Estimating Multiplicative and Additive Hazard Functions by Kernel Methods

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ESTIMATING MULTIPLICATIVE AND ADDITIVE

HAZARD FUNCTIONS BY KERNEL METHODS

*

Oliver Linton1

London School of Economics and Political Science Jens Perch Nielsen2

Codan, Copenhagen Sara van de Geer University of Leiden

Contents: Abstract 1. Introduction

2. The marker dependent hazard model 3. Estimation

4. Asymptotic Properties 5. Appendix

References

The Suntory Centre

Suntory and Toyota International Centres

for Economics and Related Disciplines

London School of Economics and Political

Science

Discussion Paper Houghton Street

No.EM/01/411 London, WC2A 2AE

February 2001 Tel.: 020-7504 7686

*

We thank three referees and the editor for helpful comments. 1 Thanks to the NSF for financial support. 2 Thanks to Bergiafonden for financial support.

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Abstract

We propose new procedures for estimating the univariate quantities of interest in both additive and multiplicative nonparametric marker dependent hazard models. We work with a full counting process framework that allows for left truncation and right censoring. Our procedures are based on kernels and on the idea of marginal integration. We provide a central limit theorem for our estimator.

Keywords: Additive model; censoring; kernel; proportional hazards; survival

analysis.

JEL No.: C14

©

by the authors. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source.

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dqg evE%ä 'S__æ'fevæE%æäer2æE%3æä zlwk eræE%3æä 'T&õ'æek'3&E%&ä

ï eS_

• é

Wkh edqgzlgwk udwh K å ?3•*E2on•ä _{jlyhv dq rswlpdo ^srlqwzlvh phdq vtxduhg huuru` udwh ri frq0}

yhujhqfh iru ek'3æE%æäc ekøE%äc dqg ek•E%ä ^l1h1/ wklv lv wkh vdph udwh dv wkh rswlpdo udwh ri frqyhujhqfh

lq rqh0glphqvlrqdo nhuqho uhjuhvvlrq hvwlpdwlrq/ vhh Vwrqh +4<;3,`1

Dq hduolhu yhuvlrq ri wklv sdshu zdv wkh vwduwlqj srlqw iru d qxpehu ri dfwxduldo dssolfdwlrqv/ vhh iru h{dpsoh Qlhovhq dqg Yrogvjddug +4<<9,1 Wkh zrunlqj sdshu yhuvlrq ri wklv sdshu/ Olqwrq/ Qlhovhq dqg ydq gh Jhhu +4<<<,/ frqwdlqv vrph qxphulfdo uhvxowv dqg frpsdulvrq zlwk d edfn?wwlqj h{whqvlrq ri rxu surfhgxuh1

8 Dsshqgl{

Zh ?uvw vwdwh dqg suryh wkuhh suholplqdu| uhvxowv wkdw duh qhhghg lq wkh surri ri rxu wkhruhp1 Ohppd 4 dqg Ohppd 5 duh xvhg wr hvwdeolvk wkh xqlirup frqyhujhqfh ri wkh slorw hvwlpdwru ek dqg wkh h{srvxuh eelq Ohppd 61 Wkh xqlirup frqyhujhqfh ri ee lv xvhg wkurxjkrxw wkh surri ri Wkhruhp 4 wr hvwdeolvk wkdw wkh uhpdlqghu whupv lq wkh h{sdqvlrq duh ri vpdoohu rughu1 Wkh xqlirup frqyhujhqfh ri ek dqg lwv ghulydwlyhv duh qhhghg wr hqvxuh wkdw +45, lv wuxh1

(15)

Zh xvh wkh iroorzlqj frqyhqlhqw qrwdwlrq= iru wzr udqgrp yduldeohv f?c t?c zh vd| wkdw f?* t?

