdudphwhu/ ¡; wkh orfdwlrq dqg �; wkh vfdoh sdudphwhu1 Wkh odujhu� lv/ wkh vwhhshu wkh ghqvlw| dqg wkh ghqvlw| lv v|pphwulf zkhq � ' f: Wkh glvwulexwlrqdulvhv dv d pdujlqdo glvwulexwlrq ri X lq EX;Z� zkhuh �

X=Z ' z�NE¡n �z; z�

z�IGE�;t� � � � � �; f � m�m < �

Gh�qlwlrq 459 D udqgrp yduldeoh [ lv QLJ glvwulexwhg zlwk sdudphwhuv E�; �; ¡; �� +ghqrwhg[�QLJE�; �; ¡; ��, li lwv ghqvlw| lv jlyhq e|

fEx� ' aE�; �; ¡; ��qEx� ¡� � � � K � E��qEx� ¡

� ��e ��

zkhuh qEx� ' s� n x � ; aE�; �; ¡; �� ' �¤ i TE�s� � � � � ��¡� dqgK � Ex� ' �

�U �

� i TE� �� xEtn

�� dt; lv wkh prgl�hg Ehvvho ixqfwlrq ri wkh wklug nlqg ri lqgh{ 41

Uhpdun 45: Wkh QLJ glvwulexwlrq lv d phpehu ri wkh Jhqhudol}hg k|shuerolf ghqvl0wlhv dqg lwv wkh rqo| phpehu ri wkh idplo| zklfk lv forvhg xqghu frqyroxwlrqv ^EEV<5`1

� IG+�; �,; � @ s� � � � � ; lv wkh Lqyhuvh Jdxvvldq glvwulexwlrq zlwk ghqvlw| f+z, @�� h{s+� �� ^ � �� `,: Wkh Jhqhudol}hg Lqyhuvh Jdxvvldq glvwulexwlrq zlwk sdudphwhuv ; � dqg� +ghqrwhg GIG+ ; �; �,, kdv ghqvlw| f+z, @

�� z �

� � h{s+� �� + � � . � � z,,; zkhuh K � +u, @�� U �� x � � � h{s+� �� u+x . �� ,,dx; wkh prgl�hg Ehvvho ixqfwlrq ri wkh wklug nlqg zlwk lqgh{ : Qrwhwkdw IG+�; �, @ GIG+� �� ; �; �,:

564

+l1h1 li X � �NIGE�; �; ¡ � ; � � �; X � �NIGE�; �; ¡ � ; � � � wkhq X � X � �NIGE�; �; ¡ � n¡ � ; � � n� � �,1

Li X�NIGE�; �; ¡; ��; wuhdwlqj �; ¡; dqg � dv frqvwdqwv lw iroorzv iurp wkh surs0huwlhv ri h{srqhqwldo idplo| ^FO<;`/ sjv 56065/ wkdw wkh prphqw jhqhudwlqj ixqfwlrq/M � Et� lv jlyhq e|

M � Et� ' i Tdt¡n �t� � � � � � �s� � � E� n t� � o

Khqfh/

EX ' ¡n ��s� � � � �

V arEX� ' ��

E� � � � � � ��

�EX� ' ��t

�E� � � � � � ��

exEX� ' ��s� � � � � E� n e� �

� � �

Revhuyh wkdw li V arEX� ' ¦ � dqg � ' f wkhq � ' �¦ � fdvwlqj wkh ghqvlw| dv

�¤ i TE�¦ � �qEx� ¡

�¦ � � � � K � E� � ¦ � qEx� ¡�¦ � ��

E| vhwwlqj 7� ' � � ¦ � zh rewdlq wkh v|pphwulfNIG ri ^I35`/ ghqrwhg e|NIG � � E7�; f; ¡; ¦ � �: Lwvghqvlw| ixqfwlrq lv

gEx� 's7�¤s¦ � i TE7��qE

x� ¡s7�¦ � � �� K � E7�qEx� ¡s7�¦ � ��; 7� > f; ¦ > f;�4 < ¡ < 4

dqg wkh fruuhvsrqglqj prphqw jhqhudwlqj ixqfwlrq

M � Et� ' i Tdt¡�s7� � � 7�¦ � t � n 7�o565

Qrwh wkdw li X�NIG � � E7�; f; ¡; ¦ � �; wkhq LE � n � X� ' NIG � � E7�; f; E � n � ¡�; �� ¦ � �iru � ; � lq �

U:Rq wkh rwkhu kdqg/ li V arEX� ' ¦ � ; EX ' f dqg � 9' f wkhq wkh ghqvlw| fdq

eh h{suhvvhg lq whupv ri wkuhh sdudphwhuv ¡; � dqg ¦ � dv

fEx� '

;AA?

