EDGEWORTH APPROXIMATION FOR MINPIN ESTIMATORS IN ...

1 downloads 0 Views 328KB Size Report
Cowles Foundation within the limits of the supply. ... stochastic expansion and give a theorem justifying order ? ... Our second contribution is to justify Edgeworth.
COWLES FOUNDATION FOR RESEARCH IN ECONOMICS AT YALE UNIVERSITY Box 2125, Yale Station New Haven, Connecticut 06520

COWLES FOUNDATION DISCUSSION PAPER NO. 1086

Note: Cowles Foundation Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. Requests for single copies of a Paper will be filled by the Cowles Foundation within the limits of the supply. References in publications to Discussion Papers (other than mere acknowledgment by a writer that he has access to such unpublished material) should be cleared with the author to protect the tentative character of these papers.

EDGEWORTH APPROXIMATION FOR MINPIN ESTIMATORS IN SEMIPARAMETRIC REGRESSION MODELS Oliver Linton November 1994

,#B,`+AO +jWAW 6+ WW ,5AWA+5 W 5,W+,A+W +,B+,55W #,w5

+ ,ji wA|NA $jVBi|6jA| Nu .WNAN6Wt vB,j PAjit|+e j QBjAe C hUF4h

`i i @4?i |i }ih Lh_ih @t)4T|L|U ThLTih|it Lu ti4T@h@4i|hU hi}hittL? it|4@|Lht |@| ihi LM|@?i_ M) |i }i?ih@* WW 4i|L_ _itUhMi_ ? ?_hit do `i _ihi @? Lh_ih ?3 t|LU@t|U i T@?tL? @?_ }i @ |iLhi4 t|u)?} Lh_ih ?3 _t|hM|L?@* @TThL 4@|L? Lu |i ,_}iLh| |)Ti

At T@Tih t M@ti_ L? @T|ih WW Lu 4) # |itt W L*_ *!i |L |@?! A a +L|i?Mih} @?_  a U!i* uLh tiu* _tUttL?t @?_ @ ULi_|Lh uLh i*Tu* UL44i?|t W L*_ @*tL *!i |L |@?! B*i?@ 4it uLh hi|)T?} |t 4@?tUhT|

4

 YA|iNaW|NA 5i4T@h@4i|hU 4i|L_t @**L L?i |L LM|@? ThiUti it|4@|it Lu Uih|@? !i) ^@? ||it |L| u**) tTiUu)?} |i ThLM@M*|) _t|hM|L? Lu |i _@|@c tii uLh i @4T*i ?_hit dc2oc U!i* dDoc U!i* i| @* d.oc O@h_*i @?_ 5|L!ih d.oc w?|L? d2eoc @?t! d2Soc ii) d oc Li**c 5|LU! @?_ 5|L!ih d .oc @?_ +LM?tL? d Hc bce2ce o( uLh hiitc tii ii) d 2o @?_ +LM?tL? defo Ai ?L|L? Lu ThiUti ti_ M) |iti @|Lht hit|t L? €ht| Lh_ih @t)4T|L|U |iLh) uLh *@h}i t@4T*i t3it Aihi @hi |L 4@? ThLM*i4t | |t |iLh) 6ht|*)c |i @t)4T|L|U _t|hM|L? U@? ThL_i @ TLLh @TThL 4@|L? |L |i @U|@* _t|hM|L? Lu |i t|@|t|U UL?t_ihi_ h@U|U@* i Tihi?Ui t}}it|t |@| it|4@|Lh TihuLh4@?Ui _i|ihLh@|it | |i ?4Mih Lu ?t@?Ui T@h@4i|iht Mi?} it|4@|i_ 5?Ui ti4 T@h@4i|hU it|4@|Lht |@!i @UUL?| Lu @? ?€?|i _4i?tL?@* ?t@?Ui T@h@4i|ihc L?i 4t| i TiU| @ TL|i?|@**) *@h}i t4@** t@4T*i ULt| |@| t ?L| hiiU|i_ ? |i *4|?} _t|hM|L?  5iUL?_*)c @ *@h}i ?4Mih Lu ULUit @i |L Mi 4@_i ? UL?t|hU|?} |i it|4@|Lht @ML| U |t @t)4T|L|U |iLh) ThL_it *||*i }_@?Ui Aiti ULUit U@? tMt|@?|@**) @giU| |i 4@}?|_i Lu it|4@|Lht @?_ |it| t|@|t|Ut `i t}}it| t?} }ih Lh_ih @TThL 4@|L?t |L LihUL4i |iti ThLM*i4t Aiti 4i|L_t @i uL?_ UL?t_ih@M*i @TT*U@|L? ? iUL?L4i|hUtc tii ?|ih @*@G 5@h}@? de.oc **Tt d Dc Soc @?_ +L|i?Mih} deeceDo W? L|ih Lh!c w?|L? d22c2 oc i @i _ii*LTi_ _i|@*i_ RtiUL?_ Lh_ih 4L4i?| @TThL 4@|L?t uLh @hLt ti4T@h@4i|hU it|4@|Lht ? hi}hittL? 4L_i*tc ?U*_?}G |i T@h|@**) *?i@h 4L_i* UL?t_ihi_ ? +LM?tL? d bo @?_ |i i|ihLt!i_@t|U *?i@h hi}hittL? UL?t_ihi_ ? +LM?tL? d Ho Aiti @TThL 4@|L?t ^@?|u) |i t4@** t@4T*i ULt| Lu |i ti4T@h@4i|hU it|4@|L? t|h@|i}) ? |ih4t Lu @ ui ?|ihThi|@M*i ^@?||it At ThL_it ?uLh4@|L? @ML| |i ULUi Lu 4i|L_c i} !ih?i* ihtt ?i@hit| ?i}MLhc @?_ |i ULUi Lu t4LL|?} T@h@4i|ih `i tL |@| |iti @TThL 4@|L?t U@? Mi _ii*LTi_ ? @ 4Lhi }i?ih@* uh@4iLh!c @?_ ?_ii_ |@| _t|hM|L?@* @TThL 4@|L?t U@? @*tL Mi @*_@|i_ `i i @4?i |i 5