zkhqhyhu f? ' t?E• n JRE•ää dv ? $ 4é

Suholplqdu| Uhvxowv

Zh ?uvw hvwdeolvk wkh iroorzlqj h{srqhqwldo lqhtxdolw|/ zklfk lv d yhuvlrq ri Ehuqvwhlq*v lqhtxdolw| iru vxpv ri lqghshqghqw pduwlqjdohv1 Wklv lv xvhg lq hvwdeolvklqj wkh xqlirup frqyhujhqfh ri ekc zklfk lv wkh vhfrqg uhvxow ri wklv vhfwlrqé

Ohw Elc Ic èä eh d suredelolw| wulsoh/ dqg ohw iI|j|Dfeh d ?owudwlrq vdwlvi|lqj wkh xvxdo frqglwlrqv1

Frqvlghu ? lqghshqghqw pduwlqjdohv ••c é é é c •?1 Ohw T2cð eh wkh suhglfwdeoh yduldwlrq ri •ð/ dqg ohw

T6cð eh wkh 6|û rughu yduldwlrq surfhvv ri •ð/ ð ' •c é é é c ?/ 6 ' ôc ec é é é1

Ohppd 41 Il{ f ÷ A é 4 dqg vxssrvh wkdw iru vrph IA0phdvxudeoh udqgrp yduldeoh -2?EA ä

dqg vrph frqvwdqw f ÷ g ÷ 4/ rqh kdv ? [ ð'• T6cðEAä é 6-₂ g632-2?EAäé +53, Wkhq/ iru doo @ : f/ K : f/ èh # _? [ ð'• •ðEAä è S @?_ -2?EAä é _2 $ é i Tdý_{2ESg n _}S2 ₂_äoé +54, Surri1 Gh?qh iru f ÷ b ÷ •*g/ ð ' •c é é é c ?/

~ðE|ä ' b•ðE|ä ý 7ðE|äc | è fc

zkhuh 7ð lv wkh frpshqvdwru ri

`ð ' •₂b2 ÷ •ðSc •ðS: n

[

r$u

Ei Tdbm{•ðErämo ý • ý bm{•ðErämä é

Wkhq i T ~ð lv d vxshupduwlqjdoh/ ð ' •c é é é c ? ^vhh wkh surri ri Ohppd 515 lq ydq gh Jhhu +4<<8,`1

Vr . i T ~ðEA ä é •c ð ' •c é é é c ?é Exw wkhq dovr . i TdS?ð'•~ðEAäo é •é Rqh hdvlo| yhul?hv wkdw ? [ ð'• 7ðEAä é b 2_-2 ?EAä 2E• ý bgäé Vr rq wkh vhw ø ' + _? [ ð'• •ðEA ä è S @?_ -2?EA ä é _2 , c 46

(16)

rqh kdv i Td ? [ ð'• ~ðEAäo è i TdbS ý b 2_{_}2 2E• ý bgäoé Wkhuhiruh/ èhEøä é i TdýbS n b2_2 2E• ý bgäoé Wkh uhvxow iroorzv e| fkrrvlqj b ' S _2_{n gS}é

Wklv uhvxow lv irupxodwhg iru ?{hg Ac dqg g pd| ghshqg rq A dqg ?é Li wkh frqglwlrqv ri Ohppd 4 krog iru doo Ac ?c wkhq lw fdq eh h{whqghg wr vwrsslqj wlphv ^vhh vhfwlrq ;15 lq ydq gh Jhhu +5333, iru uhodwhg uhvxowv`1

Lq wkh qh{w ohppd/ zh dvvxph dv lq wkh pdlq wh{w wkdw A lv ?{hg dqg ?qlwh/ dqg zulwhU 'U_fA é Zh dovr dvvxph wkdw wkh \?