AA=

a � E¡; �; ¦ � �qE � � �� K � E � ��

� ��

��

� � � qE � � �� i TE¡ � ��

� ��

� � � � x�; li ¡ > fa � E¡; �; ¦ � �qE � � �

� � � � K � E � ��

� ��

� � � qE � � �� i TE�¡ � ��

� ��

�� x�; li ¡ < f

zkhuh aE�; �; ¡; �� wkdw zh ghfrpsrvh lq wzr sduwv a � E¡; �; ¦ � � dqg a � E¡; �; ¦ � � lv jlyhqe| ;

AA?

AA=

� � ��

� ��

� � � � i TE � � ��

� �� E� � � ¡ � ��; li ¡ > f

� ��

� ��

� � � � i TE � ��

� �� E� � n ¡ � ��; li ¡ < f

:

¡ > f fruuhvsrqgv wr srvlwlyho| vnhzhg glvwulexwlrqv zkhuhdv ¡ < f duh qhjdwlyho|vnhzhg1 Zh zloo ghqrwh wkhvh e| NIG � E¡; �; ¦ � � dqg NIG � E¡; �; ¦ � � uhvshfwlyho|1Wkh xqlw yduldqfh srvlwlyho| dqg qhjdwlyho| vnhzhg glvwulexwlrqv zloo eh ghqrwhg e|SNIG � E¡; �; �� dqg SNIG � E¡; �; �� uhvshfwlyho|1 Wkh fruuhvsrqglqj fxppxodqw jhq0hudwlqj ixqfwlrqv duh

K �� Et� ' t¡n � � n ¡ �¦ � � �

�¦ �tE� � n ¡ � � � � i� � ¦ � t n ¡E� � n ¡ � �j �

K �� Et� ' t¡n � � n ¡ �¦ � � �

�¦ �tE� � n ¡ � � � � i� � ¦ � t� ¡E� � n ¡ � �j �

Jhqhudwlrq ri udqgrp qxpehuv iurp QLJ glvwulexwlrq Wkh jhqhudwlrq ri udqgrp ydul0dwhv wkdw duh NIG�glvwulexwhg lv rewdlqhg e| h{sorlwlqj d surshuw| vkduhg e| wkhLJ glvwulexwlrq zlwk wkh qrupdo ghqvlw| ^D;5`/ ^PVK:9`1 Vshfl�fdoo|/ li zh zulwh wkh

� Wkhuh lv �dz lq wkh vfdolqj surshuwlhv ri qhz sdudphwul}dwlrq ri NIG � � +��; 3; ¡; ¦ � , lq ^I35`1566

ghqvlw| ri IGE�; �� dv

fEz� ' �s2¤z � i TE��2E

E�z � �� z ��

wkhqV ' E�z � ��

z �ª � E��

+l1h1 V lv fkl0vtxduh glvwulexwhg zlwk � ghjuhh ri iuhhgrp, D vlplodu uhvxow grhv qrw krogiru rwkhu phpehuv ri wkh GIG glvwulexwlrqv1 Li z � dqg z � duh wkh urrwv ri wkh uhvxowlqjtxdgudwlf htxdwlrq/ wkhq

z � ' �� n V

2� � � �2� �

sV � n e��V ; z � ' � ��

�z �

zkhuh z � kdv wr eh fkrvhq zlwk suredelolw| �� n �z �

dqg z � zlwk suredelolw| �z �

� n �z �'

