Correction Simulations Marcel Brun 12-Nov-2002 1 Experiment:Test ...

oqpƒ‚ƒsvui‚ƒsvu4pƒy…„{0n¥n¥WYZ4†€hV¥[ˆ‡`k.e€a§‰4h{thWYk.Z
UTV¥WYXZ4[]\^[`_bac[]d5WYef[gZihjlk.mYn4[]doqp`rtsvu.w'svu.x.y)WzX¤XWYefWzmY{td|h}k~hV4[[`_7ac[]d5WYe€[gZ"hX¤WYZ
n4[]h}[]d5e~WYZ¥{thWŒk.ZŽ‰¥X5WYZ4†=d5{0Z¥n4k.e Xa§mYWŒh}hWYZ¥†4
k0jŠhV4[ˆ‡`kb[]‹~‡]WŒ[gZ"h¤k0j
U¤V4[£ef{0WYZ¢‡5V¥{0Z4†0[•WYZ hV¥WYX?XWYe¤‰¥mY{thWŒk.Z§X~WYX?hV¥{th¥Z4kƒ\¦\§[£‰¥X[£{ Z4[]\¦e€kbn¥[gmTjlk0d¥hV4[¨a§dk0©§{t©§WYmzWŒhWŒ[gX€ª~ol«šy…„
¢
{0XX‰¥e~WYZ4†€{fn¥WYXh}d5WŒ©§‰4hWYk.Z~†.{0efe~{¥o¬{0mŒa§V¥{iy…„¥\­V4[]d[{tjlh}[]d­Z4k0d5e~{0mYWŒ®g{thWŒk.ZŽWYX{~n¥WŒd5WY‡…V¥mŒ[]h|n¥WYX}h}d…WŒ©§‰4hWŒk.Z
¢ å ·
¢ åIêië ·
WYX¤{0ZŽ[gX}hWzef{th}k0dTk0j å ·
¢ ñ]ê o â y ³ ð Ø ì Ø å · ë ê ÙŒÙ ß ì Ø å ·
·
Ù WYX¤hV4[?ÂÉ`ƬÄòäfÌ0ƬÄÎÇtÔ Ầ ÁÁ}ÇtÁ€jòk0dThV4[a§{td5{0ef[]h}[]d â
¢¡¤£¥£§¦©¨¡¡
"!#%$'&()!*+
,.-0/12034/5-"6764-
8 9;:=A@B5CD>FE)GIHKJL>M§G N O @P=Q RTS
‘’v‘ “•”§–A”¥—.˜š›q—.˜šœ ”¥ž Ÿ—i¡
¢°¯ „Z"‰¥e©c[]dk0j©±t{td5WY{t©mŒ[gX
¢ ²´³µ0· „¥Z"‰¥e©c[]d¤k0jI‡`k.Z4¸¥†.‰4d5{thWYk.Z¥X
y…„¥d5{0Z¥n¥k.e
©§WYZ¥{td5¿f±0[g‡`h}k0dg„{td5[hV4[ÀÁÂÃ0ĬŃÆÈÇtÁ5É
¢°¹º³ ol»¥¼ƒ½]¾]¾]¾g½}»
¢°Ê „§d5{0Z¥n4k.eË©§WzZ¥{td¿~±t{td5WY{t©mŒ[0„4WYXhV4[fÆÈÌtÁÎÍ"ÂgÆ
Ï « ¼ ½]¾]¾]¾'½}«§Ðш{td5[hV4[ ²´³µ · ack.XXWŒ©mŒ[=Ç"ÒÉ]Â]Á…ÓtÌ0ƬÄÎÇtÔ•ÓtÂ5ÅgÆÈÇtÁ…É
¢
ÕgÖ׳ ª~Ø ¹º³ « ÖzÙ WYj×hV4[a§dk0©§{t©§WYmzWŒhv¿fk0jŠk0©§X}[]d5±bWYZ¥†¤« Ö
¢
¢¨Ú§Ö³ ª~Ø ÊÛ³ piÜ ¹³ « Ö Ù WŒjŠhV4[‡`k.Z¥n§WŒhWŒk.Z¥{0mca¥d5k0©§{t©§WYmYWYhv¿fk0jŠhV4[±t{td5WY{t©§mY[ Ê †.WŒ±0[gZLhV4[Ýk0©§X}[]d±t{thWŒk.ZŽ» Ö