ðE|ä h{lvw/ dqg duh erxqghg e| d +qrqudqgrp, frqvwdqw \ iru doo • é ð é ?

dqg f é | é Aé

Ohppd 51 Ohw X eh d erxqghg vxevhw ri U_n•_{/ dqg iru hdfk w 5 X/ frqvlghu lqghshqghqw}

suhglfwdeoh ixqfwlrqv }•cwc é é é c }?cw1 Vxssrvh wkdw iru vrph frqvwdqwv u?/ g?/ dqg 4?è •/ zh kdyh

m}ðcwE|ä ý }ðcìwE|äm é u?mw ý ìwmc sJo @,, wc ìw 5 Xc @?_ @,, ð è • @?_ | è fc +55,

m}ðcwE|äm é g?c sJo @,, w 5 Xc @?_ @,, ð è • @?_ | è fc • ? ? [ ð'• ] m}ðcwE|äm2_| é 42?c sJo @,, w 5 Xc @?_ @,, ? : •c u? é ?Dc sJo @,, ? : •c @?_ rJ6e D ÷ 4c +56, dqg g? é u ? *L} ?4?c sJo @,, ? : •é +57,

Wkhq iru vrph frqvwdqw Sf/ zh kdyh iru doo ä è Sf/ dqg ? : •

èh # týT wMX • s ?m ? [ ð'• ]

}ðcw_EùðE?äý \E?äð äm è ä4?

s *L} ?

$

é Sfi Tdýä *L} ?_S f oé

Surri1 Iurp Ohppd 4/ zh nqrz wkdw iru hdfk w 5 X/ @ : f dqg - : f èh # • s_?m ? [ ð'• ]

}ðcw_EùðE?äý \E?äð äm è @ @?_ _?• ? [ ð'• ] }2 ðcw_\E?äð é -2 $ +58,

(17)

é 2 i Tdý @2

2E@g??3•2 n -2äoé

Ohw " : f wr eh fkrvhq odwhu/ dqg ohw iw•c é é é c wùj ã X eh vxfk wkdw iru hdfk w 5 X/ wkhuh lv d

æEwä 5 •c é é é c ù/ vxfk wkdw mw ý wæEwäm é "1 Wkhq/ e| wkh Olsvfklw} frqglwlrq +55,/ rqh kdv

• s ?m ? [ ð'• ] E}ðcwý }ðcwæEwää_Eù E?ä ð ý \E?äð äm é s ?u?"E• n 7\äc

zkhuh 7\ lv dq xsshu erxqg iru \E?ä_ð E|ä/ • é ð é ?/ ? è •/ | è f1 Qrz/ lq +58,/ wdnh @ ' ä4_?s*L} ?*2/ dqg -2

? ' 42?7b/ zlwk 7b dq xsshu erxqg iru bE?äð E|ä/ • é ð é ?/

? è •/ | è f1 Pruhryhu/ wdnh " ' @*Es?u?E• n 7\ää1 Zlwk wkhvh ydoxhv/ zh ?qg

èh # týT wMX • s ?m ? [ ð'• ]

s *L} ? $ ' èh # týT wMX • s ?m ? [ ð'• ]

}ðcw_EùðE?äý \E?äð äm è 2@

$ é èh # 4@ æ'•cééécù • s ?m ? [ ð'• ] }ðcwæ_Eù E?ä ð ý \E?äð äm è @ $ é 2 i Td*L} ù ý @2 2E@g??3•2 n 42_?7boé

Ehfdxvh X lv d erxqghg/ ?qlwh0glphqvlrqdo vhw/ zh nqrz wkdw iru vrph frqvwdqw S•/

*L} ù é S•*L}E•_"äé

E| rxu fkrlfh " ' ä4_?s*L} ?*E2s?uE• n 7\ää/ dqg xvlqj frqglwlrq +56,/ zh vhh wkdw iru ä è • +vd|, dqg vrph frqvwdqw S2/

*L} ù é S2*L} ?é

Lqyrnlqj pruhryhu frqglwlrq +57,/ zh duulyh dw èh # týT wMX • s ?m ? [ ð'• ]

s *L} ? $ é 2 i TdS2*L} ? ý ä 2₄2 ?*L} ? HEä4_?s*L} ?g??3•2*2 n 42_?7bäo 48

(18)

