Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://lms.cs.msu.su/files/tmp/Dzhugan_diplom.pdf Äàòà èçìåíåíèÿ: Mon Nov 21 22:58:15 2005 Äàòà èíäåêñèðîâàíèÿ: Sat Mar 1 20:19:15 2014 Êîäèðîâêà:

(

&' &% $#"! ¡¨ ¨¡©¥ ©¥¤©© ¨¢¡§¸¦ ¥ ¥¤¸¢¡


s$#" (( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )))))))))))))))))) )))))))))))))))))) ( t h W D '

$ )) )) ))

) ) )

)))))))))))))))))))) ))))))))))))))))))))) )))))))))))))))))))))

$s r $q& q#qs$s s$"" ' ) ) ) 1v7CYQ6421E 652C175v6y1E X)X ) ) ) ) 652C175v6y1` 61BA695714A ()X ) ) ) ) ) y146` x82wvC`712QC4wv6 )X s$#" $r $uq(p i gHHGfRedcba CBA57 876` 68242676@1Y X)( ) ) ) ) ) ) CVUTS7QRGQPIH9GF2 CBA57 C376E ()( 7 ) ) ) ) ) B"A5& 1@%121$8"65!4210) # (( © ¸¨§¦¥¤¸¢¡


( 57 y146` x822C 1426`¦74A25 1@62 ©5!3y1© 26¨ 552626`5 57 3v6§471¤ BA5¤7 w43vA598 4uv)11¤ 149 6526v6y67¤AC7 61276`1@12` 612wv1517¤¤ x6!6`5 5 82595v6 x12xC9¦vA 352C175v6y1` `4571@vC 261vy67¤ 641¥C7 ¸ ) 352C175v6y1` 66 3vy `4571@vC x821vA62 46Cy 5 82595v6 x12xC9¦vA 3526v6y67 AC7 6y5 1 x5261v1 y67 ©5BCB52 46Cv6y 62 1v7C Q6421 y146` 124A65 B¤ CBA57 C 4¨ 1@12wv6§471¤ 5B¤26¨1 3vy 1v7CYQ6421E Cy1Y6` 35ECB5§5y1` C261vy67Y 6) 41¥C7 ¤ x122Cy ¤ ¸ 3A46Cw26`¦ `14 57¤ 5B26¨1 w4A12914 149 w451v1¤y67¤ 12264A64A" B) v y ¦ 31 C 5 C )365y256vA5`918A69454C5!C127C ¦ 68S7G1F1B3v35266592562771868w2456976vA1¤3A4Cv8yC1v¤BC2y@112B5A57 4 w 4 C@ 5B26¨1 3vy AC9x6A ¤8`6¦wv1 A5 5v6y1` 68212A1 6A C2 526vC1A 6 5B4C4A1y62 5 C¤4A6!¦`567¤ 352626`57 w4ACv¥1 66 wv6y1` YC22Cy 3 3A46C812A1 ©)87141B C2 35261v1¤y67¤ 826y657¤¤ 5v6y1` x1yCB 3vu 1v7CYQ6421E 652C175v6y1E 652C175v6y1` 61BA695714A x82wvC`712QC4wv6u S GF 3526vA598 3vy ©82262C74A17 AC7 66v1¥5C2 x6v6y1` 35¨CB5§5AACvB C26y657 6v6yC7 55¨C5vC67 ©5 1y14¤` 5 6 x6v6y1` 14A612` 12C41¥C7C7 3526vA598¤ XP 3vy 5 ¸B)A57 ©8291287 552676`5 1 `147Cy2C4A 3A463v3 426`1` x822Cy C2 gPSGFa VUTSQRGQPIHGF CBA57 C76E HHGfRedcb VUTSQRGQPIHGF 3A43v3 CBA57 5`C76` 5`8273v¦¤1¤ 66v1¥5C w7641 ©54 35487B1 3vy w4C1757667 1`5y1©¥162 x87141B CvC45 CB 76`C7 )w4526¨¤1 w4A121`1¤ 4Cy 6BC4 52 © 61 5 ¤ )`C1Bv2A8579 `352yC1¥wv5B6¨17¤x1B1448261`¦74A25 ©81A2C25§ 3v6§471¤ 57641¤ 6821`1 52 4 2 ¤ x12264A64A6 wA1v53 CBA57 76` 6526v31 y 4 5 B1C4A ©82426¨17 1A7¦B ©8¤226`¥1 17C14 ©86w78A x5¨BC 26¨ 5`35¨C04)B¦) v§ ¦ A ©822C3A w7¤641 ¤ w4A121`1 x1¥1A 463vC4Ay67¤ BA57 x829128 4Ay67A ©82626y `65261v `8yCB A x822C3A BA57 w4526¨1 12914 66v1¥) 5C2 w4A121`1 3A463v3 ©CB287 ©81A2C25§ C2 841¥C7 x126 A¦ `651vA¦ `82C )8426`@6A 65@¦7y 6A 3C676 1 12wv6459C2 1714A625 3vy ¤wv8¥57 ¦wv1¥5C2¸4C@Cvy67 5B287 681A2C25 ¤ 84B617 682215¨54A625 6825 5v5¤ @C`¦¥ ©8226¨ 5v6§471¤ v 5@w26y w4C8yCB) 3A45¤2C4A 1273v¦¤1¤ 66v1¥ 6A 3`67 662y6vA1¤ ¸ ¤ 1 ¸¨§¸¢¸¢¡
· · · · ·


X

5B5y146` 682wvC4A1 45y1©A167 1v7C Q6421E Cy146` 35¨CB5§5y1` y6 149 w4CCBA 1)21` ©822Cy ©82wvC67 C2 8¤146` Y82261vy67 3¦754A6 1B48¥¦ ©821`1¤ 35487B© 3vy 1 vC45 CB x82wvC`54¤1 w4C1757667 5 12 w4A)1243176 x122CyC A SGF 352¤9C2 6 w4C¤867 ¤ 62 1Bwv14 62 x5!3v11 y146` 26v64CB6v57 66v1¥ 1714A625 3vy 6 1@14 6`17Y )3v6§471¤ 54A1`514A 352¤26`5 64C4wv¦67 S 1@¤12269¦v1¤ CB48¥¦ 5v5 5v8¥57 x1BA6954BC§ 5 526`67 426`1` x822Cy C2 GF 35269C2 1@1212@17 5526vC¤4A1 1A 3A46C9vBC 5v6y1` 54A124CB6yC 5B7617 w4A12!¦ x124CB6y¤ C 66v1¥5C2 3A¤463v3 wv6y1` 3C2261vy67¤ 149 4C8CB1¤ ¤14A64 84¨4)wv¦6 C )`3576457B `8295vC7 1 x6v6y1` ©8`6¦y6vAA5 54A1254B6§§ 5 54A12914 65262C7A 6BC4 C 5v6y1` x1¤5 5v5 x14 54A1`526`57 54ACv¥1 ¥1 1y18 ©827157 C CB7617¤ 2 wA¥S3v3 352C1754A64 wv6 CB287 ¤1@1BAx5AA17 5 6vA59 `14 x5¨¤BC CB287 5 Cv CB2 2 y § ) vC1 4 ` ` ) P Gy Q¦x1B¤1f@17¤837A4©6828CC2©8522wCvC16775C42A6w4A561BC47`5C4BAA6575v16¤y17¦6487664¦5yv6vAA6 BgC¸4¢aTR©U8R2V2¡C 6C 2 C6 ©5BA695714A5 612A1 C2 x6v6y1` 552C1754A64 3A46C9vBC `CB287 `81A2C25§ `8295vC7 B SGF x6v6y1` ©8`6C574C`AAC7 54A1`526`57 A S¤GF65¦2BC261¨y16v5A`y146` C 5`5@¦7y A 55262C7A ¦2914 66v1¥ w4CCy 26v1y ©3526)952C7@1 ©82wvC`525`


D
r
t

65BA695`§57C@1v 56BC4 w4C1wv1 A5 121` 5`84A17¤ A ¦y37C ) 526`67 426`1` 1426`¦74A¤5 ©6A 2 714B6 w4C9C21¥1 `6x62wvCy `6y¦¥ ) 526`67 426`1` C426`¦74A25 1@Q
i n r
i,t

5`34A12y1©1y x64A12y1©1y 676 C26¨ Q B 6y@
p
i,t

1426`¦74A25 54A12y1©1y 684A17¤ w4C1wv1 A5 `6y¦¥ 1v5C7 BCB `6x62wvCy ©8226`676 64A69CB CBA57 5`C714BC§ w4¤C8C2 `6y¦¥ 3v¤6§471 w4A1`514A 45A5C ©871¤41B 41 g) y)4¸5)x5¨BC 8AB6y25 8A7¦B 5BC4Aa 1426`¦74A25¤ 826§ )14A695v1B 1@6 Q C C426`¦74A25 1@Q w4A1`514A 3C291287 3C!¦B64 Q 6y@
pi (t) (t) = p i
i,t+1 i