¢ ÞiÖ׳ efWzZco ÚÖ ½gp¤ß Ú§Ö yWYX|hV4[ Â]Á…Á}ÇtÁ¤Å…ÇtԧƬÁÄÎÒ]à4ƬĬÇtÔ•k0jŠhV4[k0©X}[]d±t{thWŒk.ZŽá Ö
o Õ ¼g½]¾]¾]¾ƒ½ Õ Ð ½ Ú ¼]½]¾]¾]¾ƒ½ Ú Ð y§WYXhV4[)À4ÌtÁqÌ0äfÂgÆvẦ Á…ɤÓ0ÂŃÆÈÇtÁ
¢°âã³
¢ å æ WYXhV4[Ýk0a¥hWzef{0mš‡`k.Z¥X}h{0Z"hT¸§mYh}[]dol[gX}hWYef{th}k0d`y§\TWŒhV4k.‰¥h§a§d[gn¥WY‡`h}k0d5X
WYX|hV¥[k0a¥hWYef{0mc¸mŒh}[]d¤jlk0d¤hV4[çè±%{td…WY{t©§mŒ[gX
¢ é WYXZb‰¥ê ©c[]d¤k0jIX{0e€amŒ[gX‰¥X}[gnŽh}k~[gX}hWYef{th}[hV¥[k0a¥hWYef{0mck0ac[]d5{th}k0d
é X{0e€amŒ[gX
n4[gX5WŒ†.Z4[gn¥jldk.e
¢íì Ø å Ù WYXhV4[Ý[]d5dk0d­î^ï­ð Ø å Ù jlk0dT{0Z"¿¥¸§mŒh}[]d å
¢°ó?ô k.õöjlk0d¤hV4[n¥WzX}h}d5WŒ©§‰¥hWŒk.Z?ª?o Ê ½ ¹ y
p

n4[]h}[]d5efWzZ4[hV4[±t{0mY‰4[ˆk0j¢¡¤£h}k ‰¥X}[ˆWYZLhV4[†0[gZ4[]hWz‡]XT[`_7ac[]d5WYe€[gZ"hX¤‰¥XWzZ4† ¯ ³¦¥ „ Þ~³ u7¾Ypt„¨§© ³ p
_bac[]d5Wze€[gZihh}k
Ê ³ piÜ ¹ Ù h)‡`k.efa§‰4h}[gX­h}k"k hV4[¤d[gX‰4©§XhWŒh‰4hWŒk.ZŽ[gXhWYef{th}[gX‰¥XWYZ¥†~{?X5{0e€a§mŒ[k0jAXWŒ®][ ¥ u
{0Z§n¨u7¾Ypk0j±t{td5WY{0Z¥‡`[¤WYZ¨ª?Ø
£Fu7 r
u4pˆu7 u ¥
¡£Au7
ç)‰¥e©c[]dk0jIn§WYX}h}d5WŒ©‰4hWŒk.Z­p]u0u
[]a§mYWY‡]{thWYk.Z¥X|jòk0dT[g{0‡5VLn§WYX}h}d5WŒ©‰4hWŒk.ZTp]u0u
¥
ç)‰¥e©c[]dk0jŠ±t{td5WY{t©mŒ[gX
Þ Au7Yp U©d5‰4[
µ
UF¿ba[Ýk0jIZ4k.WYX[WYZ­oò_§y†.{0efef{
ô p€„§‡`k.efa§‰4h}[gn§©Au7 w0x.‚
I{td5{0ef[]h}[]d
Þ ½*)y…„.k0j ð Ø ñ ê o Ú y Ù {0Z¥nhV4[|‡`k0dd[g‡`hWŒk.Z
UTV4[FZ4[`_7hŠh{t©§mŒ[|XV4kƒ\­X©hV¥[§{ƒ±0[]d5{t†0[0„ik'±0[]d¤p]u0uÝn¥WYX}h}d5WY©§‰4hWŒk.Z¥Xcjldk.e^hV4[|e€kbn¥[gmo
ê o Þ ½*)y|jlk0dTn¥W.-c[]d[gZ"h¤XWŒ®][gX é k0jIhV4[Ýh}d5{0WzZ¥WYZ4†€X}[]h]„
±t{0mY‰4[gX,+
p]u µ u tu p]u0u p/tu
Z
ñ]ê o Ú y Ù u7Yp0p µ0µ u7 u.‚tw u7 u4pgw µ u7 u0u.‚ u7 u0u ¥ r
ð?Ø