e ihi C e t @ Thi WW it|4@|Lh h |@| 4?43it |i h@?_L4 Uh|ihL? [? E c Cc

*4?@h) ?L?T@h@4i|hU it|4@|i Lu @ iU|Lh Lu hi}hittL? u?U|L?t C @?_ ? t t@4T*i t3i Ai UL?|h@t| [? t ULti? |L i?thi |@| h tL _i€?i_ t UL?tt|i?|G uLh i @4T*ic [? E c C UL*_ Mi |i ?i}@|i Lu |i t@4T*i *L} *!i*LL_ u?U|L? At 4i|L_ Lu }i?ih@|?} it|4@|Lht Lu t UL?t_ihi_ ? ?_hit do @?_ U!i* i| @* d.oc ihi? s UL?_|L?t @hi }i? ?_ih U h t ? UL?tt|i?| @?_ @t)4T|L|U@**) ?Lh4@* `i 4@!i |L UL?|hM|L?t 6ht|*)c i _ihi @? Lh_ih ?3 t|LU@t|U i T@?tL? s uLh |i t|@?_@h_3i_ it|4@|Lh |h ' ?E h  c i i €?_ h@?_L4 @h@M*it |hWW @?_ - tU |@| |h ' |hWW n -c ihi |i hi4?_ih |ih4 - _Lit ?L| @giU| |i _t|hM|L? Lu |h |L Lh_ih ?3 Ai *i@_?} |ih4 |hWW t R E @?_ @t ML?_i_ 4L4i?|t |L tL4i

t|@M*) } Lh_ih W? u@U|c |hWW t @ TL*)?L4@* ? @ iU|Lh Lu i}|i_ Lt|@|t|Ut Lu

Lh_ih T |L 6  2c ihi |i Lh_ih 6 |@| t ?ii_i_ t _i|ih4?i_ T@h|*) M) |i h@|i e  i}|i_ Nt|@|t|U Lu Lh_ih 6 t @ h@?_L4 @h@M*i Lu |i uLh4 Lu UL?ih}i?Ui Lu C

X6 '

? [  cc6 '

