"!#$

%'&)( P *,+.-0/1/325406 & 607849607840:<;>= ( /3?A@CB (X , E) DCE*F784 ( /3-.GC2 ( /H4./36 σ IJ/32 ( /3?K7KL)6.;MBN? µ@CB (X , E) OP49BN?K/ ( /H4 & 6

ν µ QR=)- & ?8?K/ ν ∈ P,

49B5407KL)/H4S;>= ( /3?8?K/32 & 6T*N+.-0/MUWVYX[Z]\W^`_,^ba & Q µ DcEd*e784S;>= ( /3?8?K/32f/3-S@ & - & ;>/360-07840/3-0/36P = (νθ)θ∈Θ /3- & 2`6 & L)/3?840/32 & ?K6949B & 6 ( /3-MQR=)-<ge*)/3-06 θ ∈ Θ GC2 ( /H4M/32Q]hi2i:e607K=)2fθ ∈ M+(X , E) 49B (i& 2 & 6

νθ(A) =

∫

Afθ(x) dµ(x) Q]=)- & ?K?K/ A ∈ E, θ ∈ Θ. jJkFDKl,m

n 6oQRhi2 (i& ;>/32e6 & ?K61406 & 607840607849:pg,q9+.?K@Y/3;>7 (i( /3?cQR=)-o/32 ( =);>7K2i/3-0/361406 & 607840607849:r;>= ( /3?s/3-t ZRuP^ t Z]vcVwVwUsxzys\WuaHZ|VY\c^F\Ds~e* & -0/32 ( /<607K?W/32=)C49/3-0* & 607K=)2 x ∈ X ( /GC2i/3-0/3-*e7sQ]hi2i:`I607K=)2i/32 Lx : Θ → [0,∞) L)7K*)/36*)/ (

Lx(θ) = fθ(x) QR=)- & ?K?8/ θ ∈ Θ. jJkFDem =)-'/36'h ( Q & ? ( x ∈ X & 2iL)7K*)/3-'?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2i/3269+.60gi/ ( /32C4'*N+.- ( 7b7 x Ni] f,w,,p θ ∈ Θ D =)- & 61hi2 ( /3-94060-0/3L)/)O & 61?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2i/325/3-1/32QRhi2i:F607K=)2 & Q θ O*e78?`; & 2=NQR60/i?K=)6W49:e-07K*)/ L(θ) = fθ(x) ON=)Lo7`@i- & :40784Whi2 ( /3-060-0e:F:)/ & Q|IgC+.2iL)7KL)gi/ ( /32 & QP=)C40/3-0* & 607K=)2i/32 x D, & 2 ( -0/4 & ;M;>/32igC+.2iL)//3- ( /36s=);*)/32 ( 62ee60607KL)6& 6dgehC40:)/@CBO & 6oQR=)-949:)/3?K?K7KL)/=)C40/3-0* & 607K=)2i/3-oL)7K*)/3-Q]=)-940:)/3?K?87KL)/S?K7K:)/3?87Kgi=e= ( Q]hi2i:e607K=)2i/3-3O=)L<; & 26 & ?K/3- (i& L)/3-02i/S=);4 & ;>?K7K2iL)/32 & Q & ?K?K/o?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/3- (Lx)x∈X D

l

S3 e F Ni ] A; & 2iL)/r607K?KQ]+.? ( /M/3- ( / 7K2 ( LbBN/32 ( /p69+.60gi/ ( /3-<= *)/3- & ?K6 @Y=b407K607K*)/)D & 2 *e78? (i& =NQR60/& -0Y/ q ( /T;>/ ( ;>7K2`hC4?K=)L & -07K60;>/32 & Q?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/32wO

`x(θ) = − log fθ(x) QR=)- & ?8?K/ θ ∈ Θ.