1426`¦74A25 ©81A2C25§ ©8295vC7 5 1@6!314A1A A@39 432 5432a 3v6§471 w4A1`514A 3C!¦B64 Q 35269C21¥1 6g58!¦y66v8A5 w4C176wv1¤A5 `6y¦1 ¤ ) yC2 x5¨B2¦§ ©8226952C7Q@1 Q y 14A612` Q 6y@ Q 0`8276`129621B w4C459A `6wy4¦C¥45197A1`16B¦1 )5C2766``673C244A61`214`31x756!¦y¦¥ 0C7x¥562@3v1C4A1A 64 2 ©821`1 14A612` Q 6y@ 64A2C74A17 `124A1243176 C2 x1226v6y67¤1 x1226`676¤ x12xC9¦vA 3A463v3 x548¥¤A ©5917¤ 6BC4 C ©CB287 1 ©81A2C25§ C2 x52626`5 654Ay6vA 3C7141B g55¨C52C@71 55¨51 3v6§471 a w4A1`514A 3C!¦y¦¥ 3A463v3 5`CBA57 5526vC7¤¦ 35269¦5 `1¤4B6)¥ ¤ vC76425 414 46yC 1 65269C2 6122CCBAy67 149 1@14 54A1243176 x1`6CCyC )w4A6 14 6C9¦vA ©824¤35¤7 1@Cv¥62 ©822617¥41¤5v1y 41 5 C421571@ 1@1226`67 825vy 41 45A5C CBA57 CB2¤6¨1 5 12wv64C1y6vA 5 C12@17¤ 1@12wvC76425 C2575 ( 6C9¦vA '' 5v5 ' 3A©5C4A1 5 `124357 1@Cv¥62 `1`CA A 4¦2B521 687141B )5B48¥¦ 4&¦y¦¥ CB&5W7 x1B26¨1 Cy@1© )82595¤v6 x12xC9¦vA x5269C2 ©821`1 vC76425 w45¦A w4A6 14 6C9¦vA ©824357¤1@Cv¥62 ©8`CA g& 5v5 &W 1298¥1a v1y ¦wv1¥62 w459vBA5 vA8`A 466`5 3526741`AAC7 5 w4A1226¨ ¦BA6954BC7¤ x6!6`5 CBA57 1@1291287 5B26¨1 35269¦v1¤ 3v y 4 5 B1C4A ©824u6¨17¤ 2 6857B 1B287 8AB6y25 Q 8`64A5A x11A2C25§ 5v64CCB) 1)¤ 65BA695`121B17BC` 3A43v3 1BA57 ©8291287 5`CB52914A5 w4A6 14 ¦y1757 ¦BA695`121B17BC` 466`5 BA57 x829128 1B287 ©81A2C25§ 5`35265¤y 5`824357 1@Cv¥62 A ©822C3A w7641¤ w4A)121`1 x1¥1A 463vC4Ay67¤ BA57 x829128¤ %$# "! ©

(

¢¤¸¨ ¡§

¸ ¨ ¦¥¨¤

¤¢§¸

¡§

¢¡



P



r

i,t

=

= ln( T

pi,t+1 ), pi,t p
i,t

ni pi (t),

-p

(, F, P)

i,t

,

t

(T ) G

(t)



t

r

n

t

i

= (r1,t , r2,t , . . . , rn,t ) G (T ) F




W 1B4C4A1y62 ©82264A6!¦A `1y37 46CyCv¥1 SGF w4A1273v¦¤1¤ 1A C2 3741`A6 5BA57 68291287 y1 SGF '' x8262yQ x824C7B©674 w4C1757667 1B2C¥ 41 4)¦¥674 82C@71 65!¤ ¦75v¦&@67 BC( 5`C2C@71 5`5!¦75v¦@67 5 12 5`C76y626` 5`82wv6§471¤ 5 5`!(yx674 ©Bwv14 62 3A46¦wv1¤A5 2 C 76 ¨ 1 ) 5 )87¦4C7645v 14A695v1B 612yC`17@ 126!3A1¤ SxG6F2y35266v5A9855v5¥@1v1y1`4669E66v1¥ xC C `7 6 4 ) 4 ¤1 S6G3F6451vA6529849 8©24766¦7& C2483E6` Cw¤A5y35a26y76C2¦Cw74C 14755yv1¦©`1716§ `661` 184` A72 9 C 2 ¨1 ¤ 3A 6 7 6 7 1@14 w4A12431¸6)g )6)4 82595v6 x12xC9¦vA 3526(v6)y67¤AC7 3217¦ wv5¤42CB w4A6
X q q q = V aR (X ) = -inf {x| P[X X N
(i)

C2595v6¸ )x62y y1576¤ C wv6§471¤ ©5!3y1© 1426`¦74A25 ©81A2C25§ B117541B 352626`5 41 43A5C 5 x12xC9¦vA x12595v6 3A43v3 54 576410 )x62y 6769 3v6§471¤ 1@6C2 57641¤ 6769 `59C21¥ )gC469AC7 y146`a CB287 3526y61¤ 6764BC7C© 1 55261v1¤y67¤ `122Cy 57¤ 0gvC76425 x82wv645761ya 3526867¤ 62 1@6 w4A1243176 x122CyC A 0g421571@ x1226`67a `6!¦y¦¥ 526`67 y1576¤ x8226v6y67¤1 C2 3A46C8459AAC7 5 ©CB287 ©81A2C25§ C2 26¨ 5`352C¥6v1B x8226v1vA¦¥1 B148¥¦ x8`6Cy51 x5wv1¥5C2 BCB 3A463v6y67¤1 87¦4B¦74A x122CyC 3v6§471 3vy SGF C2595v6 149 46¦y6vA CA17 1 3v6§471 1@122Cy 3vy w4A1243176¤ x122CyC A 526`67 y1576 x8226v6¤y67 1 C 54A621 ¤ `6¦BA57 8` B148¥¦ x82wvC`5ABC` x1BCB ©CB287 ©¤81A2C25§ C2¤ x5¨C76 1 5¤526y617 57 x5!CB521 A17 1 C2 4641 x829C212y1 5 x5B469 w4Cy ¤C2C57¤ VUT¤SQRGQ¤PIHGF 35¨¤6¨21Y ¤
©©¨ %$# %§ ©©

e C R ¥ V SQ QPIHGF CBA57 876` w26y x5232y1@6A C2 68273v¦¤1¤ 82)6H7HG4¦1R`dAcAb7¥4P¦¡yP¦¤¥¸66v5CuUT RG 6526C7¥141 w4A6 CBA57 C76E ¢©¢ "¡ )CBA57 876` 6526v6y67¤1 66!¥1 66v1¥5C2 `6¦75v¦`71§¨ C426`¦74A25 826¨ 652626`5 61262y12y1 w4C6`¦C7y1¤ `6y¦¥ `1B4C4A1 y10 CBy371¤ 1@1714 126v9 1y w4A12914 A `34A12y1©1y `84A17¤ 12C7 12wv6455v¥)57¤ 149
xt = p
i,t+1

·

·

·

- pi,t .

N

G R.

x] > }



X

N


h x6v6§471 55¨CB5§5A765y 14Ax1A 46CC741 CBA57 1@12wv6452v1 1y 46Cy)1A 62 ¤652625y6)¥1 66!C9C21 54A12545yyC¥¦A 14Ax1¤¨ )( 59CyC x1226vC4A1 5B5§5¨6¤A 41 45A5C 149 x51vA¦ 5 x52C1¥674 ©82wv64) 52v1 1y 41 w46A5C¤26v1y 352C1wv1¤A5 1@1BA6954BC7 3vy 876` x12467B21B 71¤¥8 AACvB x8275¥1 ©5 43v6y67 1 ` B C CBA57 ¦76¤ ¦BA695§5¨6¤A wy¦¥52Q¦¸C)B 43v6y67¤1 62 C4Ax1A 54¤¡ )( 354Ay6vA 687141B62 `59¦v1¤ 5 C4Ax1A 54 `6¦75426``1B170 ¦@5yA B 54A1242C57C25 14Ax1 0 w4A12y1712y1 3C2wv6451v10 0 w4A12545yyC¥¦ 0 5vA6 w4A12214121E 14 `C4Ax1A `5!¦y6vA 4637146v1y¦ C21 5vA6 x1242676@1B 3A46C8C2 CBA57 C76E ¢¢ "¡ wA5 C CBA57 76` ©8242676@1B ©8`6C8C2 3vy 149 w4C9C21 46y¦¥ BC4 AACvB 4CyC CBA57 676` B 352¤¥6)74 682`¦C7 65!¥1 66v1¥5C C1
©©¨§¦¥ %$# ¤ "¸ ¢©
· · · · Y ( ). X1 , X2 G : (X1 + X2 ) X Y :GR (X ) X G, X G, R : (X + ) = (X ) - . (Y ) X 0 : (X ) = (X ) (X1 ) + (X2 ) Y