é 2 i TdS2*L} ? ý ä 2_{*L} ?} HEä*2 n 7bäo é 2 i TdS2*L} ? ý ä *L} ?_H o é 2 i Tdýä *L} ?_•S oc zkhuh lq wkh odvw wzr vwhsv/ zh wdnh ä è 27b/ dqg ä è •SS21 Qrwh wkdw e| wkh frqwlqxlw| ri +55, dqg wkh erxqghgqhvv ri Xc wkh vwdwhphqw ri Ohppd 5 grhv qrw jlyh ulvh wr phdvxudelolw| sureohpv1 Qrwh pruhryhu wkdw +55,0+57, lpso| wkdw g?c 4?c dqg u?

fdqqrw eh fkrvhq lq dq duelwudu| pdqqhu1 Prvw lpsruwdqw khuh lv wkdw wkh vxs0qrup vkrxog qrw jurz wrr idvw dv frpsduhg wr wkh u2 qrup1

Ohppd 61 Vxssrvh wkdw wkh dvvxpswlrqv vwdwhg lq Wkhruhp 4 krog1 Wkhq/ iru dq| @ ' E@fc é é é c @_ä

zlwk m@m é o ý •c zh kdyh

+d, týT_%MUm(@_{eeE%ä ý (}@_{eE%äm ' ï}

å ýKo3ƒ@ƒün ïå qý_?K_n•n2ƒ@ƒ*L} ?

ü•*2r

+e, týT_%MUm(@_{ekE%ä ý (}@_{kE%äm ' ï}

å ýKo3ƒ@ƒün ïåqý_?K_n•n2ƒ@ƒ*L} ?

ü•*2r

é

Surri1 Zh zulwh (@_{eeE%ä ý (}@_{eE%ä ' (}@_{eeE%ä ý .(}@_{eeE%ä n .(}@_{eeE%ä ý e(}@_{E%äc d ghfrpsrvlwlrq}

lqwr d cvwrfkdvwlf* sduw (@_{eeE%ä ý .(}@_{eeE%ä dqg d celdv* sduw .(}@_{eeE%ä ý (}@_{eE%ä1 Qlhovhq dqg Olqwrq}

+4<<8/ , vkrzhg/ iru wkh fdvh @ ' fc wkdw .(@_{eeE%ä ý (}@_{eE%ä ' ï EK}o_{ä iru dq| lqwhulru srlqw %1 E|}

lghqwlfdo glvwulexwlrq zh kdyh .(@_{eeE%ä ' .}] A f ( @_g KE% ý fðEräätðErä_r ' ] (@_g KE% ý %âäeE%âä_%âc

zkhuh wkh odvw lqwhjudo lv ryhu [ 1 Lqwhjudwlqj e| sduwv ^xvlqj wkh idfw wkdw &Eæä_{E÷•ä ' f` dqg fkdqjlqj}

yduldeohv %â _{:ý$ ü ' E%}â_{ý %ä*Kc zh kdyh} .(@_{eeE%ä '}] _g KE% ý %âä(@eE%âä_%â ' ] d3•c•o_n•gEüä( @_{eE% n Küä_üc}

zkhuh wkh odvw htxdolw| iroorzv iru odujh hqrxjk ? ehfdxvh % lv dq lqwhulru srlqw dqg vr hyhqwxdoo| wkh udqjh ri ü frqwdlqv wkh vxssruw ri gé Iru qrwdwlrqdo vlpsolflw| zh qrz vxssrvh wkdw @ ' fé E| Wd|oru h{sdqvlrq eE% n Küä ý eE%ä ' o [ æ'• [ tSGƒSƒ'æå KS S-üS(SeE%ä n [ tSGƒSƒ'oå KS