ê o Ú y§jlk0d,¡£ ³ u7¾ u4p u7 u µ u7 utr00w u7 u ¥ w%r u7 u µ ‚ u7 u µ u ¥
+
ê o Ú y§jlk0d,¡£ ³ u7¾ u ¥ u7 u µ u.w u7 u4pgx0x u7 u4pgx µ u7 u4p ¥ p u7 u0u.x7p
+
+ ê o Ú y§jlk0d,¡£ ³ u7¾ u u u u u u
WŒ†.‰¥d[ˆp]{iyAXV4kƒ\­XhV4[){g±0[]d5{t†0[0„4k'±0[]dp]u0u¤n¥WYXh}d5WŒ©§‰4hWYk.Z¥XŠjòdk.e hV4[­efkbn4[gmŠo Þ ½*)y…„bk0jšhV¥[¤hV4[­n§W.-[]d5[gZ¥‡`[¤©c[]hv\§[][gZ hV4[
1
ð?Ø ì Ø åŠê ٌ٠{0Z¥nãhV4[€[]d5dk0d ì Ø å325476
Ù k0jAhV¥[€k0a¥hWYef{0mI¸§mYh}[]dÝjòk0dÝ[g{0‡…V n¥WYXh}d5WŒ©§‰4hWYk.Zš"KhÝ{0mYXk XV4kƒ\­X
hV¥[€[gX}hWYef{th}[gn•[]d5dk0d
[]ddk0d5X 859 ²;: %

³ pt )
WŒ†.‰¥d[ µ ©cyTX5V4kƒ\ hV4[€‡`k0dd5[g‡`hWŒk.Zã±%{0mz‰4[gX+ ê o â y­jòk0dÝn¥W.-c[]d[gZ"hÝX{0e€a§mŒ[=XWŒ®][gX{0Z¥nã‡`k.Z4¸n4[gZ¥‡`[gX\TV4[gZ㉥X5XWYZ4† ) ³ pt
1
1 WŒ†.‰¥d[ ¥ XV4k'\TXhV4[X5‡]{th}h}[]d­a§mYk0h]„4©[]hK\F[][gZLhV4[h}d5‰¥[Ý[]ddk0dT{0Z¥n hV4[Ý[gX}hWYef{th}[gnLk.Z4[gX]
1 WŒ†.‰4d[ µ ddk0dT{0Z§n ‡`k0dd[g‡`hWŒk.ZL‡]‰4d±0[gX¤\­WŒhV¥hV4[‡`k0dd[g‡`h­±%{0mz‰4[Ýk0j ) ³ p
1 WŒ†.‰4d[ ¥ ¡ ‡]{th}[]dTa§mYk0h]„¤¡¤£ ³ u7¾ ub„ 騳µ u
1 WŒ†.‰¥d[Ýr~XV4k'\TXhV4[Ýh}d…‰4[{0Z¥nŽ[gX}hWzef{th}[gn []dd5k0d5Xjlk0d¤hV4[fp]u0u~n§WYX}h}d5WŒ©‰4hWŒk.Z?jòk0d¤hV¥[e€kbn¥[gmAo Þ ½*)y…„§\TWŒhV 騳 µ u
qZ hV¥WzX¸¥†.‰4d[ˆ\F[¤‰¥X}[ˆhV¥[‡`k0dd[g‡`h­±%{0mz‰4[
hV¥[n¥W.-c[]d[gZ¥‡`[ˆ©[]hK\F[][gZ¨hV4[[gX}hWYef{th}[gn¨[]ddk0d5X 859 ²;: %09 ê o â y{0Z§n ì Ø å325476
Ù
0.12
0.1
0.08
µ ρ
= 0.4 , Repl. = 100 , N = 3 , ρ = 0.1
xi................................
corrected1.........................
corrected2.........................
corrected3.........................