`37641 `5BA695§174AC4CB B w45y157 4¦@1` Cy@125 12 657C26¨A 64A25wv1 57) ¤ y1©1¤y x5171© 4Cy 687141B 5¤5@64C74A 681@714 4637!11¤ SGF )x6v6§471¤y1¤ ©5!3vC4A1A 17667 ¦``¦A `69 6wv1¥ w4C1757667 54A1`5y1©¥162 462 3v6§471 1@6A CBA57 35487B1 3vy 14 CBA57 1@1291287 35487B1 3vy 1@1`6¦75¤7667 CvC45¤CB 76`C¤7 BCB CBA57 ¦76` w4C574C`AAC7 5vA¤6 12wv6454Ax6u )¦vA8`A ¦`1C7y 4596715417¤ 14¡ )4514A1A 21 ©87141B 5 x6v6§471 y1 ©¦y 1QSGF C``¦A `69 6wv1¥ 3v6§471 S F Cy@1B 876`57 54A657¤ ¤2¤1E )54A12545yyC¥¦A `14Ax1A 46CyCv¥¤1 6 21 54A124AC9 ¤ 1 2 g CBA57 876` 68242676@1Y v6yC7 ) `Aa x176` x1242676@1B 3A463v3 62 SGF )X 46C8¥¦ 1226vy6` w4A1241v 1@6 6 4a 84A1© 68v634 466`5 w7641¤ 6526v6)yg67¤AC7 Cy@1B 6C9¦vA BA5¤7 46C) )526¨11y62 ¦`141¤ 5 w7641¤ 3526v6y67¤AC7 14A1© 8¤54 682C7 w4595vC7 461` 62 SGF )( ) ) Q `69 5`5w26`a 5`34A1243176 5`5Bw26vC` A 54x1517¤ 4¦@1` 68714g1B(w76(41¤ ©5wv1¥ ©821`1 46C8459¦ 62 SGF )(

: X ( )

¥


1 0"! )'"('&%$#"! ©¨ ¤§¦¥¤¸¢

4A1© x8v634 466`5 w7641 6526v6y67 AC7 Cy@1B 6C9¦vA 6B54BC7 C2 `12262C74A)17 AC7 BA57 46C526¨1 B 124C¤6yC 66v1¥ 6¤ BC4 2 )w4A1243176 g ¤ `69 x6w26`a x¤wv1¥62 A 54x1517 1 4¦@1` 687141B 57641¤ 6wv1¥ w4C8459¦ 463v11¤ HHG¦Redcb `1C7¥1 `5BC© ¤ 5 Q CvC` ¦ BC4 62 57641 54 54A621 w4A1243176 3`67 6 14 5 82wv6459C2 12wv11y¸ w)(4)¥ 4¦@1` 5C2 57641 ¤5 ¦Bwv1BA1¤¤ `822C74A 8 3A463vC4Ay67¤ 461A 414¡ )6!¥11 vC45¤CB w4C17576¤67 62 `C2 46¦461A SGF 6)4
V aR (X ) = 0,
¡

3vu ) w4A1243176 A ' 5 '' w4A1243176 A 43C4A1A 57641 5WC2)Cy@1 CvC2(5`)12 ¦251v1 1BwvW 14 `59¦)v1¤ 8` 5 4Awv6C3¥1 ©51A 35262¤v1¤A5 & (© 4) 1 3A46CB41 B5!`¤6C () 4 w4A1243176 A C w4A12v1 C26C@1 46y¦¥ C21 '') w4A1243176 ¨ )C26C@1¤ w48¥ C2v1y C74C 3C7141B ¤ 2 ¤ ( `1vC5`12 35¨vC5@5v¥1 w4A6 AC2 1¦ 149 `51v1¤y670 )HHG¦Redcb 5 SGF 652612411vAC4x55!B ¦x7557w4vA1¥vw4C17765`5776¤6x71446A¦¥76¤74`5741`3A2AC17¦ 6 C 21 1@14 5 1@12y1 3vu )SGF `6) 9 CB¤A57 x176` x1254C76A21B 66v1¥ 3A463v3 HHG¦Redcb
X S hortf all (X ) = M(X |X > q ). N X q = V aR (X ), X X


6wv1¥ 149 551vA¦ 57¤ 652Cy51 61BA6954C`64C` 6121vA¦ w4A6 Cy@14 x62y 6769 3v6§471¤ 1@6C2 57641¤ 6769 SGF 5526v6y67 1 57¤ 5 BCB `59C21¥ 54A12545yyC¥¦A `14Ax1A 46CyCv¥1 4 21 54A124AC9 a SG¤F ©82264Ax1A 1BC)4gA1y62 ©5@12` 265v HHG¦Redcb 149 46¦y6vA C4Ax1A 14)9 46¦y6vA X C4Ax1A 5 BCB BC(4 )D 14A26C7 61@174A 3A4632v1 8 Cy@1B 54A12545yyC¥¦A C4Ax1A `6C9¦vA `82w)v6y67¤ 3A463v3 54A12y1712¤y1 x12wv6451v1¤ 14Ax1¨ )X
(0) = 0 X 0 (X ) 0 (0) = ( 0) = (0) S hortf all (X ) q 0.

S hortf all (X ) = M(X |X > 0) = 50.

1-



TM

©52y1¤ 66v1¥ 5¨C57C1B 6yCvB `6wv1¥ 4514A1A CB5§5¨6¤A 1@" ) U¡TeRPVUTS ¢G¸ed )) 55@1v1y146` 3A46¦wv1¤A5 y146` x822Cu )55¨C57C1B 682266Q12wvC5¨2621¤AB¡ )( )`626`67 1A 3A4326`5 55¨C57C1B 459C2 5 5`827C215¨C4A62 3A43v3 8y37 68226`67 682wvC67 149 4C8CB1 352C1y6vAA 5`CA6 5`82C7 A 1y37 ©8226`67 5 3526yv¥C2 6A¤ 3A4¦wv1 A ) 8¨574C` 5B26¨1 x1254B6§§ 5 x1226!6`A62 35269¦v1¤ 5 3vy 14 3¤ 2C526¨1 Cy1576¤ 55263417¤ C2 822314A1¤ 55¨C57C1B 5vA" 5
^ t+1 = 1 T -1
T -1 s=0

BCB 3A463vA598 C¨574C` 3C2215¨C57C1B `1C7¥1 `5BC© ) C12@17¤ Cy1576¤ 55263417¤ C2 6BC4 C x526yv¥C2 C2B1 1@6!3wv1BA © 825vy Cy1576 55263417 C2 5`822314A1 3A43v3 x64A12y1©1y 55¨C57C1B 149 3A46C@Cv¤1¤y67¤ 6y1©y¤1¤ `6x64A17¤ `1¤`CA `14 ¸ )5`CA6 5`82C7 A C¨574C` 3C2215¨C57C1Y )( )x526!¦¤1y ©54 5 ©87141B62 41 ©82y1¥1A 1y146` ©821vA 66v1¥ 1y © C`¦ 54A1`5A5C62 5 3526v6y67¤AC7 54A12wvC`712 1AA6¨17¤ 54A127C215¨C4A 1 x5261v1¤y67¤ © ©84A17¤ ©8`CA 41 1AA6¨17 ©5BA6954AC©14A 6764BC7C© 1 ©35261v1¤y67 ©8295vC7 C2 3A4¦75C¥ 3526v6y67 ¤AC7 1746`C7C 5B26¨1 8y146 3v6§471 ¤ 54A1`514A 352626`5 55¨B2¦§ `1y5¤ 5 x5¨C57¤C1B 8¨574C`a E35)26v6y67 ¤C7 1@122Cy A 1746`C7C¤ 5B26¨1 5`Cy146` gw5v 3A4C95vC7 5v6y1` `1C7¥1¤ `5BC© 7 ¤1 Q)g 06v6§471¤ 1426`¦74A25 31vy4aA639552v61vB6yx657!ACC78@1C2Bw1v¤C`¨761¥2v14A3Q21B6¦wQv542CB 71 4 g(a 6y5 `1BA69545vC2C 3vy 6526C78 ©35261v1¤y67¤ ©822Cy 570 3A46C8A5¤C v¦2 12C7 662y67A 149 w4C459A `6y¦¥ 1v5C7 BCB 66vC ) x6¨574C` x12215¨C5)7C1B 5 `82C7 `52y67A 1A ¦21BC ¦`12wvC¤`712 1 8u6v6y67 AC7 2 54A12y1©1y CBA57 1714BC§ x52626`5 5526v6y67 AC7 `12wvC`7¤2 `1276`1¤@12` 1 1 35261v1¤y67¤ ©CB`C7 3A`5y1©C2 8` Cy146` 1@¤122Cy 552C1wv1¤A5 570 ¨¤ § ¦¥%¤%¦¥¤ ¢ ¦¦¥¨¤ ¨¦§¸¸¡ ¨¦¨¢¡¢¸¡

t

r

n





V aR = -nT µ + z





nT n ,



r

T t-s rt-s

z = 0.05% z = -1.644853627

V aR

µ

t




' 3A46C459A `1C7¥1 ©6A 5 BCBa v¦2 12

`5BC 12C7 CAA6¨AB 65269C2 C 1426`1` ©8246962 5 C7 5©7)4X6``5AC 65269C2 3526v6y67¤AgC7 1@12wvC`712 3vu
1 = ^ n 1- ^ t+1 = 1 - T t 1 = ^ n
T -1 s=0 i=1 i=1 n n