/3-./3-7K:F:)/dhi2i7K*)/3-949/3?Y/32i78L)gi/ ( =); ge*)=)-0*e7 ( 6 ( /3-49: & ?49+.6060/H4.;>7K2ehC4QR=)- & 2?K=)L & -0786I;>/32wOY;>/32 =);$; & 25QN?KL)/3- ( /32i2i/ :)=)2`*)/32e607K=)2 /3?K?K/3-d/ q/3-dh ( /32fY/36 ( 2i7K2iLp7 ( /36060/: & @i7K60/3?JD ( /3/32 ;>/ (p& 6.6 & L)/1?K=)L & -07K60;>/3-/3- & 669+.60gi/ ( /3-.Q]=)-.4 & ;>;>/32C4 & 6060/o/3:40@A/3-07KI;>/32e60/3- =NQ]60//3-@i-0= ( hi:e60/3- & QY69+.60gi/ ( /3-Q]=)- ( /3?K/3:40@A/3-078;>/32`60/3-sIc=)L ( 784940/.@i-0= ( hi:e60/3-Q]=)-0* & 2 ( ?8/H4607K?w40hi;>;>/3-.*)/ ( /32?8=)L & -07K60;>/360- & 2C4Q]=)-0; & 607K=)2wD

! #"$%& #'(")*+,"$)$-#../1032#41657)$.+4#%

8 7*e78?1496 & -060/ ;>/ ( & 6L)7K*)/ /32 -9+.:F:)/ /3:F40/3;M@i?K/3- @CB ?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/3-M7/32( 7K;M/32C407K=)2wO49Bp; & 2f: & 2fQ Bp/32fQR=)-02i/3;>;M/3?840/ & Qg`*)=)- (i& 2#49B (i& 2i2i/<Q]hi2i:e607K=)2i/3-40/3-h ( D9 u;:H^eX=<^ t?>;@BA %'&)( X1, . . . , X10 *N+.-0/1h & QRgC+.2iL)78L)/)Oi-0/3/3?K?K/14060=): & 49607840:)/o* & -07 & i?8/)O& ?K?8/S/3:F49@A=)2i/32e607 & ? Q]=)- ( /3?K60/S;>/ ( hi:)/32 ( 6d;>7 (i( /3?K*,+.- ( 7 λ ∈ (0,∞) D %'&)(( /32 L q0=)-060/=)C40/3-0* & 607K=)2*N+.-0/ (X1, . . . , X10) = (x1, . . . , x10) g`*)=)-

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10

0.086 0.306 0.291 0.147 0.419 0.779 0.146 0.037 0.527 1.226

( /36g`*)/3-06 Xi g & -69+.60gi/ ( ;>/ ( gi/32C40F2607K? % /3Y/H40L)hi/3;MBN?K/36 m @CB (R, B) Oig & - ( /324078;hi?K6 & 2i/QR=)- ( /3?K782iL & Q (X1, . . . , X10) 69+.60gi/ ( ;M/ ( gi/32C40F2r6078? m10 @CB (R10, B10) O=)L ( /32i2i/407K; hi?K6 & 2i/169+.60gi/ ( /3-L)7K*)/36*)/ ( @i-0= ( hi:e60/36

fλ(x1, . . . , x10) =

10∏

i=1

1

λe−xi/λ, (x1, . . . , x10) ∈ (0,∞)10.

/3-0;>/ ( /3-?87K:)/3?K7Kgi=F= ( Q]hi2i:F607K=)2i/32r40* & -0/32 ( /607K?w=)C40/3-0* & 607K=)2i/32 x = (x1, . . . , x10)

Lx(λ) =

(

1

λ

)10

exp(−10∑

i=1

xi/λ) Q]=)- λ ∈ (0,∞),

=)L>;M7K2`hC4.?K=)LNIJ?87K:)/3?K7Kgi=F= ( Q]hi2i:F607K=)2i/32>/3-`x(λ) = 10 log λ +

1

λ

10∑

i=1

xi Q]=)- λ ∈ (0,∞). jJkFDkem

7R# e e bNi ] A k BfGCL)hi-pkFDKlg & -<*e760/3L)2i/36ML)- & QR/32 =)@ QR=)->49B *)/3??K7K:)/3?87Kgi=e= ( Q]hi2i:e607K=)2#40=); ;>7K2ehC4?K=)LNIJ?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2<Q]=)- ( /325:)=)2i:F-0/360/S=)C40/3-0* & 607K=)2wD

0.0 0.5 1.0 1.5 2.0

0.0

0.1

0.2

0.3

0.4

λ

`(λ)

L(λ

)

0.0 0.5 1.0 1.5 2.0

510

15

λ

`(λ)

L(λ)