0.06
0.05
µ ρ
= 0.4 , Repl. = 100 , N = 3 , ρ = 0.1
correction1............
correction2............
correction3............
Error difference
0.06
0.04
0.02
Correction
0.04
0.03
0.02
0
−0.02
0.01
−0.04
10 20 50 100 150
n
0
10 20 50 100 150
n
{iy
©y
σ ρ
=0.05, n=20
1
0.8
Estimated ε[ψ opt
]
0.6
0.4
estimated...........................
corrected...........................
0.2
0
0 0.2 0.4 0.6 0.8 1
ε[ψ opt
]
¥

1 WŒ†.‰4d5[Ýr dd5k0d ô k.e€a§{td…WYXWŒk.Z
0.35
Error for n=20
0.3
0.25
Error
0.2
0.15
0.1
0.05
ε[ψ opt
]
λ n
(p) for σ ρ
= 0.01
λ n
(p) for σ ρ
= 0.03
λ n
(p) for σ ρ
= 0.05
E[ε[ψ n
]](p)
0
0 10 20 30 40 50 60 70 80 90 100
Model p
r

¢ ŸŠ”¤£¦¥¨§v”§–©LŸ›;”¨©0”¥—iœ¦§v–A˜©§vŸ×–
p]u µ u tu p]u0u p/tu
Z
u7 ‚bp µ w u7 ‚bp µ w u7 ‚bp µ w u7 ‚bp µ w u7 ‚bp µ w
ó
ó'ê Ù u7 r0r0r u7 .‚%u u7 .‚tw u7 0x%r u7 ‚%utrix
ð?Ø
o Þ ½*)y|jlk0d,¡¤£ ³ u7¾ u4p u7 0w µ u7 0x.‚'r u7 ‚0‚ ¥ u7 ‚0‚ µ¥ u7 ‚ 7p ¥
o Þ ½*)y|jlk0d,¡ £ ³ u7¾ u ¥ u7 rix0w ¥ u7 µ r0 u7 ‚ µ u u7 ‚ ¥0µ ‚ u7 ‚ ¥ u ¥
o Þ ½*)y|jlk0d,¡¤£ ³ u7¾ u u7 r0r0r u7 .‚%u u7 .‚tw u7 0x%r u7 ‚%utrix
WŒ†.‰¥d[t{iyTXV4kƒ\­XThV4[¤{g±0[]d5{t†0[gXg„k'±0[]dÝhV4[?p]u0u n¥WYXh}d5WŒ©§‰4hWYk.Z¥X§jòdk.ehV4[efkbn4[gmFo Þ ½*)y…„ck0j hV4[¤n¥W.-c[]d[gZ¥‡`[gXT©[]hK\F[][gZ
1
ó ŠU¤V¥[¤±0[]dhWY‡]{0m©§{td5XFXV4kƒ\ hV4[T±t{td5WY{0Z§‡`[olkƒ±0[]d|hV4[¤p]u0un¥WzX}h}d5WŒ©§‰¥hWŒk.Z¥X…yk0jšhV4[Tn¥W -[]d[gZ§‡`[gX]
hV¥[Th}d5‰4[­{0Z¥nf[gX}hWzef{th}[gn
WŒ†.‰¥d[%©yTXV4k'\TX¤hV4[¤{g±0[]d5{t†0[gXg„kƒ±0[]dhV4[ p]u0u?n¥WzX}h}d5WŒ©§‰¥hWŒk.Z¥X|jldk.eºhV4[¤e€k7n4[gmFo Þ ½*)y…„ck0j hV4[ˆh}d…‰4[{0Z¥n¨[gX}hWYe~{th}[gn
