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D qrwh rq wkh uvw prphqw ri h{wuhph rughu vwdwlvwlfv iurp wkh qrupdo glvwulexwlrq

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(1)

D qrwh rq wkh uvw prphqw ri h{wuhph rughu vwdwlvwlfv iurp wkh

qrupdo glvwulexwlrq

e| Pd{ Shw}rog

Ghsduwphqw ri Vwdwlvwlfv/ Vfkrro ri Hfrqrplfv dqg Frpphufldo Odz/

Jùwheruj Xqlyhuvlw|/ Er{ 993/ VH0738 63 JÙWHERUJ/ Vzhghq1 Hpdlo= Pd{1Shw}rogCvwdwlvwlfv1jx1vh

Nh| zrugv dqg skudvhv= dv|pswrwlf> eldv> Fudpìu> h{wuhph> qrupdo> rughu vwdwlvwlf1

Devwudfw

Lq wklv qrwh vrph zhoo nqrzq dv|pswrwlf uhvxowv iru prphqwv ri rughu vwdwlv0 wlfv iurp wkh qrupdo glvwulexwlrq duh wuhdwhg1 Wkh uhvxowv ruljlqdwhv iurp wkh zrun ri Fudpìu1 D eldv fruuhfwlrq iru qlwh vdpsoh vl}hv lv sursrvhg iru wkh h{shfwhg ydoxh ri wkh odujhvw revhuydwlrq1

4 Lqwurgxfwlrq

Ohw t E E? ghqrwh wkh =wk ydoxh iurp wkh erwwrp lq d vdpsoh ri ? l1l1g1 revhu0 ydwlrqv/ vxfk wkdw h1j1 t E?  t E? E? lv wkh odujhvw revhuydwlrq1 Wkh s1g1i1 ri t E lv zhoo nqrzq

s t

E

E+ ' ?-

E  - E?  - E8 t E+ 3 E  8 t E+ ?3 s t E+ +4,

exw frxog kdugo| eh xvhg wr qg h1j1 h{shfwhg ydoxhv ri rughu vwdwlvwlfv1 Wr

uhdfk dq dv|pswrwlf uhvxow iru t E? / frqvlghu wkh f1g1i1 8 t

E?

E+ ' d8 t E+o ? 1

(2)

8 t

E?

E+ $

  li 8 t E+ '  f li 8 t E+ 

lw lv reylrxv wkdw vrph wudqvirupdwlrq ri t E? lv qhhghg lq rughu wr dyrlg wulyldo olplwlqj uhvxowv1 Vxfk wudqvirupdwlrqv ri rughu vwdwlvwlfv kdyh ehhq h{whqvlyho| vwxglhg e| h1j1 Iuìfkhw ^5` dqg Jqhghqnr ^6`1 Khuh/ vrph dv|ps0 wrwlf uhvxowv e| Fudpìu ^4` iru qrupdoo| glvwulexwhg revhuydwlrqv duh suhvhqwhg dqg d eldv fruuhfwlrq ri wkh h{shfwhg ydoxh iru qlwh vdpsoh vl}hv lv sursrvhg1

5 Dv|pswrwlf uhvxowv iru wkh qrupdo glvwul0 exwlrq

Dffruglqj wr Fudpìu/ iru   ?*2 dqg ? $ 4

t E  > n j

% s

2 *? ? 

 *?E*? ? n *?EeZ  2` E  2 s

2 *? ?

&

+5, zkhuh ` E kdv wkh ghqvlw|

} E ' 

E?  - i T E E?   n    i T E c 4  4 Vlqfh . 

` E? 

'  E ' fD..22 dqg T 

` E? 

' ‚E ' Seeb / vhh Mrkqvrq ^7`/ wkh odujhvw revhuydwlrq kdv wkh prphqwv

.  t E? 

 > n j s

2 *? ?  E*?E*? ? n *?EeZ  2  fD..2

2 s 2 *? ?



+6, T 

t E? 

 j 2  Seeb

2 *? ? +7,

Lw lv ri vrph lqwhuhvw wr vwxg| krz wkh uhvxowv lq E2 zhuh rewdlqhg1 Ohw

~ ' E+  > *j dqg ohw ~ E eh wkh fruuhvsrqglqj rughu vwdwlvwlf1 Vwduwlqj

(3)

A E ' ? 

  8 ~ 

~ E 

' ? ] "

~

E

s 

2Z i T 

 2 *2  _ ' ' s ?

2Z ] "

~

E



  i T 

 2 *2 

_ ' s ?

2Z   

~ E n B  i T 

 

~ E n B  2

*2 

e| wkh phdq ydoxh wkhruhp1 Iurp wklv 

~ E n B 

fdq eh vroyhg

 ~ E n B 

 s

2 *? ?  *?E*? ? n *?EeZ n 2 *? A E

2 s 2 *? ?