7KL)hi-kFDKl "!$#$ &%'(!)* + ,&(-./-+%-#102-/34 546 /3;r+.-0: & 6 (i& 6 & :ehi2f7K2 ( LbBN-17 `x(λ) 78L)/32i2i/3; 49hi;>;>/32 ∑xi D /361/3-1/36 & ?8;>7K2I( /3?K7KL)6Q +.2i=);>/32 & 6 ( /36:ehi2 /3-.*F784940/ & 40@A/3:F60/3- & Q ( /32L q=)-060/1=)C40/3-0* & 607K=)2wO ( /3-.g & -A/36z ( 2i7K2iL Q]=)-?K7K:)/3?87Kgi=e= ( Q]hi2i:F607K=)2i/32wD

9 u;:H^FX < ^ t > @87 %'&)( X *N+.-0/ri782i=);>7 & ? Q]=)- ( /3?K6;>/ ( ?8+.2iL ( /9+9FOKl ;: =)L hi:)/32 ( 640h=<</H494 & 2 ( 40F2i?K7KL)gi/ ( p > 0 D %'&)(p( /32pLHq0=)-060/=)C40/3-0* & 607K=)2p*,+.-0/ X = 34.055 40=); 7/3:F40/3;M@A/3?clNDKlNDi ( /36 X g & -.69+.60gi/ ( ;M/ ( gi/32C40F2p607K?P69+.?K?8/3;MBN?K/36 τ @CB N0 OeL)7K*)/36*)/ (

fp(x) =

(

66, 170x

)

px (1 − p)66,170−x Q]=)- x = 0, 1, . . . , 66, 170,

j]g`*)=)- ( /36hi2 ( /3-Q]=)-94069B)4 & QW69+.60gi/ ( /32/3-2ehi?wh ( /32QR=)- ( /36 & 2iL)78*e2i/1=);>-9B ( /Nm9OC40/3-.*F7& 6?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/32r49* & -0/32 ( /607K? ( /32 LHq0=)-060/=)C40/3-0* & 607K=)2/3-Lx(p) =

(

66, 170x

)

px (1 − p)66,170−x Q]=)- p ∈ (0, 1) ,

=)L<;>7K2`hC4.?8=)LNIJ?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/32M/3-`x(p) = − log

(

66, 170x

)

− x log p − (66, 170 − x) log(1 − p) QR=)- p ∈ (0, 1) .

BfGCL)hi-pkFDfg & -<*e760/3L)2i/36ML)- & QR/32 =)@ QR=)->49B *)/3??K7K:)/3?87Kgi=e= ( Q]hi2i:e607K=)2#40=); ;>7K2ehC4?K=)LNIJ?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2rQR=)- ( /32 :)=)2i:F-0/360/ =)C40/3-0* & 607K=)2wD 8 7s49/3-/32;>/3L)/36o49: & -0@Y/3-0/; & -0:)/3-0/3660=)@5/32 ( 7w/3:40/3;>@Y/3?'kFDKlND

S3 e F Ni ] A

0.46 0.48 0.50 0.52 0.54

0.00

000.

0010

0.00

200.

0030

λ

`(λ)

L(λ)0.46 0.48 0.50 0.52 0.54

010

020

030

040

050

0

λ

`(λ)

L(λ

)

7KL)hi-kFD*+ ( "! #$*&% "!+)8 - " 2-%-#10 3+4 4

6 /3;M+.-0: & 6 ( /32 40*N+.-0/H406607K?KLb+.L)2iL)/3?87KL)/ ( /3? & Qs?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2i/32wO`i7K2i=);>7 & ?K:)=NI/ <7K/32e60/32wO`@CBS497K2<*e7|4/3-h ( /32>Y/36 ( 2i7K2iLIc7Cge*)/3-06Q & ? ( Q]=)-x VP_NX#^e\ & Q?K7K:)/3?87Kgi=e= ( IL)- & Q]/32wDdBN- x /3-.Q & 4060? & L)6/3-i7K2i=);M7 & ?K:)=F/ <7K/32`60/32q0= /32; hi?K607K@i?87K: & 6078*:)=)2C406 & 2e63O49B 60/3L)2i/ ( /; & 2 L)- & QR/325QR=)-

p 7→ px (1 − p)66,170−x QR=)- p ∈ (0, 1) ,

*F7K?K?K/; & 2pQ]B @i-9+<7844 & ;M;>/L)- & Q40=);GCL)hi-kFDTIi?8=)6*F7K?K?8/ y I & :40/32 *,+.-0/S49: & ?K/3-0/36& 2 ( /3-0?K/ ( /H4HD