(19)

zkhuh %W_{Eüä ' E%}W

fEüäc é é é c %W_Eüää duh lqwhuphgldwh ydoxhv vdwlvi|lqj m%Wæ ý %æm ÷ Kmüæmc iru æ ' fc é é é c _1

Wkhuhiruh/ zh kdyh .eeE%ä ý eE%ä ' [o æ'• [ tSGƒSƒ'æå KS S-(SeE%ä ] d3•c•o_n•ü S_{gEüä_ü} n [ tSGƒSƒ'oå KS S-] d3•c•o_n•i(

S_eE%W_{Eüää ý (}S_{eE%äj gEüäü}S_{_üc}

zkhuh _o [ æ'• [ tSGƒSƒ'æå KS S-(SeE%ä ] d3•c•o_n•ü S_{gEüä_ü '} Ko o-o [ æ'• Yo_eE%ä Y%o æ ] _• 3•r o_{&Erä_r} e| dvvxpswlrq D7/ zkloh ññ ññ ññ_tSGƒSƒ'oå[ K S S-] d3•c•o_n•ü

S_i(S_eE%W_{Eüää ý (}S_{eE%äj gEüä_ü}

ññ ññ ññ é [ tSGƒSƒ'oå KS S-] d3•c•o_n•ü

S_m(S_eE%W_{Eüää ý (}S_{eE%äm gEüä_ü}

' JEKo_äc

ehfdxvh ri wkh frqwlqxlw| ri (S_{eE%ä iru mSm ' o ^zh fdq dsso| grplqdwhg frqyhujhqfh ehfdxvh}

m(S_eE%W_{Eüää ý (}S_{eE%äm lv erxqghg iru doo %c üoé Lq frqfoxvlrq}

ññ ññ ñ.eeE%ä ý eE%ä ý Ko o-o [ æ'• Yo_eE%ä Y%o æ ] _• 3•r o_{&Erä_r}ññññ ñ' JEKoäc

dqg wklv uhvxow krogv xqlirupo| ryhu % 5 U ehfdxvh (S_{eEéä lv xqlirupo| frqwlqxrxv ryhu Ué Wkh eldv}

whup iru (@_{ee lv ri rughu K}o3ƒ@ƒ _{ehfdxvh wkh sduwldo lqwhjudwlrq fodlpv vrph ri wkh nhuqho prphqwv1}

Zh qrz wxuq wr wkh vwrfkdvwlf sduw ri eeE%äé Zh fodlp wkdw týT

%MU meeE%ä ý .eeE%äm ' týT%MU

ññ ññ ñ • ? ? [ ð'• ] _A

f dgKE% ý fðEräätðErä ý . EgKE% ý fðEräätðErääo _r

ññ ññ ñ ' ïå +ë_{*L} ?} ?K_n• ê_•*2, é +59,

Wkh srlqwzlvh uhvxow ^zlwkrxw wkh orjdulwkplf idfwru` lv jlyhq lq Qlhovhq dqg Olqwrq +4<<8,1 Wkh xqliruplw| ^dw wkh frvw ri wkh orjdulwkplf idfwru` iroorzv e| vwdqgdug dujxphqwv/ wkh nh| frp0 srqhqw ri zklfk lv wkh dssolfdwlrq ri dq h{srqhqwldo lqhtxdolw| olnh wkdw rewdlqhg lq Ohppd 5

(20)

deryh1 Zh zulwh eeE%ä ý .eeE%ä ' S?_ð'•lS

?cðE%äc zkhuh lS?cðE%ä ' l?cðE%ä ý .l?cðE%ä zlwk l?cðE%ä '