�� n �z � +Vhh ^PVK:9`,: Khqfh

Z ' z � � �� n z � � � � �IGE�; ��

zkhuh U lv wkh xqlirup glvwulexwlrq rq wkh xqlw glvn dqg P lv wkh fruuhvsrqglqj sure0delolw|1 Wkh QLJE�; �; ¡; ��, udqgrp yduldwhv lv rewdlqhg e|

X ' ¡n �Z nsZNEf; ��:

715 Vwxghqwzv w glvwulexwlrq

X�tEv; ¡; �� +l1h1 X kdv wkh jhqhudo vwxghqwzv t glvwulexwlrq zlwk sdudphwhuvv; ¡ dqg �� li lwv ghqvlw| lv jlyhq e|

f � � � � � Ex� ' �BE � � ; �

� ��sv E� nEx� ¡� �

v� � � � � � �� ;�4 < ¡ < 4; � > f567

Glvwulexwlrq Sdudphwhuv pdj1 PVH dw ohyho ��I YdU! � � f:fD f:f2D f:f�

GLEf; �; i TE �� 3134 316 3198 315<<< 317976 31:637

3134 314 31;8 317898 31:39: 41444;

NIGE�:2D � ; f; f; �� 3134 316 3198 31;3<9 415<:7 5145:33134 314 31;8 31<563 417:<4 51757<

NEf; �� 3134 316 3198 315743 316755 317;543134 314 31;8 314<<: 315;68 316<<7

t � ED; f; �� 3134 316 3198 41764: 5165<< 61<<563134 314 31;8 516993 61;836 918<:9

Wdeoh G114= 833 vlpxodwhg gdwd iurp JDUFK+4/4, zlwk qrupdo lqqrydwlrqv

568

Glvwulexwlrq Sdudphwhuv pdj1 PVH dw ohyho ��I YdU! � � f:fD f:f2D f:f�

GLEf; �; i TE �� 3134 316 3198 316:3: 318:6; 31<35:

3134 314 31;8 31<93< 417;:9 516736

NIGE�:2D � ; f; f; �� 3134 316 3198 315647 316:3; 3193:<3134 314 31;8 319<<3 414536 41;699

NEf; �� 3134 316 3198 31743; 318;66 31;54;3134 314 31;8 413864 417<83 514397

t � ED; f; �� 3134 316 3198 3137;7 313:;; 3146833134 314 31;8 315558 316953 319537

Wdeoh G115= 833 vlpxodwhg gdwd iurp JDUFK+4/4, zlwk vwxghqwzv w lqqrydwlrqv

569

Glvwulexwlrq Sdudphwhuv H+DPVH, dw ohyho ��I YdU! � � f:fD f:f2D f:f�

GLEf; �; i TE �� 3134 316 3198 31336 31337: 3133:6

3134 314 31;8 31336< 313394 3133<9

NIG � � E�:2D � ; f; f; �� 3134 316 3198 31336: 31339 3133<;3134 314 31;8 31336; 313395 313434

NEf; �� 3134 316 3198 31336: 313386 3133:73134 314 31;8 31337 31338: 3133;

t � ED; f; �� 3134 316 3198 313;78 3146:8 3156893134 314 31;8 313<48 3147< 315886

Wdeoh G116= H+DPVH, zlwk JO lqqrydwlrqv iru v|pphwulf glvwulexwlrqv

56:


GLEf; �; i TE �� 3134 316 3198 31336: 31338: 3133;<

3134 314 31;8 313376 31339: 313438

NIG � � E�:2D � ; f; f; �� 3134 316 3198 313389 3133< 31347:3134 314 31;8 313387 3133;9 313474

NEf; �� 3134 316 3198 313354 31336 3133763134 314 31;8 31335; 31337 313389

t � ED; f; �� 3134 316 3198 718<:9 :17;4< 451;5363134 314 31;8 719346 :17;:< 451;638

Wdeoh G117= H+DPVH, zlwk qrupdo lqqrydwlrqv iru v|pphwulf glvwulexwlrqv

56;


GLEf; �; i TE �� 3134 316 3198 31336; 31338; 3133<5

3134 314 31;8 313385 3133; 31345:

NIG � � E�:2D � ; f; f; �� 3134 316 3198 313367 313388 3133<3134 314 31;8 313376 31339< 313447