9 u;:H^eX=<^ t>;@> %c&)( =b4Y/360- & L)60/ ( /32 & Q 40:NBN-02i/d/3:40@Y=)2i/32`607 & ?KQR=)- ( /3?K782iLb40;>= ( /3?iQR- &/3:40/3;>@Y/3?FDKlFOige*)=)-/32 hi2 ( /3-0?K7KL)L)/32 ( /)Ohi=)C40/3-0*)/3-0/36* & -07 & Y/3? Y /3-./3:40@Y=)2i/32`607 & ? IQ]=)- ( /3?K6;M/ ( ;>7 (i( /3?8*,+.- ( 7 λ OP;>/32C4 ( /32 =)C49/3-0*)/3-0/ ( /* & -07 & Y/3? X /3-Q]-0/3;>:)=);>;>/3640=);

X = minY, 300 .

=)- ( /3?87K2iL)/32 & Q X g & -7K:F:)/S69+.60gi/ ( ;>/ ( gi/32C49e2 607K? % /3Y/H40L)hi/3;MBN?K/363OCQR=)- ( /32 g & -@Y=b407K607K*@ihi2i:F694 & 2 ( 40F2i?K7KL)gi/ ( 7Sk :+: D B ( /32 & 2 ( /32 407 ( /g & - Q]=)- ( /3?K7K2iL)/32 & Q Xgi/3?K?8/3-7K:e:)/<69+.60gi/ ( ;>/ ( gi/32C40F2 607K?/36S69+.?8?K/3;MBN?JOAQR=)-49B40:Fhi?K?K/ QR=)- ( /3?87K2iL)/32 *,+.-0/:)=)2=</32e60-0/3-0/36d@CB<69+.?8?K/3?K7KL)6; & 2iL)/@ihi2i:e60/3-3D /32 ;>/ ( ?K7 ( 6=)@GC2 ( 40=);>gi/ ( : & 25; & 25L)= ( 6GC2 ( / /36 ( =);>7K2i/3-0/32 ( /;MBN?JDCEd*e7|4*e7Y/360- & L)60/3-

µ = m(0,300) + ε300 ,

7R# e e bNi ] A & ?K6949BT49hi;>;>/32 & Q % /3A/H49L)hi/3;MBN?K/3694-0/H4060-07K:F607K=)2M6078?P7K2e60/3-0* & ?K?8/36 (0, 300) =)L /360@ihi2i:F6I40;MBN?K/367'@ihi2i:F60/36 300 OP=)L>g`*F784.*F7'40/3-@CB Q]hi2i:e607K=)2i/32

gλ(x) =

1λe−x/λ QR=)- x ∈ (0, 300)

e−x/λ QR=)- x = 300

0 /3?K?K/3-943O49B 7K2 ( 40/3-.; & 2?K/36 & 6

P (X ≤ x) =

∫ x

−∞gλ(x) dµ(x) QR=)- & ?K?8/ x ∈ R .

/36r*e78?407KL & 6MQR=)- ( /3?K782iLb4QRhi2i:F607K=)2i/32 Q]=)-<Q]=)- ( /3?K7K2iL)/32 & Q X /3- ( /32 4 & ;>;>/540=);QR=)- ( /3?K782iLb4QRhi2i:F607K=)2i/32 QR=)- gλ · µ DdLp@CBrL)-0hi2 (f& Q/32`6 ( 7KL)gi/ ( 494+.602i7K2iL)/32QR=)-dQR=)-I( /3?K7K2iLb4Q]hi2i:e6078=)2i/3-3OC;MB Q]=)- ( /3?K7K2iL)/32 & Q X ( /3-Q]=)-og & *)/ 69+.60gi/ ( gλ ;>/ ( gi/32C49e2 607K?µ DE*F784*e7Yg & -.h & QRgC+.2iL)7KL)/1=)C40/3-0* & 607K=)2i/3- X1, . . . , Xn QR- & ( /324 & ;>;>/ & Q]49:,BN-02i/1/3:4I@A=)2i/32e607 & ? Q]=)- ( /3?K7K2iLiO)49Bog & - ( /32M4078;hi? & 6 & 2i/Q]=)- ( /3?K7K2iL & Q ( 784940/.* & -07 & i?K/.2 & 60hi-0?K7KL)*F78469+.60gi/ ( ;>g`63Di@i-0= ( hi:F60;MBN?K/36 µ ⊗ . . . ⊗ µ D =)- & 6.GC2 ( /S/364078;>@A/3?86 h ( 60-0F:MQ]=)- ( /32407K; hi?K6 & 2i/ 69+.60gi/ ( /3- ( /36>gi/32C4078L)60;M+49407KL)6 & 6>=)C40/3-0*)/3-0/ & 6 ( /32 /36 ( 7K;>/32C497K=)2 & ?K/69+.60gi/ ( : & 2549:e-07K*)/H449=);