? 6 )Ef c c f6 c

E

ihi i? 6 j @hi _i|ih4?t|U i}|t t@|tu)?} Uih|@? Lh_ih Lu 4@}?|_i UL?_ |L?t tTiU€i_ Mi*Lc *i f c c f6 @hi ?_iTi?_i?| h@?_L4 @h@M*it( tii wii d2fo uLh @ hii Lu L t|@|t|U 4@|ih@* h tiUL?_ UL?|hM|L? t |L t|u) ,_}iLh| h Lhi |)Ti _t|hM|L?@* @TThL 4@|L?t uLh |i _t|hM|L? Lu |hWW c @?_ i?Ui Lu |

ThiUti*)c i }i UL?_|L?t ?_ih U Uih|@? X TLttitt @? Lh_ih ?3 ,_}iLh| _t|hM|L?@* @TThL 4@|L?  _i*|@4i|L_ @h}4i?| |i? i?thit |@| |hWW |ti*u @t @? Lh_ih ?3 ,_}iLh| @TThL 4@|L? h Lh! L? Nt|@|t|Ut i |i?_t @**@ih|c a@?tti? @?_ Vih@hMihi!i dHo L _i@* L?*) | ?i}|i_ Nt|@|t|Ut Lu tiUL?_ Lh_ih s e |L Mi UL?tt|i?| @| @ h@|i Mi||ih |@? 6Lh ? UL?tt|i?U)c i }i?ih@**)2 ?ii_ C ?*e At hi^hi4i?| t t@|t€i_ M) @ ?4Mih Lu ?L?T@h@4i|hU it|4@|Lht Lu hi}hit tL? u?U|L?t Lh |ih _ih@|it ?_ih t4LL|?itt @?_ _4i?tL?@*|) hit|hU|L?tc tii  **ih d2bo @?_ O@h_*i dHo `i _L ?L| }L ?|L |i _i|@*t Lu |i ?L?T@h@4i|hU it|4@|L?c @?_ hiuih |i hi@_ih |L |iti hiuihi?Uit uLh 4Lhi _tUttL? 6

`i Lh! |hL}L| | @ € i_ _it}? uLh UL?i?i?Ui At 4@) Lu|i? Mi t|€i_ M) @?U**@h|) UL?t_ih@|L?t W? 5iU|L? 2 i _i€?i |i t@4T*?} tUi4i i t@** Mi @?@*)3?}c @?_ }i @ ?4Mih Lu i @4T*it W? 5iU|L? i _ii*LT |i t|LU@t|U i T@?tL? @?_ ihu) |@| h W? 5iU|L? |i TL*)?L4@* t|@|t|U |hWW @t |i t@4i _t|hM|L?c |L Lh_ih ?3c @t |

e i ThLi |@| i}|i_ L t|@|t|Ut Lu Lh_ih @i @ @*_ Lh_ih ?3 ,_}iLh| @TThL 4@|L? ** ThLLut @hi UL?|@?i_ ? TTi?_Uit W @?_ WW S

L|@|L?t 6Lh @?) iU|Lh % ' E% c c %& A c *i| m%m ' d &' %2 o*2 Mi |t ,U*_i@? ?Lh4c *i uLh @?) hi@* t)44i|hU 4@|h c *i| b4?E @?_ b4@ E Mi |i t4@**it| @?_ *@h}it| i}i?@*it hitTiU|i*) `i ti E @?_ xE uLh |i t|@?_@h_ ?Lh4@* _i?t|) @?_ U_u hitTiU|i*) 6?@**)c uLh @ €?|i ti| 7c *i| r7 Mi |t U@h_?@*|)

2 tt6V|NAt BAa .!B6V,jt Ai LMtihi_ _@|@ if j?' @hi T@h||L?i_ @t f ' EtA c ~A A c ihi |i _iTi?_i?| @h@M*it t @hi g+  c @?_ |i hi}hittLht ~ @hi g5   Ai _@|@ @hi ?_iTi?_i?| @UhLtt c @?_ @hi _itUhMi_ M) |i uL**L?} UL?_|L?@* wiMit}i _i?t|) u?U|L?tG st ƒ~ E+( c CE~ c  ' c 2c c ?c ihi t @    iU|Lh Lu ?!?L? T@h@4i|ihtc *i C ' ECc C2 c c Cu A t @? u   iU|Lh Lu ?!?L? hi}hittL? u?U|L?t h LMiU|i t |L LM|@? it|4@|it Lu e Lu C }i? Thi*4?@h) ?L?T@h@4i|hU it|4@|it C e ? |i tTiU@* `i UL?t_ih @? it|4@|Lh Lu |@| 4?43it |i Uh|ihL? [? E c C