1
ó
1 WŒ†.‰4d[ A{iyõ)W -[]d[gZ§‡`[©c[]hv\§[][gZ h}d5‰4[{0Z§nŽ[gX}hWYef{th}[gn ó „7©y ó ‡`k.e€a{td5WYXWŒk.Z
WŒ†.‰¥d[ b„ ¨XV4k'\TX¤hV4[ h}d5‰4[ ó {0Z¥n hV4[¥[gX}hWYe~{th}[gn ð Ø óƒê Ù d5[gX}ac[g‡`hWŒ±0[gmŒ¿0„ jlk0d¤hV¥[£p]u0uãn¥WYXh}d5WŒ©§‰4hWYk.Z•k0j¤hV4[¥e€k7n4[gm
1
Þ ½*)y…„š\TWYhV é ³öµ u7 1 WŒ†.‰4d5[€‚~XV4k'\TX)hV4[¤X5‡]{th}h}[]da§mŒk0h­©[]hK\F[][gZ„šhV4[h}d…‰4[ ó {0Z¥n£hV4[¤[gX}hWYef{th}[gn ð Ø ó ê Ù „šWYZL©mY‰4[0„
o
ô k0d5d[gmY{thWŒk.Z©c[]hv\§[][gZhV¥[Fh}d5‰¥[§{0Z¥n[gX}hWYe~{th}[gn¤‡`kb[]‹~‡]WŒ[gZ"hIk0j§n4[]h}[]d5e~WYZ¥{thWŒk.Zjòk0d n¥WYjò[]d[gZ"h
UTV4[FZ4[`_7hŠh{t©§mŒ[|XV4kƒ\­X©hV¥[
X5{0e€a§mŒ[ÝXWŒ®][gXg
k0dd[gmY{thWŒk.Z Z ³ p]u Z ³Ýµ u Z ³ tu Z ³ p]u0u Z ³ p/tu
ô
ó'ê Ù u7 w0x µ u7 x ¥0µ u7 x0w ¥ u7 x0x µ u7 x0x0w
ð¥Ø
ê o â y|jòk0d,¡£ ³ u7¾ u4p u7 x µ ‚tw u7 x.‚ ¥ u7 x0w0x0w u7 x0x µ w u7 x0x%r
ê o â y|jòk0d,¡£ ³ u7¾ u ¥ u7 x7p/ u7 x0w.‚ u7 x0x%r0r u7 x0x0w7p u7 x0x0w0w
‘’¡
UTV4[¤Z¥[`_bh)h{t©§mŒ[=XV4k'\TX)hV4[€{g±0[]d5{t†0[0„k'±0[]d~p]u0uŽn¥WYXh}d5WŒ©§‰4hWYk.Z¥X|jòd5k.é
hV¥[¤e€k7n4[gm|o Þ ½*)y…„k0jhV4[=h}d5‰4[={0Z¥n£[gX}hWYe~{th}[gn
‡`kb[]‹~‡]WŒ[gZ"hX¤k0j n4[]h}[]d…efWYZ¥{thWŒk.Z¥jlk0dTn¥WYjò[]d[gZ"hX{0e€a§mY[XWŒ®][gX]„¥‰¥X5WYZ4†=h}d5‰4[Ý[]dd5k0d¤k0jIhV4[n4[gXWŒ†.Z¥[gnŽ¸§mŒh}[]d5Xg
θ difference
0.3
0.25
0.2
0.15
0.1
0.05
0
−0.05
−0.1
difference............................
corrected1............................
corrected2............................
corrected3............................