`1C7¥1 `5!¦y6vA 3A43vA598 521 5B71¥8 ©35269C2 C )12264A46411A 14A1© ©8v634 6595vC2 5 3526v6y67 AC7 5746``5AC 65!¦5764BC7C© U U a AA6¨AB 5 UU ¥ VUa 35746``5¤C 3A43v3 1@12wvC`712 41 3526vg6yT 6d7ReAICV7 A 65 2621vB41g CP2¢ 5P`5!C8CB¦ 6B5746`121B 5`CB54A54C4A 5`821¤2A 4B C )5w44AA1122wwvvCC``7711221 652618v41A6y67 65A¦69w5v§5A¨56¤SAGF 86`v56y11©`¥11624A25x5w2v61v¥6y6C7¤BACC7 xy BB © 2 ¤ ¤ 4 1¤ )w4A12wvC`712 C2 652C1754A6© )X )g¤S ¢ © ¡RT¡TRUG¥P¡UdePRP¤ HG¢dTRT¥¢d P UU ¸ w T G a 54A12954ACy6BA17646@ x121vA¦ x12215AA67@6714C x1T22P6e!P¥e1d¥R1I v6¥yP1¸`H 3eAP4¢6P3¢v3 5v6y1` x12266Q12wvC5¨2621¤AB `6526!¥1¥ )35¥1y1¤1yC7¤ 55¨B2¦§ 55¨C5`5ABC` Cy146` w!1`1 A 76`57 C2 w4Cv6yA 121` 14 a `1AA6¨17 `82215AA67@6714C 3A46C8A¤5 1 `1C7¤¥1 `59¦v5C2 ¡ ¡H ) g( Q w4A12w¤v54Cv1 3C`6C8C2 G 6BC4 67¦¤4C7645v x11A2C25§ a g35RT¨HTCR57dC 1@6 `17141B 57 65269C2 612wvC`54¤1 54xC2 121` Cy37 1@1226`67 1@1yCB 3vy ¤6¤5¨257¤ ¸ x5¨C57C1B 5B26¨1 w4A12914 C2 4635v 149 5B71¥8 C76`C7 1@1254B6§§ )526w26`¦ B 46y6 35269C2 6526w26`¦ 821714A x1@¦7y ¨ ) `1y37 `8226`67 A `5!3y1©A517¤ `352626`5 B wv6y1` 662wv6454A¦9 `64 3v64512` 65269C2 6w26` `6 CBy371 1@176 1@62y67A 1@6!3wv1BA 5 55AA67@6714C 662y67A 61226B6) x1¥¤A 463vC¤4Ay67¤ 149 1 g(a `1C7¥1 `5!¦y6vA 12C1C7¥167¤ w48¥ 461` 6526C78 662y6vA10 6y@ `1C7¥1 `5!¦y6vA 3A463vA598 C¨574C` 3C2215¨C57C1B 3C2266Q12wvC5¨2621¤AB 526`67 41 3C!3A5C 5v6y1` x14 ¸ )x526yv¥C2
T ^ ^ t+1 = t + (1 - )rt rt ,

s r

T t-s rt-s

¯ Xi - X ^ ,

¯ Xi - X ^

3

4

,

. 0 < < 1.


( B 3A45y1©A w4A12wv64C1y6vA1 C2 ©8`6C8yCvBC2 ©3526952C7@1) ©8226v6y67¤1 57 149 12CCB1u ) ¤2C5AA¦C@ 3A4¦wv1¤A5 14AC9 C7y3 55¨B2¦§ 64A69CB ¸¤ 149 C1BC4
-

4637146v1y¦ `64¦ CB4C4¤A 46Cy1

17y3 C 3526v6y67¤AC7 54A1241v¤ 35¨B2
1 fT (x) = T hT V
i=1 T

¦§ ©

`1C7¥1 `5!¦y6vA 3A463v6y67¤1 3526v6y67¤AC7 35¨B2¦§ g3C29171¥8a 3CBA69575¤` ¦ ¢ ¦
X F (x), X1 , ..., XT = 0, 1,
T

)1@12wvC`712 41 1B6vCy 154BC 54A1`514A x52626`5 6526v6y67¤AC7 149 1y37 ©8226`67 352C1y6vAA5 ©C76`57¤ C2 122C17574Avv517¤ 46y¦¥ D 6Cv@ ¸ 3526v6y67 AC7 ¦ 14A1© ©8v634 6595vC2 C2 46C8CB¦ X 66!C867¤ 1) 2wv6459C2¤ CAA6¨AB 65269C2 149

35269C21¥1 `652C1wv1¤A5 3526v6y67 AC7 a ¨ ) x1226`676¤ x12xC9¦vA x14 CB71¥8 Q g w4A¦ ¤ 5¨B2¦§ ¦24A6562 3C!6`5 3C226`676 ¤ 3C2xC9¦vA © `1C7¥1 `5!¦y6vA 3A05174A 4 3526v6y67 AC7 35¨B2¦§ 3CBA69575 ` 3526v6y67) AC7 35¨B2¦§ 3CBA69575 ` 3A45174A ¤5B54A54C4A 35267 5B914¤ A¡ 6)4 6822Cy¤ 65BA695714A5 3A4¦wv1¤ A5 352314A1A ©8226`676¤ x52626`5 352)C) 175v6y1` 3vy 6y146` `14 ¸ ¤ %¡¥¨¤ $#!# © ¢
,

1 FT (X ) = T

i=1

76`57 C2 1y62 1@1¤4 A 149 y5

`17y3 `87141B62 A 5B476A w454A6!¦A1 121` 652C5Cv@ 652C5Cv@A 3A45y1517 35262C74A¦ 3v x6v542CB x5269C¨ )5526v6y67 1 57 54A12y¦7¤ 4 2 x84C926¤¦4A 4u66)`5 3526v6y67¤AC7 35¨B2¦§ 3C¤BA695¤75¤`¡
FT (x) (x - Xi ).

fT (x)

{hT }

hT - - 0, T hT - - . - -

T

V (x)dx = 1,

V

T

x - Xi hT

,

V :RR

+

-x2 2



w4A12wv64C1y6vA1¤ 5


{fT (x)}

f (x)

V

f (x)

exp

51vA¦ 6y@


((
m
T

x52Cy51 ©5BA6954C`64C` `1714B6 x12215¨C57C1B 5 3A463v6y67¤1 6526v6y67¤AC7 612wvC`712 61276`1@12` 61226y1786 )3526v6y67¤AC7 1@12wvC`712 652C175v6y1E ) 64A2C74A17 `12y1©A5 3v6§471¤ 1B48¥¦¥5v8¥57¤ 6526v6y67¤AC7 `63vA598 `)64C )68 ¤©822CA5¤1 x5¨C57C1BQx5¨C57C 1y146` 5 x1¥v 3¦wv1 A5 x1226`676 x112 3vy SGF 65269C2 `63vA598 66vCu )6526v6y67¤AC7 612¤vC`712 ¦!¤6`5 ¦226`676¤ w ¦2xC9¦vA x1226`676 x12xC9¦vA x12y1©A5 552C1C7¥167 `12x625v62 4514A1A 36y x526v6y67 ¤C7 14A1© 4Ax1A `65262C7©1A A x5¨C¤57C1BQx5¨C57C A Cy146` 14A¥1) y¦ x5!C¤4691A y146` x812 261vy67¤ ¤ ¸ 6w4C4A ¸ ) ©C4B¦y17¤ ©82``C7@17 ©5@12` 1 2C15vC67 6¦ x87141B `4571@vC x821vA 62 46¦4A6!¦A 3¤526v6y67 ¤AC7 1@12wvC`712 352C175v6y1` 3vy 821714A x1@¦7y ¨ 6 ¦ 4 5 2 )3SCGF6952659C`2 4B6CyB¥ 5Av4564C29C¦vB1 65236v96C§47616!5¦4A41A`651441A1Ax5523652266`v65y673¤5A2C67v3y6¨7B2A¦C§ B A 57 3 6 ¤7 35¨B2¦§ ¤06526v6y67 AC7 617¤141B62 `C¤2 4¦yCy 3v6§471 1@122C175v6y1`A x5269C2 55¨C¦4B¦v ¤ 3v6§471¤ 1@6C2 ¦B26¨1676 `5y1¤517 8` 55¨C¦45A x122C175v6y1`A x1y C)B 3v CB287 352626`5 3¦75¤v6y1` `6!¦¤y¦¥ 3v6§471¤ x5269C2 6526v6y67 AC7 `63vuA5)98 8 CBA57 1714BC§ x52626`5 3526v6y67 AC7 6764BC7C© 1 3526¤1v1¤y67¤ 1@1BCB5E )3A46Cv6y 62 1v7CYQ6421E 6y146` ¸ ¤ 2 )8y146` 65@¦7y w4326`57¤ 69¦v g&'' ©5wv1¥ `6CBA 54A127614A1y 3217¦ x5269C2 3vy SGF x5269C2 ) 6)4a x6v542CB ©5wv1¥ 3526y1©C2 3v 2595v6 ©82xC9¦vA ©8226v6y67 AC7 11BC25y1 ©8`5A5C62 ``¦A 3526v6y67uAC)7 54AC9 x12wvC7426¨ 352CA5¤1 3¤vy 45y1©y1 1171© 8`67164 x12wv6y67¤ x12¤wvC7426¨ 5 46¦y6vA BCB 6526v6y67¤AC7 6¤12wvC`71 ¥%¸§ %¡¥¨§ ¢ ¢
= (m1 , .., mn )