NEf; �� 3134 316 3198 3133: 3133<< 31346<3134 314 31;8 3133;< 313459 3134:;

t � ED; f; �� 3134 316 3198 31483: 315785 3175353134 314 31;8 31476; 31567 31734

Wdeoh G118= H+DPVH, zlwk Vwxghqwzv w lqqrydwlrqv iru v|pphwulf glvwulexwlrqv

56<

zkhuh B lv wkh Ehwd ixqfwlrq1 v lv wkh qxpehu ri ghjuhhv ri iuhhgrp/ ¡ lv wkh orfdwlrqdqg �; wkh vfdoh sdudphwhu1 Iru v > 2;wkh phdq dqg wkh yduldqfh duh

EX ' ¡; V arEX� ' vv � 2�

�

Wkh zhoo nqrzq vwxghqwzv t glvwulexwlrq lv lghqwlfdo wr tEv; f; �� iru gl�huhqw v: Li wkhorfdwlrq lv }hur dqg wkh V arEX� ' ¦ � zh kdyh wkh iroorzlqj gh�qlwlrq1

Gh�qlwlrq 45; D udqgrp yduldeoh [ +ghqrwhg t � Ev; f; ¦ � �, li lwv ghqvlw| lv jlyhq e| �

f � � � Ex� ' �BE � � ; �

� �¦sv � 2E� nx �

¦ � Ev � 2�� ; v > 2; ¦ > f

Uhpdun 45< Lw lv zhoo nqrzq wkdw wkh wdlov ri wkh Vwxghqwzv w ghqvlw| +l1h1 tEv; f; ��,ehfrphv wklfnhu dv v ghfuhdvhv dqg rqh pljkw eh txlfn wr frqfoxgh wkdw wklv surshuw| lvfduulhg ryhu wr wkh t � Ev; f; ��; wkh v|pphwulf xqlw yduldqfh ghqvlw|1 Wklv lv qrw wuxh irudoo ydoxhv ri v > 2-: Dv v $ 2; wkh wdlov ri wkh v|pphwulf xqlw yduldqfh ghqvlw| ehfrphvoljkwhu1 Vhh �jxuh F1114 +orzhu sorw,1

� f �� +x, @ t�

�� f �� +xt �

�� ,

573

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.5

1

1.5

2

-10 -9.5 -9 -8.5 -8 -7.5 -7 -6.5 -6 -5.5 -50

0.2

0.4

0.6

0.8

1x 10

-3

v=2.1

v=3

v=2.5

Iljxuh G114= Xsshu judsk= Ghqvlwlhv ri t � zlwk v ' � +oljkw vrolg olqh,/ v ' 2:D Ewklfnvrolg olqh, dqg v ' 2:� +grwwhg olqh,1Orzhu judsk= Wkh ohiw wdlo ehkdylru ri wkh ghqvlw| ri t � Ev; f; �� iru gl�huhqw v:

574

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5GLNIGσ

2

Nts

-10 -9 -8 -7 -6 -50

2

4

6

8x 10

-4

-10 -9 -8 -70

0.2

0.4

0.6

0.8

1x 10

-4

Iljxuh G115= Ghqvlwlhv dqg ohiw wdlo ehkdylru riGLEf; �; i TE ��

� ��; NIG � � E�:2D � ; f; f; ��; NEf; �� dqg t � ED; f; ��:

575

Glvwulexwlrqv Sdudphwhuv pdj1 DPVH dw Ohyho ��I YdUSdudphwhu+v, fkrlfh ! � � f:fD f:f2D f:f�

SGGL � 413: 3134 316 3198 315839 316;<< 319494

41338 3134 314 31;8 317895 31:398 41444;

SGGL � 31<8 3134 316 3198 315:7 3175:6 319::

415 3134 314 31;8 31734: 319394 31<5<

SNIG � 3138/ 415 3134 316 3198 5661::: 6:4153:; 9391443;

3134 314 31;8 319677 4133:6 41977;