U@ti ihi [? E c C t @ t@4T*i @ih@}i  U | L*_ Mi ?_ih |i ?_iTi?_i?| t@4T*?} @tt4T|L? W? u@U|c i @U|@**) Lh! | tL*|L?t h Lu |i @ttLU@|i_ €ht| Lh_ih UL?_|L?tG ?3

? Y[ [ ' Y Z

e Ef ( hc CE~   ' fc Z ' c 2c c 

E2

?_hit do @?_ U!i* i| @* d.o ML| _ihi |i €ht| Lh_ih *4|?} _t|hM|L? Lu hc ? t*}|*) _gihi?| ti||?}t u@?3@}* d o @?@*)3it |i }ih Lh_ih ThLTih|it Lu tU 7

it|4@|Lht ? @ T@h@4i|hU UL?|i |c i ihi C t !?L? `i i |i?_ t |iLh) |L i?UL4T@tt |i ti4T@h@4i|hU U@ti

, @4T*it Ai uL**L?} i @4T*it €| ? |L Lh uh@4iLh! @?_ @hi Lu T@h|U*@h ?|ihit| uLh iUL?L4i|hU @TT*U@|L?tG

, @4T*i  OL4Lt!i_@t|U L?*?i@h +i}hittL? t ' }E~  n L c  ' c 2c c ?c ihi L t @ iU|Lh Lu _ t|LU@t|U ihhLht | 4i@? 3ihL @?_ UL@h@?Ui 4@|h Pc *i ~ @hi € i_ hi}hittLht Ai €?|i _4i?tL?@* T@h@4i|ih Lu ?|ihit| t Pc *i }E t Lu ?!?L? u?U|L?@* uLh4  tU@*@h ihtL? Lu |t it|4@|L? ThLM*i4 t @?@*)3i_ ? @hhL** @?_ O@** dfo ,t|4@|it Lu P @hi ThL_i_ M) @?) tL*|L? Lu ?

[ ?3 sdit  }eE~ jit  }eE~ jA  Po ' fc '

ihi |i !?L? sE t tU |@| .dsEL LA  Po ' f , @4T*i 2 @h|@**) w?i@h +i}hittL? + ' q A t2 n wE~  n 0 ( t2 ' }2E~  n # c  ' c 2c c ?c ihi ~ @hi € i_ ? hiTi@|i_ t@4T*itc *i 0 @?_ # @hi 4i@? 3ihLc ?_iTi?_i?| @?_ 4|@**) ?_iTi?_i?| h@?_L4 ihhLht Ai €?|i _4i?tL?@* T@h@4i|iht Lu Th4@h) ?|ihit| @hi qc *i }Ec ihi }E~   .d+ oc @?_ |i iU|Lh }2 E @hi ?!?L? hi}hittL? u?U|L?t +LM?tL? d bo @?_ i? d2o it|@M*ti_ |i €ht| Lh_ih @t)4T|L|U |iLh) uLh @hLt it|4@|Lht Lu q ?_ih |i @*|ih?@|i t@4T*?} tUi4i ? U S EtA c ~A A @hi _ 5|@M*i it|4@|Lht Lu q tL*i ?3 ?' sd#e EDe  #eA qo ' fc ihi

.dsE# 0 o ' fc *i #e ' t2  }e2E~  @?_ De ' +  }e E~ c ihi }e @?_ }e2 @hi Thi*4?@h) ?L?T@h@4i|hU it|4@|it Lu } @?_ }2 8

, @4T*i  Oi|ihLt!i_@t|U w?i@h +i}hittL? + ' q A ~ n 0 jE~ c  ' c 2c c ?c ihi 0 @hi _ 4i@? 3ihL @?_ @h@?Ui L?ic *i ~ @hi € i_ ? hiTi@|i_ t@4T*it Ai €ht| Lh_ih @t)4T|L|U |iLh) uLh ui@tM*i Bw5 it|4@|Lht Lu q @t it|@M*ti_ M) @hhL** dboc @?_ tMti^i?|*) @_@?Ui_ M) +LM?tL? d Ho ?_ih |i @tt4T|L? |@| S E+ c ~A A @hi _ `i U@? UL?t_ih qe |@| t@|t€itG ?3 ?' sEje32 ~ E+  qeA ~  ' fc

ihi .dsEj3~ 0 o ' fc *i je2 ' }e2E~   }e2E~  it|4@|it j2c ihi }e2E~  @?_ }eE~  @hi ?L?T@h@4i|hU it|4@|it Lu .d+2m~ o @?_ .d+ m~ o