Surfhhglqj wr wkh jhqhudo fdvh ri wkh qrupdo glvwulexwlrq/ uhsodflqj ~ E zlwk

 t E  > 

*j/ vxevwlwxwlqj ` E iru  *? A E dqg qdoo| fkrrvlqj B ' f zh jhw htxdwlrq E21 Dq| rughu vwdwlvwlf fdq eh rewdlqhg lq d vlplodu zd|1 Gxh wr v|pphwu| . 

t E 

frxog eh irxqg gluhfwo| iurp htxdwlrq E 1

6 D eldv fruuhfwlrq ri wkh h{shfwhg ydoxh iru

qlwh vdpsoh vl}hv

Wkh h{dfw dqg wkh dssur{lpdwlyh ydoxh/ uhvshfwlyho|/ ri .  t E? 

duh jlyhq lq Iljxuh  iru t  Ef(  dqg ? ' Dc Sc c f1 Wkh h{dfw ydoxh zdv vlpxodwhg zlwk wkh vwdwlvwlfdo vriwzduh VDV+u, LPO1 Iru doo vlpxodwlrqv lq wklv sdshu/

wkh frpsxwhu forfn zdv xvhg dv udqgrp vhhg dqg f S uhsolfdwhv zhuh wdnhq1 Wkh dv|pswrwlf uhvxow lq htxdwlrq E  zdv xvhg dv dq dssur{lpdwlrq iru qlwh vdpsoh vl}hv1

Dv fdq eh vhhq iurp Iljxuh  dqg Iljxuh 2 wkh htxdwlrq E  jlyhv d eldvhg

uhvxow iru qlwh vdpsoh vl}hv1 Orrnlqj edfn dw wkh ghulydwlyhv ri htxdwlrq E 

lw fdq eh vhhq wkdw wzr dujxphqwv duh pdgh1 Vshfldoo| wkh fkrlfh ri B ' f

(4)

wkh eldv jlyhq lq Iljxuh 2=

B E? ' f.2.? 3f2.Db Vxewudfwlqj wklv ixqfwlrq lq htxdwlrq E  zh jhw=

.  t E? 

 > n j s

2 *? ?  E*?E*? ? n *?EeZ  2  fD..2

2 s

2 *? ?  f.2.? 3f2.Db



+8,

Wklv fruuhfwlrq zdv whvwhg iru vrph rwkhu sdudphwhu ydoxhv lq wkh qrupdo glvwulexwlrq dqg vkrzhg xs wr frpshqvdwh wkh eldv hhfwlyho|1 Lq Wdeoh  wkh xqfruuhfwhg htxdwlrq E  dqg wkh fruuhfwhg htxdwlrq ED duh frpsduhg iru vrph qrupdo glvwulexwlrqv1 Wkh pd{lpdo dqg plqlpdo txrwd ehwzhhq wkh eldv ri wkh xqfruuhfwhg htxdwlrq dqg wkh eldv ri wkh fruuhfwhg htxdwlrq lv jlyhq iru

? ' Dc Sc f1 Dv fdq eh uhdg rxw/ wkh eldv ri wkh xqfruuhfwhg htxdwlrq lv dw ohdvw 2 wlphv odujhu wkdq wkh fruuhvsrqglqj eldv ri wkh fruuhfwhg htxdwlrq1 Dovr wkh yduldqfh lq htxdwlrq Ee lv eldvhg dqg pd| qhhg wr eh fruuhfwhg lq vrph zd|/ exw lv ohiw lq wklv sdshu1

Wdeoh = Wkh pd{lpdo dqg plqlpdo txrwd ehwzhhq wkh eldv ri wkh xqfruuhfwhg htxdwlrq E  dqg wkh eldv ri wkh fruuhfwhg htxdwlrq ED iru ? ' Dc Sc f1

Glvwulexwlrq= Ef(   Ef( D  Ef( 2f

Pd{lpdo ffff  fff .eff

Plqlpdo 2 2e 2.

(5)

1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60

0 20 40 60 80 100

n Equation (3)

Simulated

Iljxuh = D frpsdulvrq ri wkh vlpxodwhg dqg wkh dssur{lpdwlyh .  t E? 



0.00 0.02 0.04 0.06 0.08 0.10 0.12

0 20 40 60 80 100

n

Iljxuh 2= Wkh eldv ri wkh dssur{lpdwlyh .  t E? 

lq Iljxuh 1

(6)

Uhihuhqfhv

^4` Fudpìu/ K1 +4<8:,1 Pdwkhpdwlfdo phwkrgv ri vwdwlvwlfv/ :1 su1/ ss1 8:81 Sulqfhwrq= Sulqfhwrq Xqlyhuvlw| Suhvv/ 6:706:91

^5` Iuìfkhw/ P1 +4<5:,1 Vxu od orl gh suredelolwì gh o*ìfduw pd{lpxp1 Dqqdohv gh od Vrflìwì gh Pdwkìpdwltxh 4;/ <604491

^6` Jqhghqnr/ E1 +4<76,1 Vxu od glvwulexwlrq olplwh gx whuph pd{lpxp g*xqh vìulh doìdwrluh1 Dqqdov ri Pdwkhpdwlfv 77/ 75607861

^7` Mrkqvrq/ Q1 O1/ Nrw}/ V1 dqg Edodnulvkqdq/ Q1 +4<<7,1 Frqwlqxrxv xql0

yduldwh glvwulexwlrqv/ yro1 4/ ss1 :891 Qhz \run= Zloh|/ 5:505;41

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