(

(

5B71¥8 65wv1¥ 3A4¦¥674 1@14 3vy BCB BC4 x548¥1A g©82431761vC`a ©82wvC)`674AB x64A1243176 5526vA598 57 ¤ 54A12y¦74 3A43v3 352C175v6y1` 1@1BA695714A5 `1B4C4A1y6 )SGF 5`CB26¨1 ¦ ` Ag 14 Q5A`A657@4A7y3AA41C248A6C213A526CB5147B5Cv76¤y1C`2 C3A1471¦@1wBvC14¤854C5wv¸¦¢6TRUPRCyU3U7e1U@a12`26C5y26C5267y564A16¤ y 4 PR C2 ¦`635v1¤ 12wv5A 548¥1A 1¥5vQ¦`1BCB x5!)¦4A46411A 54A124AC9 Cy37 1@1BA695714A5 B16741 x1¥v 3Aw4C1wv1 A5 461` 3526v6y67 AC7 35¨B2¦§ 3CBA69575¤` 3A45174A x17141B 1¤ 5¤B71¥8 64A69CB ¸ ¤ )SGF `65269C2 3A463v3 5 x87141B wv542CBQ 3A45y1©C2 1B@6v x62 1¤ 1@69 6vA1¤ gW( 676` x6w26` 1¤ w4C1wv1¤A5 46¦y26`1B67 UR¢P¢PHRRPb HG¢dTRG¢ePR¢¡ ¦d V¢G © b¡ a Cy37 1@1BA695714A5 5 x64A12y1©1y ©52y6vA1¤ 3¦wv1¤A5 3526v6y67¤AC7 5¨B2¦§ ¦BA69575¤` `5174A1¤ SGF 35269C2 35269¦v1¤ 3vy w76¤6©



((
X1 = c11 Y1 + m
1



5 66vCu ¦Bwv1BA10 ) x17¦y6¨17¤ x124267¦B67 3A43v6y67¤1 3A4C1wv1¤A1 `64C 5 2595v6 ©82xC9¦vA 55¨C5vC67 68`5A5C62 w459¦v1 46¦y6vA x12xC9¦vA x1276`1@12` x5¨C5vC67 3¤5269¦v1¤ 3vy x65A76 A5y 5 `52y67A `86v¦2 A 12wvC`712 826 35¤ 6v6y67¤AC7 w4A1241v¤ 65!6`5 2
11 =
n

`51v10 Cy@1© ) A AA AA A 3C2wv1@¦674 3C2wvC21@C5yy1 Q C¨574C` `6957¤ 6y5 `1BC4 w45C4Ay67¤ 1)2g1` 66 x6271B ©824C7yCB Cy¤146` w!1`1 A 14 3C226v6y67 1 12wv6451v1 3C226y17862 Q C¨574C` BCB BC4a ¤ 651vA¦ 3A4¤632v1¤8 x1714¤1B 3vy 352C1C7¥167¤ ¦¨574C` ¦BC4 `5174A1¤ 82595v6 x12xC9¦vA x12wvC`712 x1276`Q 35269¦v1¤ 3vu `6526v6y67¤AC7 `8247Cy2C4A 1A 82595v6 x12xC9¦vA 5 gXa 352C1C7¥167 1@12x625v `64¦¤ w459¦v1¤ 121` `6526v6y67¤AC7 A ¦2595v6 ¤¦2xC9¦v¨ 6y@ )¨6¥v14AQ714B6 Q
xT = (x1 , ..., xn ) X = m + CY 1 (det)- 2 exp (- (x - m)T -1 (x - m)), (x1 , ..., xn ) = n 2 (2 ) 2 n Xi Y
i

w4A1241v¤ 1@"


x6¨574C`

C

X = (X1 , ..., Xn )

14 1B@6v 84265¨5§§1Y )gXa `652612411A ©82wvC`712 ©8276`12y1 82595v6 `1C7¥1 `5BC© C¨525y6 v6y67¤AC7 5 8`)5A5C62 82595v6 682xC9¦v¨
c
ij

(x1 , ..., xn ) = (y1 , ..., yn ) = ~

c

11

=

M[c11 Y1 (c21 Y1 + c22 Y2 )] = 21 , Y1 , ..., Y C 11 21 . . . n1 21 22 . . . . . . Xi = mi + i Y
n1 11



1

DX1 . 12 22 ... n2 0
n2

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

1 1T n exp (- y y ). 2 (2 ) 2

N (mi , i )

1n 2n ... nn

0 0 ..

nn

Y1 , ..., Y

i

.

X2 = c21 Y1 + c22 Y2 + m
n

X

N (0, 1).

= CC

2

T

~

n


X(
§¨¤¤ ¨ ¥§¨¨¥¤¸ ¨¨§©©¨§ ¥ §¥¦§¦¥¤¸ ¢

¡



121`1 Cy@6A 14 3526v6y67 AC7 1@124AC9 1@128767 62 3v 6526v6y6)7 AC7 612wvC`712 ¦!6`5 ¤ ¦226`676¤ 1426¤`¦74Au5) 1B4C4A1 ¤652C1C7¥167¤ 612x625v62 Q @C x8760 2 x64A1`5A5C ©82x625v62 ©6A 5 1B4C4A1 3526v6y67 AC7 ¦ 1@6!314AC2 14A1© ) ©8v634 ©5BA1AA¦C@62 `65262C7©1A A 67¦4B¤74A x1BA1AA¦C x1B62 1B4C4A1 6526vC4Ay67¤ w4C1wv1¤A5 Q 36y5 3C212A 1B4C4A1 `6526v6y67 AC7 `82wvC`712 62 54AC941 3A4632A3)¥1 14¡ ) C59x14A)¦62 5 C26v1vA¦¥1 1©1¤v¤ 14AC9 C¨574C` 3C2215¨C57C1B 1BC2y 2wv6 1 )6512462261`v¦B7441A2652y©667A BCC¨B574C6`3v3CC242A1y56¨C571CBA15B7 BCCBB265¨1142©68`2¦27C4yA2©55B©A89571yA45 6 A 3A4 7¤ 5 64 5 4C9¦v1 1298¥1 5B26¨1 54 6B54BC7 C 154BC 1B4C4A1 55¨3v6771B 5 54A12wv5¤Cv1 5B26¨1 4¦¥674 14¦4¤ 4A5) ©81A2C25§ 59CyC 65@12E 4 52 )3526v6y67¤AC7 1@12wvC`712 5`5!6`5 62 5`CB4C4A1 ¦y6` 54A1`5A5C x176` x112 3A463v3 C¨574C` 3C2215¨C57C1B 3w9 68226`676 65BA1AA¦C@ 6812 C2 5B4C4A1 46326`C 617141B 652C1C7¥167 612x625v62¤ 3A46C@Cvy67 5v6¨ x14 3v w4C15764BC7C©1 `1C7¥¤ 1 `87141B62 1@6 3A`6C48¤ 1 5 6526v6y67uAC7) 6124A6`1A 61276`1@12` 46`5 5B4C4A1 5w9 15¤4B¤C wv6§471¤ `57¤41`AAC7 8` 641¥C7 x122Cy ¸ ) ©8226`676¤ 652C1C7¥1670 )( 5 57¤ `6957¤
· cij = 0 y (x) i k =1 0

C1BC4 Cv¦`71§ 3C242677¦B67 3C
c c
21 22

M[c21 Y1 + c22 Y2 ] = 22 .

cij =

cik c

jk

=

=

ij -

= 0, 1

12 12 = , c11 11

j j -

22 -

k =1

j -1

k =1

j -1

cik c

2 21 . 11

j

c

2 jk

jk

i

,

n.

x

!¥



12wv64C1y6v¨


D(
! ¦ ¡

1



3526yv¥C2 y1576 x87141B62 C 1@176 3vy 154B¤C 3vy 1B4C4A1 CB71¥8 3A466`5 AC2 ¦ 149¤ `51v1¤y670 )154BC ©5Bwv1BA62 xC9¦vA x5!¥ ga ¦y5 B 3A46¦C7¥167¤ 6y@ g a 6526C78 `1C7¥1 `8295@1vC2! ga 5¨C5746`C7C¤ ¦!¦y6vA w4C1wv1¤A5 12¦2 v¦2 12C7 62 662y67A Cy@1B 6C9¦vA ¸ `1C7¥1 `5x64A17 5C2 wA1v3v4A6!¦A1 `C2595v6 `82wvC`712 B 652C1)C7¥167¤ 8¥149 BC4¤C2C7¥8 w48¥ 461` gDa C`71 gha 149 `6C9¦v1¤ gDa 3vy 6526C78 33vC4Ay10 )x5¨3v6771B ©821`1 35262C7©1A 3vy `5y1©¥162 55¨B2¦§ BC2 gWa `1C7¥1 `5!¦y6vA ¦226`676¤ `5v6y67¤ x5269C2 ©5wv1¥ 3vy 3526v6y67 AC7 1@12wvC`712 w4A1241v `69 662)26vy6` 46C8¥¦ w4A6 14 84A1©¤ 68v634 466`5 14 ¤ ©5wv1¥ 57¤ 14A6` 466`5 5vA6 149 `546`C )g154BC ©5@12` 3vy 14A6` 466`5 651vA¦a `¦`5ABC` x82264A25y6 466`5 Cy@1B ©3C9¦vA ©6A 1 3A4C1wv1¤A5 461` 35¨C5746`C7C¤ 3CBC© ) UPUT ¢dF 68226`676¤ 68`6C8C2 BC4 46C8A5¤1 gDa 6526v6y67¤AC 57 B 3A45`674A C2v1y 149 `546`Ca 55¨B2¦§ )¦g2y1517¤ 46¤C9C21¥1 C 5B175`712 C42C4A21B Q 6y@ gDa
©8

