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1.5 0.0
0.5
Density
1.0
MLE(X) STEM(Y) PLE MnE
−1.0
−0.5
0.0
0.5
1.0
1.5
1.5
φ^
0.0
0.5
Density
1.0
MLE(X) STEM(Y) PLE MnE
−1.0
−0.5
0.0
0.5
1.0
1.5
1.5
2.0
φ^
1.0 0.0
0.5
Density
MLE(X) STEM(Y) PLE MnE
−0.5
0.0
0.5
1.0
1.5
φ^
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15
φ8
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15
1
1
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1
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2
o
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Ï3
3.0 2.5
2.5 2.0
2.0 Density
1.5
1.5
PLE MnE
0.0
0.0
0.5
0.5
1.0
1.0
Density
PLE MnE
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.6
0.8
1.0
θ^2
2.0
2.5
2.5
θ^1
2.0
PLE MnE
1.5
Density
0.0
0.0
0.5
0.5
1.0
1.0
Density
1.5
PLE MnE
−1.0
−0.8
−0.6
−0.4
−0.2
0.0
0.0
θ^1
0.2
0.4 θ^2
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2
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û
0.010 0.005 0.0 -0.005 -0.010 -0.020
-0.015
bias of the estimate
MLE(X) MnE_3 MnE_4 MnE_5 MnE_6 MnE_7 MnE_10
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
phi
â ·ÉE¿jº»¸U³j¾9ǵ¤¹ç¼È¥°kº<·g¸d¹E¹ ¹ ½¤¶·c°¸¤¹½¤Ë~¹¥ÎG·¼»·ØË~åiÎÂ_ç Ä ½)¹%º¾°Ôº·¹E¿*¸¾°P² ¿ ¸dJ¸ ¹½ K °P¹¥º φ
z²Ý¥jvþUjÿݲæå £z²àÝ K T²³£kk´ o iÛz²C ¥² ¤k ã²Ý 2 ¥ ²³´ ²Ý ³Ô £Ô´ Ð ^ x b k´ ´dT²³Tär ¢ £k+ûj û £ ¢ ¥³£k_ éû ûì ßkT²ÌåÖ¥´ dK µd² φ ß T²² Ý :_ k´ T´Ã²Â²Ýäþr .ÿïþY ÿ φ ݲ³+å ¥²³£k+å £jdx²Ý k± ´ é. Ù¤jb k´ ´dT£ ²³ Tã²Ý ± % _ ²³T±+´ µd
T C ¤Ù k¥ßk²ß _ k²Ì ݲ³ìY þ ÿ o æ¤Ù j ¥²ï µ ç²³ £j φ8 0 FÖµ jß Âݲ³x £k ²Y dT ²³T :± ¢³£Ô { û ßk²Ìå"j´ d µd² ß T ²8 ²iÝ Ó £ _« dT²³T]Ù¤j®µd²kµd¥£k¥²³ k å ¥ ¥ ²k2² ´¦²³ µ² ß ¥²³: ²Ý _ k´ T ´ dT²³T ] ¤Ù k¥Òµ²³Ôâ. h K 10jÐ k _ ® ¥[ ¥Ûi ² ² £ ®b k´ ´ xÓ£kN xµd¥²³ #ª ² äݲ³zU þ jÿ dT²³ Ô´>þ>ÕÔÿ ݲ³zþH.ÿëݲ³ 8 ¢¥³£ £k _§ý ûjÊk_k²ÌY jï ¥å Fèµkµdd 8²Ý:þH.ÿ µd² ß _d´ò² £Ôk²j kφ þ ÿ >_ ^ I± 40% Y rµd² ß T²ké ± ² rKÔ´ dEß 3dkj´ d 2²à«Ý jrå ¥£ ì²Ý ¤Ù d_ §¥ ßkßÔ²áÖ ¥ ³ ² ¦ ¥ ì èJµd dkµ N 8 µ ³²³ d¦µ² ß Ô ´ ² ± 10% ²kTå"ÔIx d ´ k£j²³k ¥²³æݲ³ ¯:µ ³²³Td û
0.16 0.14 0.12 0.10
Standard Error of the estimate
MLE(X) MnE_3 MnE_4 MnE_5 MnE_6 MnE_7 MnE_10
-0.6
-0.4
-0.2
0.0 phi
â ·ÉE¿jº»¸U³j¾@³Íµ¤¹ç¼È¥°kº<·g¸d¹E¹ ¹E½I¸¼&°P¹*ºÍ°kº<º¸º»º ¾jk° º
o ¯
0.2
0.4
0.6
¹%º§¹½GË~¹ÍÎÒ·¼»·æË~åiΪÂbç Ä )½ ¹%º
0.030 0.025 0.020 0.010
0.015
MSE of the estimate