L?T@h@4i|hU ,t|4@|L? `i ti ?L?T@h@4i|hU it|4@|Lht |@| @hi *?i@h ? tL4i _iTi?_i?| @h@M*ic i e & E~  ' C 

[ 

& )& Et c  ' c c ? ( & ' c c uc  

E22

& j @hi _i|ih4?t|U uLh tL4i u?U|L?t )& E tU |@| .d)& Et o ' C& E~ c ihi i

t4LL|?} i}|t _iTi?_?} L?*) L? ~c ~2 c c ~?  At ?U*_it 4Lt| UL44L?*) i4T*L)i_ it|4@|Lht Lu hi}hittL? u?U|L?t @?_ |ih _ih@|itc tii O@h_*i @?_ w?|L? dSo W| t UL?i?i?| |L hi@hh@?}i E22 @t e&  C e & E~   C& E~  ' T n  c K   & & 

ihi T& '

S & & S & & & & & &   L | L ' ) Et   C E~ c @?_ & '   C E~   C E~ 

`i 4@!i |i uL**L?} @tt4T|L?tG  E 6Lh i@U &c |ihi i t|t K 4 @?_ k& : fc tU |@| 4@ $? m& m K?k&  & 9' fj ' E 6Lh i@U &c |ihi i t|t  & @?_ >& : f tU |@| riEc  G  &m   E?n2>&  @?_ ?2>& m &

E wi| l 4?d4?& i>& jc 4?& ik& joc | l: *e 9

2 Ai ti^i?Ui Lu u iU|Lht iL j?' @hi ?_iTi?_i?| @?_ 4i@? 3ihL wi| P  .dL LA oc |i? P t ML?_i_ @@) uhL4 3ihL @?_ ?€?|) 6h|ih4Lhic tTD .dmL ma o 4 uLh tL4i a : e +i4@h!G L?_|L?t E  E @hi t@|t€i_ uLh !ih?i* i}|t i? ~ @t ML?_i_ tTTLh| @?_ C t tUi?|*) t4LL| 5ii uLh i @4T*i  **ih d2bo wi| J Mi |i tT@Ui Lu @**L@M*i u?U|L? @*i_ ?t@?Ui T@h@4i|iht @?_ _i€?i |i ?Lh4 n  n? L? J M) nMn? ' 4@

 $u

4@ mM E~ m $?

`i 4@!i ti Lu |i uL**L?} *i44@G wi44@ 2G wi| *e  w l tt4i |@|  2 L*_c @?_ |@| ac lc @?_  t@|tu) a : E n 2*El  w Ai?c uLh tL4i B : fc e  Cn : U*L}?o ' JE?32 c hd?wnB nC ?

;S : f

6Lh i @4T*iG i? w ' *ec  ' *2 @?_ l ' * c i 4t| @i a  H wi44@ 2 t ThLi_ ? TTi?_ W 5ii +LM?tL? deo uLh t4*@h hit*|t ? @ T@h@4i|hU UL?|i | Ai 4TLh|@?Ui Lu |t *i44@ t |L it|@M*t i? L?i U@? hit|hU| @||i?|L? |L @ t4@** ?i}MLhLL_ Lu C

@h@4i|hU ,t|4@|L? `i ?L UL?t_ih |i ThLTih|it |@| [ 4t| TLttitt `i ?ii_ tL4i 4Lhi ?L|@|L? 6Lh |L_4i?tL?@* @hh@)t E4@|hUit  ' E@& c i _i?L|i |i c & c uLh tL4i > *2 Ai hit*|?} @TThL 4@|L?t i _itUhMi_ @t RtiUL?_ Lh_ih h ThTLti ihi t |L t|u) @? Lh_ih ?3 _t|hM|L?@* @TThL 4@|L? M@ti_ L? Lh_ih ?3 @TThL 4@|L?t |L @** |i hi*i@?| U4*@?|t W? u@U|c i Lh! | |i tTiU@* U@ti uLh U |i €ht| uLh U4*@?|t tUi 5ii |i i @4T*it Mi*L uLh @? i T*@?@|L? Lu |i T*@tM*|) Lu |iti UL?_|L?t Ai |h?U@|i_ t|@|t|U |hWW t @ TL*)?L4@* ? |i @hh@) 7er  W? Lh tTiU@* U@tic rc 7 er @?_ 7er 7e?Z ?Zk ?Zk @hi iU|Lht Lu 3ihL 4i@? i}|i_ Lt|@|t|Ut Lu Lh_iht T |L 