gλ(x) =

(

1

λe−x/λ

)1(0,300)(x) (

e−x/λ)1300(x)

=

(

1

λ

)1(0,300)(x)

e−x/λ

QR=)- & ?K?8/ x ∈ (0, 300] /3-0;M/ ( Q]BN- ( /32 497K;hi?86 & 2i/Q]=)- ( /3?K7K2iL & Q X1, . . . , Xn 69+.60gi/ (

fλ(x1, . . . , xn) =

n∏

i=1

(

1

λ

)1(0,300)(xi)

e−xi/λ =

(

1

λ

)

∑ni=1 1(0,300)(xi)

e−∑n

i=1 xi/λ

/36060/=*)/3-949+.6060/H4d607K?/325?K=)L)?K7K:)/3?87Kgi=e= ( @CB

`x1,...,xn(λ) =

(

n∑

i=1

1(0,300)(xi)

)

log λ +1

λ

n∑

i=1

xi . jJkFD m

/32i2i/dQ]hi2i:e607K=)2p/3-=)@i60/3L)2i/367YGCL)hi-kFDkFOg`*)=)-.=)C40/3-0* & 607K=)2i/3-02i/1/3- & :F60h & -0L)-0hi@i@A/32QR- & @ihC40?K/H40@i78?840/3:40@A/3-078;>/32`60/367Y/3:40/3;>@Y/3?FDKlFD 8 7Y49/3- & 6.?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2i/325j]=)L?K=)L)?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2i/32m1:ehi2 & Q]gC+.2iL)/3- & Q ( /L q=)-060/ =)C40/3-0* & 6078=)2i/3- x1, . . . , xn

9 S3 e F Ni ] A

100 200 300 400 500

9810

010

210

4

λ

`(λ)

L(λ)

7KL)hi-kFDk &%2"!) ; -% +% 02 * +!;% # )- -%-#10 4*5,4 & *!&!+- +$ ! "002- 0&% % 0 %-+% 0 8)# - ( 2-%-#10 4*5,4

L)/32i2i/3; ( /3-0/H440hi; ∑ni=1 xi =)L L)/32i2i/3; & 2`6 & ?K?K/36 & Qs7K:e:)/I </32C49hi-0/3-0/ ( /o=)C49/3-0* & 607K=NI2i/3-3D 6 /3;M+.-0:7H*F-07KL)6dge*)=)-69+.6dY/H40?8+.L)60/ ( / ( / 60=r?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2i/3-SjJkFDkem.=)L

jJkFD m/3-;>/ ( gi7K2 & 2 ( /32wDPEd*F784.; & 2540:F-0hi/ ( /S=)@Q]=)-1</32C40hi-0L)-9+.2C40/32 7w/3:F49@A/3-07K;M/32I60/363OF49B (i& 2 & 6782iL)/32>=)C40/3-0* & 607K=)2i/3- 7C@i- & :40784 </32C40hi-0/3-0/ ( /H43O49B*F7K?K?K/; & 2<Q BS4 & ;>;M/?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2540=);7 ( /361/3:40@A/3-078;>/32`6dg`*)=)-; & 2f40?K/36d7K:e:)/ g & * ( / 69+.2i:e64078L& 61</32C49hi-0/3-0/)D -07<9:)/363O ( /3-QBN-060//32<69+.60gi/ ( @CBS60hiA=)-0LNQ]=)-0; = *)/3- 7i/36-0/32`6; hi?K607K@i?K78: & 607K*F6'h ( 60-0F:AO/3-L)= ( 6 & 66 & L)/>607K?4078LiD & 2iL)/<69+.60gi/ ( /3-3Oc; & 2 406$ ( /3-@CB 7@i- & :40784HO49:e-07K*)/H4o2 & I60hi-0?K78L)6d@CBp60hiY=)-0LNQR=)-0;$I;>/32 ( /32;rB ( / & 640:F-07K*)/T69+.60gi/ ( /3-02i/M@CB>Q]hi2iL)/3-0/3-7K:F:)/49+.-0?87KL L)= ( 63Oi2CBN-4B (i& 2i2i/S69+.60gi/ ( /3-d40: & ?wL & 2iL)/H44 & ;M;>/32wD