MLE(X) MnE_3 MnE_4 MnE_5 MnE_6 MnE_7 MnE_10
-0.6
-0.4
-0.2
0.0 phi
0.2
0.4
0.6
â ·ÉE¿jº»¸U³j¾ïǵ¤¹ç¼È¥°kº<·g¸d¹E¹§¹E½iË ÌÎ E¹ ½iË~¹ÍÎ Ò·¼»·Ë~åiΪÂbç Ä ½)¹%º¾j°kº<·g¹E¿¥¸«¾°P² ¿ ¸dJ¸ ¹½ K °P¹¥º φ
o§
0.9 0.8 0.7 0.6
Efficiency of the estimate, comparing to MLE(X)
MnE_3 MnE_4 MnE_5 MnE_6 MnE_7 MnE_10
-0.6
-0.4
-0.2
0.0 phi
0.2
0.4
0.6
â ·ÉE¿jº»¸U³j¾ðǵ¤¹ç¼È¥°kº<·g¸d¹E¹¨¹½ ¦¸d²°j¼À·¾$¸ÒÎØÁ ·¸d¹*Á ·¸d¸Â¹½¤Ë~¹ÍÎG·¼»·Ë~åhΪÂ_ç Ä )½ ¹%ºÒ¾jk° º
o
0.10
0.06 0.04 0.0
0.02
Actual Type I Error
0.08
20x20 Field 30x30 Field 40x40 Field
â ·ÉE¿jº»¸U³j¾¢ôµ 2 °È¥È0º»¹Í· ¼Ã°j¼À·¹¹Ñ½)¹%º^¼»·j¸ÃÁ³º<·¼ ·gÁ°P²¾j°P²c¿j¸d¸¦¹½ ê å ã¶(½ ¹çº^¼Ì¸d¸¼ ÷¹½Ð·g¹¥ºj¸¬È¸d¹¥ºj¸d¹¥³Á ¸ 0.0
0.02
0.04
0.06
0.08
0.10
Theoretical Type I Error
6 (+89!:/!:(+89!;(+<=! "!$#%'&)(+*-, .0/21%'3+!$#'&4#51 7
± ± Ñ¥d ßk ²³b6 Hdb:¥kφ0_6²Ý ßç ²CdT ¤k ² ²³d²³£Ô]µ _+j¤ í ß$²CdT²Ý°¥k± ´ßk´dÔµ 8 ê+zµ £k¦jiß . µ¤ ç²"²Cb ßJßÔ²"µd´Ô£k ²ªdb ¦k¥q ßçC² dT i. dx¤µ²³kT´qã£k_²àÝ¥ k²Ö²"´Öí) ²ãßk²"µ´k£k_d ¢ .²³b k´ ´¨¥ k²Ö²"´§ ²Udbd¯²³µ Ø _ô db:ßk²"µ´k£k ²Ú k ÷ ´k¥_ Ô j£ ¥²³Ç #z²» Cå d¯ +kC² Þ d´ú jß d ¯{ßk_d£k´Ô²àí× Ð Ù ÑhÐÙ ´k§² äkC² å N ¥k ÷ ´kb T kj£ ²³ kßk ² Ô ¬¯ xhÑ ÐÑÙ µ d´ ÷ kz´k¥_Tk£§²Ç0êx£Ô £k ²³Ô£kN ݲ³]å ¥²³ £kµ T£k³_ h . µ ¥k µd²³kb :¥{Ó£k ¦µd¥ ²³ Ò²²³ à¯T!² £Ô£ Ò`µ kí)Ó£ _ d´ ßkßÔ²áÖ ¥²³§²³ :Ó£kN °Þ ݲ³q ÂiÑ Ðõ Ù ¥² ± > ¢ ³ Ô £ xû ¯.¤_£kÞ ´ ÂßEdTݲ kµã²Ý* ± ßkßÔÞ²áÖ ÐÙ ´8²³þr.ÿ dh 2 G j® hÑ Ð#Ù G Ô â.³£k_ ¬¯ zk² j ß dT²³T _ d´¦²³J `µ kîí Ó£ d´ ßkßkT²ÌÖ ²³ +ßk¥C² d´ Ô_ i µ £ $å ¥£ dݲ³ k x´k Ïd_ d æ µ ]¤Ù Ô oô 0
2
2
2 1
0.05 0.0 -0.05
20x20 30x30 Field 40x40 Field
-0.10
-0.15
Relative Error of Approximation
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0.8
1.0 0.8
0.6
0.6
0.4
φ^
0.4
φ^
0.2
0.2
0.0 −0.2
0
5000
10000
15000
20000
0
5000
10000
15000
20000
25000
30000
20000
25000
30000
20000
25000
30000
20000
25000
30000
Index of the Sampler
−1.2
−0.6
−1.0
−0.4
−0.2
θ^1
−0.6 −0.8
θ^1
−0.4
0.0
−0.2
0.2
Index of the Sampler
0
5000
10000
15000
20000
0
5000
10000
15000 Index of the Sampler
θ^2 −0.4
−1.5
−0.3
−1.0
−0.2
θ^2
−0.1
−0.5
0.0
0.0
Index of the Sampler
0
5000
10000
15000
20000
0
5000
10000
15000 Index of the Sampler
θ^3 −1.5
−0.2
−1.0
−0.1
θ^3
0.0
−0.5
0.1
0.0
0.2
Index of the Sampler
0
5000
10000
15000
20000
0
Index of the Sampler
5000
10000
15000 Index of the Sampler
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