T@h| uhL4 @ _i|ih4?t|U M@t |ih4 |@| U@? Mi @?_*i_ M) @?@*)|U 4i|L_tc |iti U@? Mi h||i? ? |i uLh4 ?

[[

'

:

[ " ' ?3*2 }Ef  n ?3l

4 )Ef c f  n ?31

[[[ &::

Z& DEf c f c f& c Ee

ihi 1 : l L| ) @?_ D @hi Tih4|@|L? ?@h@?|c i )E%c + ' )E+c % @?_ DE%c +c 5 ' DE5c %c +c | .d)Ef c f omIo ' .dDEf c f c f& mIo ' fc ihi I _i?L|it @?) ThLTih tMti| Lu i|ih if c f j Lh if c f c f& jc *i .d}Ef o ' fc c c & ' c 2c c ? Atc |iti |hii tiT@h@|i TiUit @hi 3ihL 4i@? @?_ 4|@**) Lh|L}L?@* M) UL?t|hU|L? Oihic i4 j @?_ iZ& j @hi ti^i?Uit Lu ?L?h@?_L4 i}|t _iTi?_?} & j ? tU @ @) |@| L? i

SS SSS : 4 )Ef c f  @?_ &:: Z& DEf c f c f&  @hi

ML| RE W? |t hiThiti?|@|L?c |i _LM*i @?_ |hT*i t4t @hi !?L? @t _i}i?ih@|i i}|i_ L t|@|t|Ut Lu Lh_iht |L @?_ |hii hitTiU|i*) h @h}4i?| t M@ti_ L? it|@M*t?} |i uL**L?}G E,#B, Ai iU|Lh " ' E"c "2c c "o A TLttittit @? Lh_ih ?3 ,_}iLh| @T ThL 4@|L? E,#B,2 Wu " TLttitt @? Lh_ih ?3 ,_}iLh| @TThL 4@|L?c |i? E" _Lit |LLc ihi E t @ TL*)?L4@* | ML?_i_ ULiUi?|t ,tti?|@**) |t t|h@|i}) Thti_ ? @||@U@h)@ @?_ BLt d o OLiihc |i) _i@* L?*) | L t|@|t|Ut Lu Lh_ih c i t?}*i t4tc uLh U E,#B, @t Mii? it|@M *ti_ ?_ih i@! UL?_|L?t Ai) @*tL it|@M*t E,#B,2 Aih @h}4i?| t|u)?} 49

E,#B,2 U@? Mi i4T*L)i_ ihic @*|L} @ h}LhLt ThLLu Lu |t L*_ hi^hi UL? t_ih@M*) 4Lhi Lh! `i ThLi E,#B, uLh Lh 4Lhi }i?ih@* U*@tt Lu t|@|t|Ut +@?_L4 @h@M*it Lu |i |)Ti Ee @hi hi*@|i_ |L t|@?_@h_ L t|@|t|Ut Lti ThLTih|it @hi i** it|@M*ti_ W? T@h|U*@hc U!i*c BL|3ic @?_ @? ~i| dSo @?_ @**@ih|c a@?tti? @?_ Vih@hMihi!i dHo }i UL?_|L?t ?_ih U |i uLh4@* ,_}iLh| @TThL 4@|L? t @*_ uLh t|@?_@h_ L t|@|t|Ut Lu Lh_ih 2 5|@?_@h_ L t|@|t|Ut @i ULhhiU|L? |ih4t |@| @hi R E?3*2c *i Lh i}|i_ L t|@|t|Ut @i ULhhiU|L? |ih4t |@| @hi R E?3l c ihi ? }i?ih@*