10 20 50 100 150
Sample size
Coefficient of Determination
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
E[θ n
]
E[θ]
ω n
(p) for σ ρ
= 0.01
ω n
(p) for σ ρ
= 0.03
ω n
(p) for σ ρ
= 0.05
10 20 50 100 150
Sample size
{iy
©y
{0Z§nŽhV4[X‡]{th}h}[]d)a§mŒk0h©c[]hv\§[][gZš„¥hV¥[h}d5‰4[ ó {0Z¥n hV4[ˆ‡`k0dd[g‡`h}[gnL[gX}hWYef{th}[gn ê o â y…„§WYZ¥d[gn„
‘’¡
¢ Ÿ—.—.”§v˜©§vŸ×–
ê o â y|jòk0d,¡£ ³ u7¾ u u7 w0x µ u7 x ¥0µ u7 x0w ¥ u7 x0x µ u7 x0x0w

1 WY†.‰4d[ AU©d5‰4[{0Z¥n [gX}hWYef{th}[gn ó
{0Z¡ ô k0dd[gmz{thWŒk.Z Z ³ p]u Z ³Ýµ u Z ³ tu Z ³ p]u0u Z ³ p/tu
ó'ê Ù u7 w ¥ w u7 x%r u7 x0w µ¥ u7 x0x ¥ u7 x0x.‚tw
ð¥Ø
ê o â y|jòk0d,¡ £ ³ u7¾ u4p u7 x µ0µ¥ u7 x.‚tx u7 x0w.‚ ¥ u7 x0xtu u7 x0x ¥ r
ê o â y|jòk0d,¡£ ³ u7¾ u ¥ u7 w0x.‚ u7 x µ u7 x0x ¥ p u7 x0x.‚ µ u7 x0x0w%r
©c[gX}hh}d5‰4[ ó
{0Z¡ ô k0dd[gmY{thWŒk.Z;ol©c[gX}h µ u£¢~y Z ³ p]u Z ³µ u Z ³ tu Z ³ p]u0u Z ³ p/tu
Ø óƒê Ù u7 7p0p ¥ u7 0x.‚0‚ u7 w%r07p u7 x µ x ¥ u7 x.‚ 0x
ð
o â y§jlk0d¡£ ³ u7¾ u4p u7 7pgx u7 w0w µ ‚ u7 xtutrix u7 w0x%r0r u7 x7p/0w
Iê
ê o â y§jlk0d¡ £ ³ u7¾ u ¥ u7 riw0w.‚ u7 ‚tw ¥ u7 x%r"‚'r u7 x.‚ 0x u7 x.‚0‚'r
1
θ for n=20 for true ε[ψ 0
]
0.9
Coefficient of Determination
0.8
0.7
0.6
0.5
0.4
0.3
0.2
θ
ω n
(p) for σ ρ
= 0.01
ω n
(p) for σ ρ
= 0.03
ω n
(p) for σ ρ
= 0.05
E[θ n
]
0.1
0 10 20 30 40 50 60 70 80 90 100
Model p
UTV4[)Z¥[`_bhh{t©§mY[X5V4kƒ\TX§hV4[ {0Z¡
ô k0dd[gmY{thWŒk.Z ©c[]hv\§[][gZ¥hV4[h}d5‰4[Ý{0Z¥n¥[gXhWYef{th}[gnŽ‡`kb[]‹~‡]WŒ[gZ"hk0j©n4[]h}[]d5e~WYZ¥{thWŒk.Z jlk0d
n§WŒjò[]d5[gZihX{0e€amŒ[XWŒ®][gX]
ê o â y|jòk0d,¡£ ³ u7¾ u u7 w ¥ w u7 x%r u7 x0w µ¥ u7 x0x ¥ u7 x0x.‚tw
UTV4[)Z¥[`_bhh{t©§mY[X5V4kƒ\TX§hV4[ {0Z¡
ô k0dd[gmY{thWŒk.Z ©c[]hv\§[][gZ¥hV4[h}d5‰4[Ý{0Z¥n¥[gXhWYef{th}[gnŽ‡`kb[]‹~‡]WŒ[gZ"hk0j©n4[]h}[]d5e~WYZ¥{thWŒk.Z jlk0d
n§WŒjò[]d5[gZihX{0e€amŒ[XWŒ®][gX]„¥‡`k.efa§‰4h}[gn k.Z§mŒ¿~jlk0d¤hV4[ µ u£¢
Iê o â y§jlk0d¡£ ³ u7¾ u u7 7p0p ¥ u7 0x.‚0‚ u7 w%r07p u7 x µ x ¥ u7 x.‚ 0x

1 WŒ†.‰4d[‚< ¡ ‡]{th}h}[]dTa§mYk0h©[]hK\F[][gZLh}d5‰4[{0Z¥n [gX}hWYef{th}[gn ó o5¡¤£ ³ u7¾ ub„ 騳µ uiy
σ ρ
=0.05, n=20
1
0.8
Estimated θ
0.6
0.4
estimated....................
corrected....................