1. % 57.+"*41)4+0 "$)$+,"$)$-#../1032#4 5).+41,4

sQ]=)-0i7K2 ( /3?840/;>/ ( :)=)2i:)=)- (i& 2C40=*)/3-0*)/ q/3?|40/3-QR=)-060=)?K:)/ ( /d*e7Y4060=)-0/o*,+.- ( 7K/3- & Q'/32M69+.6Igi/ ( 40=); h ( 60-0e:Q]=)- & 6 ( /32 @CBNLb+.? ( /32 ( /.=)C40/3-0* & 607K=)2* & -s p hi2 ( /3-/36s:)=)2i:e-0/364 & 2 ( 49e2i?K78L)gi/ ( 40;MBN?JDb=1496$N-0-0/69+.60gi/ ( O q0=1A/ ( -0/)D dh *)/32 ( /3-s*F7i407K60h & 607K=)2i/32,*F7eg & -

A KF , e FbNi ] A )

:ehi232f=)C40/3-0* & 6078=)2wOY;M/32 ; & 2iL)/ 4 & 2 ( 40F2i?K7KL)gi/ ( 49;MBN?JD ( QR- & /3640e;M;>/360-07 & -0L)hI;>/32`6;MB *F7'40/?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2i/32r40=); /36h ( 60-0F:rQR=)-ge*)=)-L)= ( 6 ( /QR=)-949:)/3?K?K7KL)/4 & 2 ( 40F2i?K7KL)gi/ ( 40;MBN?./3->7:)=)2i:)=)- (i& 2C4M;>/ ( =)C49/3-0* & 607K=)2i/32wD )=[406$N-0-0/ 69+.60gi/ ( Owq0=406$N-0-0/:)=)2i:)=)- (i& 2C43D % 7K:)/3?87Kgi=e= ( Q]hi2i:e607K=)2i/32rh ( 6 & ?K/3-497KL & ?K6949BS=); g`*)=)- p ( //32i:)/3?K60/@ & - & ;>/360/3-0*N+.- ( 7K/3-/3-3OC40/367'?8F40/36 & Q ( /32LHq0=)-060/S=)C40/3-0* & 607K=)2wD /3-/3-/36*N+.? ( & QsQ & ? ( L)-0hiY/3-.7 ( /32i2i/T40F;>;>/360-07KY/360- & L)602i7K2iLiD 8 7w*F7K?Y:ehi26078?K?8+.L)L)/( /32>7K2`60hi78607K*T*N+.L)63Ob=)LS7K:F:)/Y/360- & L)60/ ( /32M40=); /36?K=)L)7840:60*e782iL)/32 ( / & -0L)hi;>/32`63D =)L*e7K?g`*)/3-06<@i-9+<7|406 -0/H40hi?K6 & 6 *e7*F7840/3- L)7K*)/406$N-0-0/ =)L 406$N-0-0/ *,+.L)6<607K? ( /32/3-Q & -07K2iL& 6>?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2i/32 /3- ( /32 @i-07K;M+.-0/5C+.-0/3- & Q7K2QR=)-0; & 607K=)2=); 4 & ;r40@i7K?K?K/36;>/3?K?K/3;=)C49/3-0* & 607K=)2 =)Lo;>= ( /3? D 8 7F*F7K?Aj]7F;>= ( 49+.602i7K2iL6078?`/32 ( /3? & 2 ( -0/.406 & 607849607K:)/3-0/Nm=);>g`FL)L)/3?K7KL)6hi2 ( LbB & 6 ?8+.L)L)/2i=)L)/36-0/3?K7KL)7 )4067C?K7K:)/3?K78gi=e= ( A/3L)-0/3Y/36 ( /;>/360= ( /3- *F7h ( ?K/ ( /3-w@CB & L)L)-0hi2 (S& Q?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2i/32wO 40: & ?)*Fhi- ( /3-0/H4'@CB ( /3-0/H4c/Y/3:F607K*F7K60/363O=)L ( /49: & ?'*)/ q0/H4d=)@ ;>= (5& 2 ( -0/ ;>/360= ( /3-I ( /36d/3-7K:F:)/S/36 & -0L)hi;M/32`6o7W407KLr40/3?K* & 6( //3-QRhi2 ( /3-0/36@CB = *)/3-0*)/ q0/3?840/3-=*)/3-?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2i/32wDn 6>@i-0=)i?K/3; Q]=)- Q]=)-060=)?K:e2i782iL)/32 & Q1?87K:)/3?K7Kgi=F= ( Q]hi2i:F607K=)2i/32[/3- & 6>69+.60gi/ ( /3-r7K:F:)//3-/32`6 ( 7KL)6L)7K*)/363DWEd*e7|4 νθ = fθ · µ OW49B5: & 2 *F7+.2 ( -0/ fθ @CB5/32 µ IJ2ehi?K;M+.2iL ( /)Oc=)L( /32r-0/H40hi?860/3-0/32 ( /Q]hi2i:e6078=)2 fθ /3-.406 &)( 7KL/32r69+.60gi/ ( Q]=)- νθ ;M/ ( gi/32C40F2r6078? µ DeEd*e7|4/32360@ihi2i:e6940;r+.2iL ( / x0 /3- E IJ;MBN?K/3?87KL>;>/ ( µ IJ;MBN?2`hi? O4B : & 2f*F77s@i-0782=<7K@i@A/3649+.6060/