l *2 @?_ i?Ui |i ULhhiU|L? |ih4 ? Ee t Lu *@h}ih Lh_ih AihiuLhic i 4t| i |i?_ |t ThiLt Lh! i*L i it|@M*t |@| |i uLh4@* Lh_ih ?3 ,_}iLh| @TThL 4@|L? |L |i _t|hM|L? u?U|L? Lu " ?_ii_ @t ihhLh Lu Lh_ih ?3 `i ThLi Lh |iLhi4 uLh tU@*@h @*i_ _ h@?_L4 @h@M*it Ai i |i?tL? |L ?L?_i?|U@**) _t|hM|i_ @?_ 4*|@h@|i t|@|t|Ut t @*tL @*_ ?_ih |i } *ii* @tt4T|L? #D Ai ThLLu Lu |tc Liihc hi^hit tL4i@| 4Lhi ?L|@|L? |@? i @*hi@_) @ic @?_ i *i@i |t uLh u|hi Lh!e  wi| ? Er ' .der"o @?_ #Er ' .der}Ef o ihi  '

s

c @?_ @tt4i

# riEc  G 4 9' fj ' E?n2l ( riEc c & G Z& 9' fj ' E?n21  Aihi t @ €?|i UL?t|@?| B tU |@| ?*2nl m4 m B @?_ ?*2n1 mZ& m B `i hi^hi |@| l : *H @?_ 1 : D*H #2 .dm}EfmSo 4( .d)Efc f2mSo : 4( .dmDEfc f2 c f mSo # *4 tTƒrƒ c uLh tL4i > *2c 4@) Mi tUi?|*) **4?@|?} Aiti tiUL?_ Lh_ih @TThL 4@|L?t U@? ui@tM*) Mi UL?t|hU|i_c tii w?|L? d2c22c2 o @?_ ThL_i tiu* ?uLh4@|L? @ML| |i t4@** t@4T*i Mi@Lh Lu |i ti4T@h@4i|hU it|4@|Lh |@| t *iu| L| Lu |i €ht| Lh_ih |iLh) h Lh! _i4L?t|h@|it |@| |i _t|hM|L?@* @TThL 4@|L? _ihi_ uhL4 |iti @t)4T|L|U 4L4i?|t t @*_ Ai _t|hM|L?@* @TThL 4@|L?t U@? Mi i4T*L)i_ |L 4ThLi Uh|U@* @*itc tii itih @?_ 5T@_) d o ?i U@? @*tL _tUtt }ih Lh_ih iUi?U) ? @ 4Lhi h}LhLt ti||?} |@? t ThL_i_ M) |i @t)4T|L|U 4L4i?|t @*L?i ? |t TL?|c i 4i?|L? |@| |i U*@ttU@* hit*|t Lu u@?3@}* d eo hi}@h_?} |i 54

hi*@|L?tT Mi|ii? €ht|c tiUL?_c @?_ |h_ Lh_ih iUi?U) ? T@h@4i|hU 4L_i*t _L ?L| L*_ i? C @t |L Mi it|4@|i_ Ai tiUL?_ Lh_ih ThLTih|it Lu h @hi _L4?@|i_ e `i U@??L| ii? ?uLh4*) h@?! _gihi?| hi}hittL? u?U|L? it|4@|Lht @UULh_?} M) C

|L 4i@? t^@hi_ ihhLh AihiuLhic |ihi ** ?L| Mi @ ?uLh4*) E? C Mit| it|4@|Lh Lu 

55

6AA,5  +LM?tL? de o it|@M*tit ihh),tti? ML?_t uLh @ U*@tt Lu ti4T@h@4i|hU it| 4@|Lht Aiti tL |@| |i ELht| TLttM*i h@|i @| U |i ?Lh4@* @TThL 4@|L? t @TThL@Ui_ t t*Lih |@? |i ?3*2 h@|i t@* uLh T@h@4i|hU ThLUi_hit Ai L?|i @h*L i_i?Ui Thiti?|i_ ? Oti @?_ @?t! dbo @?_ ii) d o t}}it|t |@| Tih uLh4@?Ui Lu |iti it|4@|Lht 4@) Mi ^|i }LL_ ? @ h@?}i Lu UhU4t|@?Uitc M| tii @*tL 5|LU! debo @?_ 5|L!ih dDfo uLh *itt @||ih?} hit*|t 2 *|L} |t t Lu|i? ?L| hi^hi_ ? Uih|@? @_@T|i t|@|L?t rc 7 er @?_ 7er   L|i |@| i 4@!i i T@?tL?t Lu _gihi?| *i?}|t uLh 7e?k ?Zk ?Zk