0.2
0
0 0.2 0.4 0.6 0.8 1
θ
‚

UTV4[¤Z¥[`_bh)h{t©§mŒ[€XV4k'\TX)hV4[¤[gX}hWYef{th}k0d5Xk0jFhV4[¤[`_bac[g‡`h}[gnã[]ddk0dÝWYZ§‡`d[g{0X}[0„š‰¥XWYZ¥†?‡`dk.XX}sȱ%{0mYWzn¥{thWŒk.Z¨[gX}hWYef{thWŒk.Z„c{0Z§n
hV¥[‡`k0dd[g‡`hWŒk.ZL±%{0mY‰¥[gXjòk0d­n¥WŒjl[]d[gZih¤X5{0e€a§mŒ[ÝXWŒ®][gXg„
Z
ê o Ú yFjlk0d¡£ ³ u7¾ u4p u7 utr00w
+
ê o Ú yFjlk0d¡£ ³ u7¾ u ¥ u7 u4pgx0x
+
+ ê o Ú yFjlk0d¡£ ³ u7¾ u u
1 WŒ†.‰¥d[ˆw€XV4k'\TXhV4[X5‡]{th}h}[]d­a§mYk0h©[]hK\F[][gZLhV4[Ýh}d5‰4[ˆ[]ddk0dT{0Z¥n¥hV¥[[gX}hWYef{th}[gnLk.Z4[gX]„¥‰¥X5XWYZ4†€‡`dk.XX}sȱ%{0mYWzn¥{thWŒk.Zš
1 WY†.‰4d[ˆw {0Z¥n4k.e ¡ amYWŒh}hWYZ4†4„ ¡¤£ ³ u7¾ ub„ 騳µ u
WŒ†.‰¥d[ xLXV4kƒ\­XˆhV4[?h}d5‰4[ {0Z¥n [gXhWYef{th}[gn []ddk0d5Xjòk0d¤hV4[£p]u0u£n§WYX}h}d5WŒ©‰4hWŒk.Z£jòk0d¤hV4[¥e€k7n4[gm­o Þ ½*)y…„ \TWŒhV 颳´µ u7„
1
\­V4[gZŽ‰¥XWYZ¥†=‡`d5k.XXqsȱt{0mYWYn¥{thWŒk.Z¥[]d5dk0d¤[gX}hWYe~{thWŒk.Z
¡¢@šQ­MM£¢¥¤§¦5B©¨¤§GcB5Q­E >M§GcB…C¤§GšBQ­E QãGšP=>º>@š@šQ­@
’v‘
=—.—iŸ—•–A˜c §
µ u
u7 u0w
ð?Ø ì Ø å ê ÙŒÙ ß ì Ø å 25476
Ù
σ ρ
=0.05, n=20
1
0.8
Estimated ε[ψ opt
]
0.6
0.4
estimated...........................
corrected...........................
0.2
0
0 0.2 0.4 0.6 0.8 1
ε[ψ opt
]
’¡ ¢ Ÿ
w

1 WŒ†.‰4d5[ˆx dd5k0d ô k.e€a§{td…WYXWŒk.Z
UTV4[€Z4[`_bhÝh{t©mŒ[€XV4kƒ\­XÝhV4[€h}d5‰4[€{0Z¥nã[gXhWYef{th}[gn;‡`kb[]‹~‡]WŒ[gZ"hXÝk0j|n4[]h}[]d5efWYZ§{thWŒk.Zãjòk0dn§WŒjò[]d5[gZihÝX{0e€amŒ[€XWŒ®][gX]„‰¥X5WYZ4†
n¥{th{?X}a§mYWŒh}hWzZ4†4
d…{0Z¥n4k.e
Z
ó
o Þ ½*)y|jlk0d,¡ £ ³ u7¾ u4p u7 0w0w
o Þ ½*)y|jlk0d,¡¤£ ³ u7¾ u ¥ u7 tu.w0x
hV4[ ô k0dd5[gmY{thWŒk.Zš„ {0Z¡ ô k0dd[gmY{thWYk.Z {0Z¥n {0Z¡ ô k0dd[gmY{thWŒk.Z jòk0d­hV4[©c[gX}h µ u£¢ ó ©[]hK\F[][gZ
UTV4[Z4[`_7h¤h{t©§mŒ[gX­XV4kƒ\