fθ(x) =

0 g`*F784 x = x0

fθ(x) g`*F784 x 6= x0.

& 2C40/36 g`*)=)- (i& 2 ( /32 =)@i-07K2 ( /3?K78L)/>?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2 Lx0 40* & -0/32 ( /r607K?=)C40/3-0* & I607K=)2i/32 x0 49B h ( OCg & -*F7w2`h5/3-9496 & 6060/36 ( /325;>/ ( /3252`p?K7K:)/3?87Kgi=e= ( Q]hi2i:e607K=)2 Lx0

( /3-/3-:)=)2C496 & 2`6 :in ?K?8/3-40/3-.h ( @CB /32g`*F7K?K:)/32 40=); gi/3?8406 & 2 ( /32;MB ( /1*F7;MBN6060/oGC2 ( /QR=)-1L)= ( 63D 8& -07 & 607K=)2i/3- & Q ( /36060/ & -0L)hi;M/32`6L N-/32 ( *F7 ( /3-0/ & 61*e77*F784940/ /3:F49/3;>@i?K/3-: & 2 +.2 ( -0/1@CB, .?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/3-g`*F784*e7Y;MBN6060/ N2C40:)/ ( /363D dL ( /3-Q]=)-40: & ?; & 2*,+.-0/* & -940=); ;>/ (& 6<6078?K?8+.L)L)/ /32 :)=)2i:F-0/36>?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2#2i=)L)/3240/3?K*4069+.2 ( 7KL A/36z ( 2i7K2iLiDw; & 2iL)/;>= ( /3?8?K/3-3O ge*)=)-sCB ( /.@ & - & ;>/360/3-0;r+.2iL ( /32 Θ =)L1-0/3@i-9+40/32e6 & 607K=)2C40-0hi;>;>/36X /3- ( /3?K;M+.2iL ( /3- & QC/3hi:F?K7 ( 7840:)/-0hi; O /3- ( /36W; hi?K7KL)6 & 6s*N+.?KL)/ 69+.60gi/ ( /3-02i/ fθ 4B (i& 2& 6 & QRi7K? ( 2i7K2iL)/32

(θ, x) 7→ fθ(x)

40=); QRhi2i:F607K=)2 & Q60=* & -07 & i?K/g & -'@C+.2i//3L)/32C40: & A/3-3OQD /3:43D & 6 ( /32/3-':)=)2e607K2`hi/3-06W/3?K?K/3-?K7KL)/Q]-0/3; C∞ DC49B Q & ? ( /3- ( /3-782iL)/32 ( /3-:ehi2i2i/ ( -$N;>;>/1=); & 6; & 2i78@ihi?K/3-0/1;>/ (( 784940/.-0/3L)hi?8+.-0/69+.60gi/ ( /3-3D /32>ge*)=)-07 ( /36Y/36 ( 2i7K2iLb4Q]hi? ( /.*)/ ( -0/3L)hi?8+.-0/69+.60gi/ ( /3-