e W @4 @@hi Lu L?*) L?i T@Tihc BL|3i dDoc |@| _i@*t | ,_}iLh| @TThL  4@|L? uLh 4*|@h@|i L t|@|t|Ut ?_ih Th4|i UL?_|L?t W? u@U|c |t T@Tih L?*) it|@M*tit |i @*_|) Lu @? Lh_ih ?3*2 ,_}iLh| @TThL 4@|L?

56

,#Wj W hLLu Lu wi44@ 2G ) |i L?uihhL? ?i^@*|)c e  Cn : S *L} ?o  hd?i nC ?

? u [ [ &' '

e & E~   C& E~ m : S *L} ?oc hd?i mC  

uLh @?) i : f | e & E~   C& E~ m : S *L} ?o  hd?i mT m : S *L} ?*2o n hd?i m m : S *L} ?*2oc hd?i mC   & &

M) |i |h@?}*i ?i^@*|) Ai?c M) @tt4T|L? 2Ec |i tiUL?_ |ih4 t 3ihL uLh *@h}i ? *tLc T& ' R E?3>& c @?_ M) |i @h!L ?i^@*|)c hd?i mT& m : S *L} ?o 

.dmT& ma o ai ?  ES *L} ?a

) |i @hU?!iU3 ~)}4?_ ?i^@*|)c .dmT& ma o t ?uLh4*) Lu Lh_ih ?3>& a  tii +LM?tL? d Hoc wi44@ . AihiuLhi ? u [ [ &' '

e & E~   C& E~ m : S *L} ?o ' JE?32 c hd?i mC  

ThL_i_  n ai  a l 2 U L*_t ?_ih |i t|@|i_ UL?_|L?t hLLu Lu AiLhi4 G

W? |i ti^i* *i| S @?_ b Mi }i?ihU UL?t|@?|t Ai

UL?_|L? 8 UL?tt|t Lu tiih@* T@h|t U @hi t|@|i_ ? |i ThLLu `i 4@) h|i '? E| ' fc @t |'

[ 

 E|  )E|c

E

Zk r Zk r k Zk Zk r ihi i E|j UL?tt|t Lu [? 7e?k c ?3l [? 7e?Zk | c ?3*2 [? [?Zk |k| c  2 ?E*2nl  [? 7e?Zk |k| c Zk erc er c e r @?_ @?_  - ?3[? [?ZkB |k| |B c @t i** @t *?i@h UL4M?@|L?t Lu -'Z c k Zk Zk

er  ZkB

wi| M? E f ( S ' i| G m|m S *L} ?jc @?_ @tt4i |@| | 5 M? E f( S uLh tL4i S `i tL Mi*L |@| hdm)E|m S *L} ?o '   JE?3 c 57

E2

i ) 4@Tt M? E f( S ?|L |ti*u | } ThLM@M*|) W? u@U|c i it|@M*t |@| uLh tL4i UL?t|@?| Sc hdm E|m : S *L} ?o ' JE?3c uLh i@U  5?Ui hdm

E 

S S    E|m : S *L} ?o   hdm? E|m : S *L} ?oc uLh tL4i UL?t|@?|

S M) hiTi@|i_ @TT*U@|L? Lu |i |h@?}*i ?i^@*|)c E  t tUi?| uLh E2 |L L*_ Ai?c t?Ui )E| t @ UL?|?Lt u?U|L? Lu | |@| 4@Tt @ UL4T@U| tMti| Lu W+R ?|L |ti*uc i U@? @TT*) hLih 

L|i |@| +

,

?

[  _   ?3 m[k(&, mn0 : S?2BEn0E*L} ?3En0 c '

e 5 uLh tL4i S @?_ uLh tL4i Bc 0 : f 5?Ui hdC * QC o ' JE?3c M) wi44@ 2c | *e

tUit |L it|@M*t |@| %

&

?

[ h ?3 m[k(&, mn0 : S?2BEn0 E*L} ?3En0 ' JE?3

ED

'

6?@**)c M) @h!L