BCB C2C15746`C7C w48¥ 461` 7141B w4A1241v¤ x526v6y67¤AC7 ©828767¤62 AA¤CvB `6`w1¸

C f ( x)

f

p( x)d x = C

x

y

y2 1 p(y )dy = e- 2 dy . 2

y = sig n( x) |f ( x)|.

f ( x)

y

f ( x)

pdf

f ( x ) x2

|f ( x)|

f

1 2

0

e-

+

1 2



f ( x )

d x,

| x|

x

2

1

·

m = M[sig n( x) |f ( x)| ].

p( x)d x = C

1 p(y )dy = exp 2

|f ( x)|

e-

T

(sig n( x)

-

x ,
(y -m)2 2

(1) 1

(1) x2

dy .

|f ( x)|-m)

,...,

(1) xT

d x,

| x|

N

p( x)

pdf

pdf


W(
erf (x) =
x u2 e 0

¨

©¨¤¥ ¨ ¥ ©¦

§

¦ ¥¤ ¸

gX(a BCB ¦¨574C` ¦2215¨C57C1B `5v6y67 1 w76 6 154BC 68226`676¤ 5v5¤v6y67¤¤1©8)` BC4 5 1@1yCB 3vy )C¨574C` 3C2215¨C57C1Y )66vCy BC4 5 1@1714 3vy Q C54BC 1@176 3vy x5269C2 71¥C2 `66`5 1@14 6vA1 154BC ¤5 1@1yCB 1B4C4A1 3526v6y67¤AC7 54A1241v¤ 3vy g((a 0g(() a g'a 352C1C7¥167¤ `6326`570 )B1¥51 55¨B2¦§ B 3C24C7¥1 35¨B2¦§ Q 6y@ g((a 6y5 g((a 6526C78 w4CA5¤676¤ `61` 8E g((a ¡ `59¦v1 x1226`676 5¨B2¦§ BCB 3C574C`AAC7 5 g'a 65262C7¦ 612wv¤5¨2676§§5y 3¤¦757@642 C g(a 3526v6y67 AC7 35¨B2¦§ 3C2543v¦`¦ ( x65A76 A5y 5 `652Cy51 `5BA6954C`6¤4C` A 6526v6y67¤AC7 612wYvC)`712 46¤`5 3C226`676¤ 526174A1¤ 10 6 g'a `66`5 C54BC 1@Q 3v )652Cy51 61BA6954C`64C` 616v¦2 46`5 154BC 5B4C4A1 Cy@1B xC9¦vAu`5741`AAC 526v6y67 AC7 ¦`1BA1AA¦C@ B w4A1241v¤ ¦yCB `6¦C7¥167¤ )`1y146` 6¤8 `822CA5¤ ) 84A1© 68v634 466`5 5 `82wvC`712 3A463v3 62 6C9¦vA `6!¥1 ¸ )C54BC 1@Q 1B4C4A1 3526v6y67¤AC7 w4A1241v¤ Q 6769 `59C21¥ ) 6526vC4Ay67 61BA695746`C7C 62 5v5 61BA695746`C7C 3¦wv1 A5 5v5 5BA69575 ` ¤`63v6y67 1 687¤41B 154BC 1B4C4A¤ 1 1 3526v6¤67 AC7 x64A1241v ¤ `66`5 8`¤`1C7¥1 `5BC y © ) 1@1 Q 3vy Q 66vC¤y BC4 5 1@1714 3vy C54BC
(2) (2) (2)



x1 , x2 , . . . , xT

x1 , x2 , . . . , xT

¤

erf

·

y ,y ,...,y

Pj ( x) Pj ( x)

(1) 1

er f

(1) 2

-1

(1) N

Fj (x)

j

y ( x) =

x

Fj ( x) =

N

y2 1 Pj ( x)d x = e- 2 dy . 2

Fj ( x) =

du)

¢



V = M[y y T ], 2erf y
- -1 x (2)

y 1 1 + er f ( ) 2 2

y (j ) (j = 1, . . . , N ) y1 , y2 , . . . , y (2Fj ( x) - 1), Pj (x )dx ,
(2)

N

j

.

(2) N

V

(N )

(N )

Pj ( x)

y

(N )

N


h(
dy ( j ) = 2 Pj ( x(j ) ) e d x(j ) |f ( xa )|] y y
(j ) i
(y (j ) )2 2

¥¤

¨ ¥¦¤¨¤
¸

©¥

¢

`82C7 `1426`6v `8Q A 8¨574C` x12wvC21@C5y wv645v6y67¤1 3A463v3 `12C5¥1B3 6C9¦vA `122Cy ¸ ) 352C1C7¥167¤ 2C5¥1B3 Q 6y@
dy dx

149 `6C9¦v10 )C54BC 1@1yCB 3vy g((a ¦v¦`71§ 3¦wv1 A5 5 5¨B2¦§ BCB 3C574C`AAC7 154BC 1B4C4A1 3526v6y67¤A¤C7 1@1276`1@12` 3vy 6526C78 `59¦v1¤ gh(a ) ©8226`676¤ ©812 8¨574C` x12215¨C57C1B 5526vA598 57 3526v6y67 AC7 14A1© 6526y61¤ 612254A5 46C8459¦ 652C1C7¥167¤ 6¤12x625v62 6¤`6¦wv1¤A 1 ) 1B4C4A1 5526v6y67 AC7 `12wvC`712 `1276`1@12` ¥1 35261v1 y67¤ 1@1`6¦wv1¤A5 14AC9 4¤1 2695v41 y1©y1¤ x82261vy67¤ 149 `54¤6`C ) 8¨574C` wv645v6y67¤1 Q F 6y@ gh(a 14 612wvC`712 61276`1@12` Q 6526v6y67¤AC7 BCB BC© )5`CB4C4A41 ¦y6` 54A1`5A5C ¦76` ¦12 BCB `6C574C`AAC7 8` ¦¨574C` ¦2215¨C57C1B ©5 C 6526v6y67 AC7 612wvC`712 65!6`5 68226`676 6812 6769 154BC 1B4C4¤A1 6526v6y67 AC7 5vC15764BC7C©¤ 1 8E )55¨C`5AB17¤¤C x1BCB52 12C1wv1¤A5 ¤v8¥ 62 71¤ ©5A 1u 1 )1B4C4A1 3526v6y67¤AC7 1@1276`1@12` 6526vC4Ay670 6y@ gW(a 6v¦`71§ 1¤ 3A43vA598 8¨574C` x12215¨C57C1B 8426`6v gha BCB 12C15746`C7C¤ w48¥ 1v@1` 617141B 3526v6y67¤AC7 3vu gD(a 82C7 x17141B 8426`6v
Va
b

·

Vab = M

ma = M[sig n( xa )

sig n( xa ) |f ( xa )| - m

^ P (y ) = (2 )

1 Vab = T

^ dy P (x) = P (y ) , dx

-

N 2

V

1 1 |V |- 2 exp(- y T V 2

xy

a

i=1

V

T

y

sig n( xb ) |f ( xb )|) - m

( a ) ( b) i yi

xi

(j )

j

.

¡

-1

y ),

b

,

x






1

(

1

1

¡

¡

1

¡

¡

¡

11

)5`Cy146` 5`8295vC7 682269¦v1¤ `1v6¨ 3v6§471¤ 5 BC4 1426`¦74A25 ©82wv6y41 BCB VUTSQRGQPIHGF 5B26¨1 826y657¤ 4¦y¦¥ 6Cv@ x6!¦y6vA ¸ ) 5 ©CB2¦A57 C2 652 826y657 8``C7@14A5@ 65!¦4A46411Aa 8295426y5gD8y3X7 149 `6C9vBC 352C175v¤6y1` `64¦¤ 1@12269¦v1¤ Cy37 5 RPe¢¡ Cy37 1@12y1©A5 35269C2 3C52C7¨ )82595v6 x12xC9¦vA x5269C2 ( 82C175v6y1`A 5v8¥ 3526v6y67¤AC7 54A1241v¤ x112 1¤ 1@14 6vA10 X Cv6yC7 5 5`352C1C7¥167¤ x12269¦v1 3526v6y67 AC7 54A1241v¤ 5 35)2X6)v6y67¤AC7 54A1241v¤ x12254A5 354A4641¤1A B5§C7@ 2¤y657 6 W 6B2¦A57 C x P 1 82595v6 x12xC9¦vA 3526v6y67 AC7 w4A1241v `5¤9¦v1 a 1)vR¦e`¢7¡§ wA3¦wv10 x1226`676¤ B 3A`6276¤ 652C1¤C7g¥1(67¤ 6124C7¥1 3¦wv1¤A5 )RPew¢7¡¤ 6 6© ¤ )6526v6y67¤AC7 612wvC`712 466`5 ¡ ¸ 3C226`676¤ 149 4Cy764y1¤ Dv5 D U Ud ) UU ¥ VU 35269C2 2 C7 1Bwv1BA62 wA1WDW9¦v)¦( CT17R1e@I1V`v1DYXXh5(7(6W4(57B 1¤P¢5BP54A54C4A 6569)C2 149 `546`C )g( )A57a x1226`676¤ x112 652C1754A64 `6y6170 )g((a 8v¦`71§ 5 65262C7¦ 612426y26¨A2C74 `6C67 5 w4A1241v¤ ¦2543v¦`¦B `6¦wv1¤A x1226`676¤ x5269C2 71¥C2 x5!¦4A46411A `59¦v1¤ 68 1@12)2¡C 5 ¤¸ A1 352C1C7¥167¤ 1@12x625v62 w!1`1¤ A RPe¢¡ x5269C2 Cy37 5 w76¤6© )Xh'(( W'') UTUdReIV (DW((WhX)( UUP¢¥PVU x64A12y1©1y 3vu )'W(W(W')h UTUdReIV X((XDX ) 26C7 RPe¢¡ 3vy UUP¢¥PVb A57a 1@12wvC`712 41 3A46C95v41 12wv6459C2 x64A12y1©1y 5v5 1B4C4A1g( 6)6 6526v6y67¤AC7 ¦`141 4vC ©8y7674 ¦vA59 B 3A45A1241 62 C4vC C4¡ a C7Cvv1y 75 1@1BAx5¤vC74AC B117541B y37 `6`w1 ¦12A1 C 352C)g1¡ v 6y1` 3vu )352C175v6y1` 76`570 )X y6 Q C C¨574C` 3C2215¨C57C1B Q 6y@ )C¨574C` 3C29525 g (a 55¨C5vC67 3526v6y67¤AC7 1@1276`1@12`
1
j =1 N