?3•UA

f gKE% ý fðEräätðErä_ré Qrwh wkdw lS?cðE%ä duh lqghshqghqw dqg phdq }hur udqgrp yduldeohv

zlwk 6? ' týT%cðmlS?cðE%äm ' S•?3•K3E_n•ä iru vrph frqvwdqw S•( wkxv 6? lv xqlirupo| erxqghg eh0

fdxvh ?K_n• _{$ 4 e| dvvxpswlrqé Iroorzlqj Qlhovhq dqg Olqwrq +4<<8,/ zh kdyh j}2

?ð ' ñ@hdlS?cðE%äo é

S2?3•K3E_n•ä iru vrph frqvwdqw S2é Ohw iîE%•c "uäc é é é c îE%uc "uäj eh d fryhu ri Uc zkhuh îE%ìc "ä lv

wkh edoo ri udglxv " fhqwhuhg dw %ìé Khqfh/ "u ' Sô*u iru vrph frqvwdqw Sôé Zh kdyh iru vrph frqvwdqw

Se týT %MU ññ ññ ñ ? [ ð'• lS ?cðE%ä ññ ññ ñ é 4@ •$ì$u ññ ññ ñ ? [ ð'• lS ?cðE%ìä ññ ññ ñ n 4@ •$ì$u_%MîE%týT_ì_c"ä ? [ ð'• ññlS ?cðE%ìä ý lS?cðE%äññ é 4@ •$ì$u ññ ññ ñ ? [ ð'• lS ?cðE%ìä ññ ññ ñ n Se"u K2_n2

xvlqj wkh gl>huhqwldelolw| ri &é Surylghg "u

u _?

Kô_nô_{*L} ?} $ fc +5:,

zh kdyh e| wkh Erqihuurql dqg Ehuqvwhlq lqhtxdolwlhv èh #v ?K_n• *L} ? •$ì$u4@ ññ ññ ñ ? [ ð'• lS ?cðE%ìä ññ ññ ñ : b $ é [u ì'• èh#ññññ_ñ[? ð'• lS ?cðE%ìä ññ ññ ñ : b u *L} ? ?K_n• $ n JE•ä é u [ ì'• i T 3 Cý b2 *L} ??K_n• 2S2_?K•_n• n_?KS_n•• b t *L} ? ?K_n• 4 D ' u [ ì'• i TíýE*L} ?b2_*2S₂ äìé

E| wdnlqj b odujh hqrxjk wkh odwwhu suredelolw| jrhv wr }hur idvw hqrxjk wr nloo uE?ä ' ?V _zlwk

V ' • n # n Eô_ n ôä*E2o n•ä iru vrph # : fc dqg wklv fkrlfh ri u vdwlv?hv +5:,é Wkh uhvxow iru jhqhudo @ iroorzv wkh vdph sdwwhuq> gl>huhqwldwlrq wr rughu @ fkdqjhv g wr g@ _{dqg dggv dq dgglwlrqdo}

edqgzlgwk idfwru ri rughu K32ƒ@ƒ_é

Wr hvwdeolvk +e, zh ?uvw zulwh ekE%ä ' eJE%ä*eeE%ä dqg kE%ä ' JE%ä*eE%äc zkhuh JE%ä ' kE%äeE%äé Wkhq e| wkh fkdlq uxoh zh kdyh

(@_{ekE%ä ý (}@_{kE%ä '} [

ƒSƒnƒ_ƒ'ƒ@ƒ

(21)

' [

ƒSƒnƒ_ƒ'ƒ@ƒ

VSc_i(SeJE%ä ý (SJE%äj(_e3•E%ä n

[

ƒSƒnƒ_ƒ'ƒ@ƒ

VSc_i(_ee3•E%ä ý (_e3•E%äj(SJE%ä n

[

ƒSƒnƒ_ƒ'ƒ@ƒ

VSc_i(_ee3•E%ä ý (_e3•E%äji(SeJE%ä ý (SJE%äj

iru yhfwruv Sc _ dqg srvlwlyh ?qlwh frqvwdqwv VSc_é E| ixuwkhu dssolfdwlrq ri wkh fkdlq uxoh dqg/ Ohppd