n4[]h}[]d5e~WYZ¥{thWŒk.Zš„7jlk0d 騳µ u7
hV¥[Ýh}d5‰4[ˆ{0Z¥n¥[gXhWYef{th}[gnL‡`kb[]‹~‡]WŒ[gZ"h¤k0j
k0dd[gmY{thWŒk.Z {0Z¡ ô k0ddg {0Z ô k0d5dg µ u£¢Ë©c[gX}h
ô
óƒê Ù u7 x%r.ui‚ u7 x ¥ 7p u7 ‚bp µ w
ð¥Ø
o â y|jòk0d,¡£ ³ u7¾ u4p u7 x.‚%u µ u7 x.‚%u µ u7 xtu ¥
×ê
o â y|jòk0d,¡£ ³ u7¾ u ¥ u7 x ¥ ‚ u7 x ¥ ‚ u7 ‚ 0w0x
×ê
0.35
Error for n=20
0.3
0.25
Error
0.2
0.15
0.1
0.05
ε[ψ opt
]
λ n
(p) for σ ρ
= 0.01
λ n
(p) for σ ρ
= 0.03
λ n
(p) for σ ρ
= 0.05
E[ε[ψ n
]](p)
0
0 10 20 30 40 50 60 70 80 90 100
Model p
µ u
u7 ‚bp µ w
ð Ø ó'ê Ù
u7 tutr
o Þ ½*)y|jlk0d,¡¤£ ³ u7¾ u u7 tutr
’¡ ¢ Ÿ—.—.”§v˜©§vŸ×– •–˜š § ¥›qŸ— ©¢¡” ¢?Ÿ
×ê o â y|jòk0d,¡£ ³ u7¾ u u7 x%r.ui‚ u7 x%r.ui‚ u7 0x0x.‚
x

1 WŒ†.‰¥d[?p]u XV4k'\TXThV¥[¤X{0ef[X‡]{th}h}[]dÝa§mYk0h]„\TV¥[gZL‰¥XX5WYZ4†fd5{0Z§n4k.eX}amYWŒh}hWYZ4†€h}k [gX}hWYe~{th}[ ó 1 WŒ†.‰4d[ p0pt„ ?XV4k'\TX­hV4[
1 WŒ†.‰4d[€p]u ¡ ‡]{th}h}[]dTa§mŒk0hjlk0d ó \­WŒhVŽ‡`dk.XXqsȱt{0mYWYn§{thWŒk.Z;o5¡ £ ³ u7¾ ub„ 騳µ uiy
ó {0Z¥n¥hV4[Ý[gX}hWYe~{th}[gn ð?Ø ó'ê Ù d[gXa[g‡`hWY±0[gmŒ¿0„4jòk0dThV4[fp]u0ufn¥WYXh}d5WŒ©§‰4hWYk.Zfk0jŠhV4[e€k7n4[gm o Þ ½*)y…„\TWŒhV 騳µ u7„¥\­V4[gZ
h}d…‰4[
X}a§mYWYh}hWYZ4†¤h}kf[gX}hWYef{th}[ ó
‰§XXWYZ4†=d5{0Z§n4k.e
σ ρ
=0.05, n=20
1
0.8
Estimated θ
0.6
0.4
estimated....................
corrected....................
0.2
0
0 0.2 0.4 0.6 0.8 1
θ
p]u

1 WŒ†.‰4d[fp0p §U×d5‰¥[{0Z¥n¥[gX}hWYef{th}[gn ó \TWŒhVŽ‡`d5k.XXqsȱt{0mYWYn¥{thWŒk.Z
1
θ for n=20 for true ε[ψ 0
]
0.9
0.8
Coefficient of Determination
0.7
0.6
0.5
0.4
0.3
0.2
0.1
θ
ω n
(p) for σ ρ
= 0.01
ω n
(p) for σ ρ
= 0.03
ω n
(p) for σ ρ
= 0.05
E[θ n
]
0
0 10 20 30 40 50 60 70 80 90 100
Model p
p0p