3A65!6`5 6769 5¨C`5AB17¤¤C

154BC 1B4C4A1 `6C9¦v1¤ 12wv64C921B
N

P (x)

V

P (x) = |V |

-

1 2

1 exp - y T (V 2

y

I

-1

- I )y

Pj ( x(j ) ), x
(j )


( 3526v6y67 AC7 54A12wvC`7126 68226)9¦v10 )w4A¤12wvC`712 C2 4A64 2 1 65261v1¤y67 4Cy764y1 84C4wv¦67 5 RPe¢¡ 3526v6¤y67¤AC7 w4A124¤1v0 ( )A5


'

(

)w4A12wvC`712 C2 4A64 5

¡ ¸

x1226`676¤ x112 3526v6y67¤AC7 w4A1241v0 ( )A5


( )82595v6 x12xC9¦vA x122C175v6y1`A C``C7@14A5 D )A5
¡

)RPe¢

¡

Cy37 1@12y1©A5 C``C7@14A5 X )A5
¡


(

(

)x64A1241v¤ 54A1`5A5C B5§C7 W )A5
¡


(( y5 6C9¦vA `6!¥1 x6!6`5 ¡ UUd HG a w7641 55¨B2¦§ x1`6C8C2 BC4 842C57C 68295vC7 3gA¢4dTR¦¢wIvf1¤A5¡B54eAP5¢4PC¢4A 35261¤74A1¤ 3vu )w7641¤ 35¨B2¦ )( 84A64 65!¦y6vA `5741`AAC7 x6v6y1` ©8`6¦y6vAA5 55¨C5764BC7C© 3vu )1254B6§)§62 5BA695`121B 149 ¦vC45¤CB ¦`122C1757667C ¦`625v5 B 45y157 ¤ CBA57 w26¤64A 3C!C8C wv6y1` 821714A x1@¦7y ¨ 1B48¥¦ ©8487B1 62 ¦2595v6 46¦5764BC7C© 5 C7 BCB 3526867¤ 65269C2 662y)67¨ )5`CB76y5 ¤`82wv6452v1¤1y A 12C3A 5 ¦4AC9C 149 C4Ay67A 682wv6452v1 1y w4C5BA85 1`5y1©¥162 55¨C52C@71 5 1B48¥¦ 35487B1¤ 3vy 1v54C© 62 C¤vC45¤CB 1@11BA57 1@122C1757667C 149 46C9C21 SGF 35269C2 65268670 )SGF x12595v6 3A463v6y67¤1 5 gHGRT¤G¡ VUTea `1vC45 CB `81BA57 ©8`6C8C2 4Ay67A ©82wv6452v1¤1y ©54 76`C 54 vy ©8 1©¤ )3wA736C4!¤ C982C1`315@63C78A7B614 53`CBA547Ay66572A6vC72w¦v6¤465122v15¤A1Ay ¸5532ACw41C75746C`6A7AC7 44 ¤ vB C 57 46y¦¥ 641¥C7 x122Cy 55@64C) 7)4A x1BC4 64A69CB ¤ 5@64C74A ¦¤7141B62 46¦75`71§ `CBA57 1 76y626` 5`CBA57 3526vC7¤¦ 3vy¸ )SGF CBA57 876` 552C1wv1¤A5 57 ¤ 57 4 1 14 0 )5v6y1` w4A1254B6§§5`Cy3A4466`3v5`35BA69654A655C74BA 5`x8¤2¤4¦7@y2C4xA1745¸617 4 7C w 7 1 121` ¦641 5@ ¦22C )w4A1243176 A 45y1©A517 6548¥1A 6y@ 5vv¦276 ¤ `1AA6¨17¤ 3A¤63v3 69Cu 5) © 5 SGF 35269C2 46C8¤7( 62 3v6§471¤ 54A1`5141A 4 6 2 652626`5 5vA6 65269C2 616(v¦2 x5!C`5257¤ AA6¨17¤ x¤82xC9¦vA 76`57¤C2 BC© )84A64 65BA6954A54C4A 68295vC7 w4C1wv1¤A5 121` 354A46411A 5B7617¤ 3vy 14 ©C25`764 ©5BA6954A54C4A 12Cy 6526v6y67¤1 ¦Bwv1BA10 ) VUTSQRGQPIHGF 526v6y67¤1 ¦`1`CA SGF 3526vA598 5v6y1` x1`6¦y6vAA5 654A46411A C2 84A64 3A43A1241 6 ¦7@ x176 5v6y1` w4A1254B6§§ 5 5v6y1` w4A12914 © 8 ¦7@ 6y C2 w45v¤6¤C7 y A 121v¤¦Y1)21` SGF x6v6y1` ©8295vC7 35262C7A 5 ¤5¤B26¨1 5576457Y 5v6y1` `1vC76425 `82wv645761y x1`6CCyC x1414AC9 x12wv645A1241 A 5B48¥¦) 68226A621 w45y1©A167 C2v1y 54A1`514A x11BA57 C2595v6 5v6y1` 352C1754A64 Cy15¤76 1@6A 55¤263417¤ C 3v6§471 54A1`514A 352626`5 64C4wv¦67 1@1226¤9¦v1 CB48¥¦ 5v5 58¥)57 x1BA¤6954BC§ 5 526`67 426`1` x822Cy C2 SGF 35269C¤2 1@1212@17 v 5526vC¤4A1¤1A 3A46C9vBC 5v6y1` 54A124CB6yC 5B7617¤ w4A12!¦¨ ¤ )x5¨B2CA ©82§C74 `3v645¦7C2 B 352626`57 `1C7¤ A wv17421B x822314A1 463v4A6!¦A1 6 BC4 C 5`C4¦454A25 5`81A2C¤ 5§ 5`6A 3526yv¥1A 3vy¤ 682wv64C3¥1 2 5576457B 682264A695v1B 5 682264A69CB 68226v6y67¤1 46C5vC2C4A¦ C71yC2 2C@71 @ ' ¦vC45 CB 1 526Cv@1A ¦`1BAwv6C 1 B 5262v1 1y 12ACv@1A BCB BC)4 w4A(124CB6y¤C C2 ¤652C1754A64 54x17¤ C2v1y wv6y1` ¤C¥¢ 3 ¦¦¥¨¤ ¨¦§¸¸¡ ¦¨¢¡¢¸¨ ¨¢¥¡§¢¦¢¦¸¢¦ ¦¦¤¸¢¡¤¡



X( B CB 3A463v6y67 1 Cy17 1@Q 5B¥51 w4A1243176 `69 6w26` x526867 6vA59 57 5v6y¤1` x1BC4 ( 3543257 1@1291¥5¸)1V w4A1243176 w45v6y67 ¤ 1 121` ¡¤¤ 3526867 3C9¦vA 1@12w¤v6y41 1@1¥v w4A1243176 C24A65 )6¤4a 4 264A65gx17141B vC76¤25 x82wv645761y x82wvC67 5v6y1` x124CB6yC62 3v) u 463vC4A1A Cy17 1@Q 5B¥51 w4A1243176a C2621vB41 46y¦¥ 1@14 552C12A1 C2 5 352C175g4A64 6vC(76425 C2 x526867¤ 66v1¥ 5v5 V 46CB1¤ wv6y1` 3C24CB6yC 149 1@14 w4A1243176 `1C7¥1 `5BC© x526867¤ V 66v1¥ 62 4ACy wv6y1` 3C24CB6yC 149 1@14 w4A1243176¸
X
k P(X = k |n, p) = Cn pk (1 - p)n-k . k