6+d,/ dqg wkh dvvxpswlrq wkdw eE%ä lv erxqghg dzd| iurp }hur rq Uc zh vhh wkdw wkh vhfrqg whup lv ri wkh vdph pdjqlwxgh dv (S_{eeE%ä ý (}S_{eE%äé Wkhuhiruh/}

týT %MU m( @_{ekE%ä ý (}@_{kE%äm é V} [ ƒSƒ$ƒ@ƒ týT %MU m( S_{eJE%ä ý (}S_{JE%äm n ï} åýKo3ƒ@ƒün ïå +ë *L} ? ?K_n•n2ƒ@ƒ ê•*2,

iru vrph srvlwlyh ?qlwh frqvwdqw Vc dqg lw vx!fhv wr hvwdeolvk wkh uhvxow iru wkh qxphudwru vwdwlvwlf (S_{eJE%ä ý (}S_{JE%ä rqo|1 Djdlq/ zh vkdoo mxvw zrun rxw wkh ghwdlov iru wkh fdvh @ ' f1 Wkh eldv}

fdofxodwlrq .eJE%ä ý JE%ä lv dv iru .eeE%ä ý eE%ä glvfxvvhg deryh1 Wkhuhiruh/ lw vx!fhv wr vkrz wkdw týT_%MUmeJE%ä ý .eJE%äm ' týT_%MUmT?E%äm lv wkh vwdwhg pdjqlwxgh/ zkhuh T?E%ä ' ?3•S?_ð'•U_fA gKE% ý

fðErää_EùðErä ý \ðErää1 Zh qrz dsso| Ohppd 5 zlwk }ðcwE|ä ' gKE% ý fðE|ääc w ' %c dqg X ' Ué

Frqglwlrqv +55,0+57, krog zlwk suredelolw| whqglqj wr rqh iru vrph frqvwdqw ò dqg= g? ' ò ü K3E_n•ä/

u? ' ò ü K32E_n•ä/ dqg 42? ' ò ü K3E_n•äe| wkh erxqghgqhvv dqg gl>huhqwldelolw| ri wkh nhuqho1 Lw qrz

iroorzv wkdw iru vrph frqvwdqw Sf zh kdyh iru doo ä è Sf dqg ? : •

èh % týT %MU v ?K_n• *L} ? mT?E%äm è ä & é Sfi TEýä *L} ?*Sfä dv uhtxluhg1

Lq wkh surri ri Wkhruhp 4 zh kdyh wr ghdo zlwk udqgrp yduldeohv ri wkh irup •|' ? [ ð'• ] _| f û E?ä ð Eüä_•ðEüäc

zkhuh •ð lv d pduwlqjdoh/ exw ûE?äð lv qrw d suhglfwdeoh surfhvv dffruglqj wr wkh xvxdo gh?qlwlrq1

Zh pxvw uhsodfh ûE?ä_ð e| hûE?ä_ð c zkhuh wkh hûE?ä_ð â_{r duh suhglfwdeoh surfhvvhv/ dqg wkhq wr dsso| vwdqgdug}

pduwlqjdoh wkhru| wr S?_ð'•U_f|hûE?ä_ð Eüä_•ðEüä1 Zh xvh wkh vroxwlrq surylghg e| Qlhovhq/ Olqwrq dqg

Elfnho +4<<;,1 Zh qhhg wkh iroorzlqj gh?qlwlrq 4<

(22)

Ghilqlwlrq D141 Wkh vhtxhqfh ri surfhvvhv iûE?ä_ð_•_cééécð_&j lv ri wkh ohdyh0&0rxw w|sh li ûE?ä_ð_•_cééécð_& lv suhglfwdeoh zlwk uhvshfw wr wkh ?owudwlrq jlyhq e|

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(23)

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(24)

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(25)

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(26)

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(27)

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(28)

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(29)

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(31)

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(33)

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