SNIG � 03138/ 415 3134 316 3198 4<:16996 6481987 84:169;<

3134 314 31;8 31639< 317<3; 31;377

SPat � 5333 3134 316 3198 315396 315<6; 317486

8333 3134 314 31;8 315389 315<57 31745<

SPat � 6933 3134 316 3198 314<47 315:44 316;4

469 3134 314 31;8 314;; 31596< 3169:5

NEf; �� Wuxh xqghuo|lqj 3134 316 3198 31574 316755 317;54

glvwulexwlrq 3134 314 31;8 314<<: 315;68 316<<7

Wdeoh G119= 833 vlpxodwhg gdwd iurp wkh JDUFK+4/4, zlwk qrupdo lqqrydwlrqv

576


SGGL � 319 3134 316 3198 313646 313796 3139<;

318 3134 314 31;8 313;<8 3145<8 3145<8

SGGL � 514 3134 316 3198 3135:: 3136< 313888

516 3134 314 31;8 313985 313<44 3145;6

SNIG � 3138/ 415 3134 316 3198 4;91::9< 5<918::; 7;715877

3138/ 415 3134 314 31;8 419<74 519<34 716<56

SNIG � 03138/ 415 3134 316 3198 48:19;;5 58514<89 746168;5

03138/ 415 3134 314 31;8 5531<476 68616484 8:<13<9;

SPat � 8333 3134 316 3198 3187;6 31:; 414344

8333 3134 314 31;8 31<<76 417479 41<<:4

SPat � 469 3134 316 3198 3155<6 31654< 3177:<

�H 3134 314 31;8 315354 315:;6 316:<7

t � ED; f; �� Wuxh xqghuo|lqj 3134 316 3198 3137;7 313:;; 31468

glvwulexwlrq 3134 314 31;8 315558 31695 319537

Wdeoh G11:= 833 vlpxodwhg gdwd iurp wkh JDUFK+4/4, zlwk Vwxghqwzv w lqqrydwlrqv

577


SGGL � 415 3134 316 3198 4194:: 518697 713686

414 3134 314 31;8 51:655 7158< 91:746

SGGL � 416 3134 316 3198 41;;<; 51;539 715:75

416 3134 314 31;8 5196;7 61<6:< 81<9:6

SNIG � 3138/ 415 3134 316 3198 7881;789 :561;576 414;4<h.336

3138/ 415 3134 314 31;8 417;7 516898 61;7::

SNIG � 03138/ 415 3134 316 3198 6;71;835 9481835; 4133;;h.336

03138/ 415 3134 314 31;8 31::54 41567< 51357

SPat � 8333 3134 316 3198 315544 316478 31777

8333 3134 314 31;8 413895 418359 514546

SPat � 6933 3134 316 3198 31573< 316746 317:<9

6933 3134 314 31;8 413987 4183<5 51453<

NEf; �� Wuxh xqghuo|lqj 3134 316 3198 3155<; 316595 3178<9

glvwulexwlrq 3134 314 31;8 31945 31;9;< 415575

Wdeoh G11;= :83 vlpxodwhg gdwd iurp wkh JDUFK+4/4, zlwk qrupdo lqqrydwlrqv

578


SGGL � 316 3134 316 3198 31549< 315<66 317398

317 3134 314 31;8 413;84 418596 514<;7

SGGL � 616 3134 316 3198 4194;8 514<9: 51<<34

61: 3134 314 31;8 41;497 51779< 61634

SNIG � 3138/ 415 3134 316 3198 56716:78 6:514899 93:199

3138/ 415 3134 314 31;8 6<51633: 9551<564 4134:4h.336

SNIG � 03138/ 415 3134 316 3198 4<:1;:8< 649179<4 84;1:37:

03138/ 415 3134 314 31;8 66415387 85<1:3:4 ;9;153<<

SPat � 8333 3134 316 3198 7178:: 916749 ;1<86

; 3134 314 31;8 515<5 617667 814646

SPat � 6933 3134 316 3198 718;<: 918347 <14697

469 3134 314 31;8 71<7:5 91<76; <1994<

t � ED; f; �� Wuxh xqghuo|lqj 3134 316 3198 618<9; 81;865 43135<7

glvwulexwlrq 3134 314 31;8 71;336 :1;44; 4616;89

Wdeoh G11<= :83 vlpxodwhg gdwd iurp wkh JDUFK+4/4, zlwk Vwxghqwzv w lqqrydwlrqv

579


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599

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