463vC4A1A 54A12914 12C7 46y¦¥ 5v6y1` x124CB6yC 3vy x526867 14A695v1B 66V!¥1 352C1754A64 6vC76425 `6A C2 149 1@14 w4A1243176¤ 14 3526867¤ 3C9¦vA 1@12wv6y41 1@1¥v w4A1243176 Q ¤ a 352C1754A64 6vC76425 C2 3526867¤ 6C9¦vA 14A695v1B Q V g3¢52C1V754A64 6vC76425 x62y 14A695v1B Q ¢ 6769 59C21¥1 5vA ) 3526867 82595v6 C469¦ 6¥ 3526867¤ 3595vC2 84BC§ 5`CA 1Bwv14 46"8459¦ 35¨B2¤¦§ 3C22Cu C 5vA6 5vA6 5v6y1` x124CB6yC62 6543257 a 6 Cy17 1@Q( 5 g5v6yS` x124CB6yC 652621vB41a)gCy17 1@Q( B1¥51 x64A124317¤ 1 5526vA598 5 GF 65269C2 `69 `5wv1¥ `CB48¥¦ ©5!¦4A46411A x548¥1A CvA59 C5vC2C 3vy 3A46¦wv1¤A5 w7641¤ 35¨B2¦§ 3C27C251 )w7641¤ 35¨B2¦§ 3C27C251 )( ) 149 65BC4 55¨B2¦§ 687141B62 © 5 R 426`1` SGF CB26¨1 © 6y@ 0( R 426`1` 3v6§471¤ 54A1`514A 652626`5 3A66C15vC67 © 5vA6 5vA6
V aR
t



t+1

L

t+1

=

P(X

L

t+1

P(X < k |n, p ) = f ( t+1 , V aRt ), g ( t+1 , V aRt ), P(X = 1, 0, f k |n, p) = g

k |n, p) = 1 -

i=0

k -1

i=0



k -1

i=0

t+1

t+1

P(X = i).

P(X = i). P(X = i). < V aRt , V aRt ,
t+1 t+1

< V aRt ; V a Rt ,

f

g


D(

)fSf

¥ ¡

`C4vC 1¤ 26¨ x52626`5 8y37 68226`67¸ h )A5

1y37 ©8226`67 5 B171¥8 ©814A64 71¥C2 5 x6v6§471¤ ©814A64 71¥C2 w4C7¥8 )1`5y1©¥162 576457B ¦`122C7¥8 1 ¤ 352C1754A64 3vy `1C7¥1 `5BC© 5 `¥ wv A 2 15 9 )x©1824AC6y4 ©x8122CB72 v16y15`BA21@v157 ©5`B6A46¦ 5731A44A653vg4CAB67!1¦A81 5`v56B1`1B5622C51v755a4A66B71¥8 y6 1 7141 6 ¤ ¤ ¤ © 64B6 AC `122Cy 66254B6§§ C21 `64 5v6y1` ¦ 5B54A54C4A 65269C2 6w26)` `6¤ 3A`6C¥51 8` SGF 3217¦ 3526867¤ 6C9¦vA `62y67A 1Bwv1BAC2 4BC) 8CB1¤ 55¨B2¦§ 65269C2 6
L
¡ ¡

CBA57 5B26¨1 x526867¤ 82595v6 46C8459¦ BA57 x8487B1¤62 x52y67¨ )BA57 x8487B1¤62 x52y67¨ )X Cy17 1@Q 5 1@Q B1¥51 x64A1243176 5`35269C2 1A 8¨5v¥C4 4¦4A6!¦A ()) ¤ 5 x(62y W(( ¢ 3vu

t+1

=

i=1

t

i=1 t

I

(i+1 - V aRi )

(

i+1


.


1

W( )5`82wvC4A1 yC2 46¦7525`1y 1v7CYQ6421E wv6y1` 576457B ¦`14 12ACv@1¨ )SGF X x82C7 vC45 CB w4C1757667 82v1y 12ACv@1A 521 BCB BC4 1B2C¥ 3vy ` 82C 66v1¥ 6!¤6 3A4512C4A x576457B 414 b6¡w26` 121` BCB 1v8¥ CvC45¤CB 1@11BA57 1@122C1wv1 A562 65269C2 8¥149 12¡w)v64Cv6 ¦`141¤ Cy1©1y 45A1257¤ 62 vC45 CB x81BA57 x¤822C1757667C )BA57 x8487B1 62 x52y67A x5w26`5C2 `66`5 ¤v7CYQ6421 `1y146` ©C469AC7 57 )SGF 52¤6v6y67 1 4¦4A46411A 1 4 A 1171© 5 C4A64 8EC4wv¦67 681BC25y1 5B06954BC7¤ 46`5 &¤W' 54A127614A1y 6217¦ 57 ¤ 5v6y1` 682261vy67¤ 6A 149 4C8CB1¤ ¸¢TRUPRV¡G 84C4wv¦6 5v6y1` 65@¦7y `69 `35¨C¦4B¦v§ `5B67 B C2wv6454A¦9 66v1¥ 1v7C Q6421 `) 1y146` CB26¨1 76`57 C SGF x6v6y1` ©8226y657¤ 3526y61¤ 6764BY7C© `E2264A69CB 1 w45y¦A 121`¤ ¦ 5§C7@ B) C1 ¦`122Cy 1 SGF 5B26¨1 65!¦4A46411A 5 '''( w7¥3426A 1¤ Cy1@ ''( 37¥CB6y A y1576 0 ) ¤ C 3v6§471¤ 54A1`514A 352626`5 B5§C7@ 26y657¤ 6B2¦A57 C & ' `82C7 2C7¥8 v8¥ U SQ GQ GF 3vy vC76425 x82wv645761 1v7C Q6421 )Cy1W46` 35¨CB5§5y1` 3C22V6T 1vRy67PIH 5 652C175v6y1` 61BA695714A5u )G Y E47Cy2C4A BCB ' 6 `65 4 269C2 1A¤ wv6y1` 3C2266Q12wvC5¨2621¢¤A¸Bed5v))14AC9¦ 552C1754A6) ¸ C S 1 )©8¸226f¤7b ©SB¡71¥8 ©¢829f5SvfC72¥w A1v¡y1 17`C6452vCC1754A64 ¥4S¤Cf7 xC1B22287 1y37 ¤¦ `6 C ¡ 5 ¤ 6 1¥ C Cy ¸ )S ¢ ¥ ¡ `C4vC 1¤ 26¨ x52626`5 8y37 68226`67¸ )A5
1



¡


h( )SGF C469AC7 x6v6y1` 54A124CB6yC 5B7617¤ 84C4wv¦6 ' )A5

©8

' 54A127614A1y 3217¦ 3vy x6v6y1` 295vC7 SGF 5B26¨1 5 3v6§471¤ 54A1`5)1&4WA x1262y 352626`5 B5§C7 )A5
¡


1

(

)CvC45¤CB 1@11BA57 352C1757667 35267 5B914 A `82wvC`54 ¤1 3A463v3 y146` 414 1@14 6`17Y 7 8w 4 4C 1 CB28 3A ! Q) ©421E7Cy1©5` @C1252CC175©¨52§v5¥C`4A665226C226`65C¤y14894C1wv¦C6B71¤6561771©6614y ¸v7CY 6 46 1 5y1 7 2C 3` ) ©822Cy ©82wvC67 C2 x6v6y1` 352C1754A64 6BC4 C 1y37 ©8226`67 ©5BA695714A5 4Ax1A ©5BA6954A54C4A 352C1y6vAA5 612A1 C2 82Cv6yA 8y18 65BC© )CBA57 1714BC§ 6526y61¤ C2 ©8`6C8yCvBC2 x526952C7@1 CQ5 ©CB287 3A©5!C5C7 C2 352C1wv1¤A5 3vy 82y1@57¤62 3Aw4CCB1 4¦@1` ©CB287 ©845C7 C2 68`6326`57¤ 1B175 5 SGF CBA57 876` 3526vA598 3vy 8`6¦wv1¤A5 5v6y1` 682215¨5yC74 149 `14 1 4¦4Awv646y5A 841¥C7 x122Cy 84C4wv¦6 ¸¨§¸¸¸¢¦¦¡


(

Ie)HdeR¢d¡VUTe)

¥¥¥¥

¥

¤

RRc

VUTe RG PIHG 6469AC7 57¤ Cy146` C`6v¥170 12C¥1¢ )! 8y146` 5 5v6y1` 5BA57 6829128 25@Cv6( )! ) u 1B5w26E )¨ ) 652626`57¤ ©5 5 CvA59 682xC9¦v¨ C12C )E )¸ 352C175v6y1` 1@1BA6954A54C4A A7¦Y 1vxC©5E )! )¡ 1BC`7" )E )¨ 8A17¤1 6826`A 5 1v7CYQ6421E y146E 1BC`7" )E )¨ )'' GT¢ed¦THG ¦d ¡RTUeP T¢¢ UPHP¸¢ Ud¡ ) U¢dTRI TeRUT PRGTeG TRHI ¢GTUUIG¢Q¢d¢ ed¦ ¡edPc dTHd¦Red ¥¢G ¸TeRG P¡¢GTeG d eGP¢TH¢d )F ) ¢PUeP¥¢ ) TRRP¢d¢Tb ) PRRP¢edb ¢¤¡¢¦¤¸¢¨¢ ¦¡¥¨

(

¤1¸ ¤¸ ¤¸ ¤¸ ¤¸ ¤¸

h W D X (



¤¸

(