S3 e F Ni ] AY/H4069BN-5j]Y=)-06940/36<Q]- & & 6 ( /L N- & 2 & ?K40/32 2i/3;>;>/3-0/NmS/3- ( /36M40*N+.-06 & 6>407KL)/ 2i=)L)/36@i-9+<78496=);5DEd*e7|4 (νθ)θ∈Θ /3- ( =);>7K2i/3-0/36 & Q µ OA407KL νθ = fθ · µ OY49B>/3-dQ & ;>7K?K7K/32 =)Lb49B ( =);>7K2i/3-0/36& Q cµ Q]=)-/36*F7K?K:NBN-0?K7KL)6 c > 0 OC=)LM*e7'40/3-?8/36 & 6

νθ =fθ

c· cµ.

Ed*e7|4d*F7s: & ? ( /3- ( /32 =)@i-07K2 ( /3?87KL)/?87K:)/3?K7Kgi=F= ( Q]hi2i:F607K=)2 j] & 40/3-0/36@CB µ m.QR=)- Lx(θ) =)L?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2i/32 & 49/3-0/36 @CB ( /36 ( =);>7K2i/3-0/32 ( /d;MBN? cµ Q]=)- Lc

x(θ) OF49BT49/3- *F7 & 6Lc

x(θ) =fθ(x)

c=

1

cLx(θ).

8 7P40/3- & 6 Lcx =)L Lx /3-@i-0=)@A=)-06078=)2 & ?K/)O`40:F-0/3*)/36 Lc

x ∝ Lx D /3-L)7K*)/3-2C+.@i@Y/;>/32i782iL& 6gC+.* ( / & 6 µ 40=); ( =);M7K2i/3-0/32 ( /.;MBN?g & -/32<=)Fq/3:F607K*Q]=)-060-07K2C49406078?K?K7K2iL.Q]-0/3;<Q]=)- cµ O49B *F7?K/ ( /H4607K? & 6:)=)2i:F?Kh ( /3-0/ & 6 ( /32 & C40=)?Khi6060/406$N-0-0/3?849/ & Q Lx(θ) 78:e:)/og & -2i=)L)/32Y/36 ( 2i7K2iLiD /36 ( /3- : & 2#Q]=)-060=)?K:)/H4T/3-<xzVP_NvcV t Us^bap;>/3?K?K/3;"?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/32C4*N+.- ( 7K/3-7w60=>@ihi2i:F60/3-PEd*e784

Lx(θ1)

Lx(θ2)> 1

49B /3- x A/ ( -0/M7:)=)2i:)=)- (i& 2C4T;>/ ( νθ1 /32 ( ;>/ ( νθ2 I ( /H4060=5406$N-0-0/>i-$N:)/32[/3-3Oq=406$N-0-0/p/3-QR=)-940:)/3?8?K/32 7:)=)2i:)=)- (i& 2C4HD 8 749/3- & 6 i-$N:)/32#7K:F:)/r+.2 ( -0/3- 407KLiOc2CBN-; & 2+.2 ( -0/3-L)-0hi2 ( ;rBN?K/36Q]- & µ 6078? cµ D

5 /#. )$4#%7,4#/# ' "$ %7. "$"$

8 784940/ 496 & 607840607840:)/ ;M= ( /3?K?8/3- /3- 7K:F:)/ ( =);>7K2i/3-0/ ( /)D n 6 /3:40/3;>@Y/3? /3- ( / 7K:F:)/I@ & - & ;>/360-07|40:)/ ;>= ( /3?8?K/3-SQ]- & /3:F40/3;M@A/3?FDKl)lND /32 ? & 2iL)6 ( /C/H4060/7.@i- & :40784QR=)-0/I:)=);>;M/32 ( /T406 & 607840607|40:)/;>= ( /3?K?K/3-^`_ ( =);>7K2i/3-0/363OY40=);-0/3L)/3? & Q/36 % /3Y/H40L)hi/3;MBN?c/3?KI?K/3-d/36d69+.?8?K/3;MBN?JD /32 ( / /3-o=)Lb49B ( =);>782i/3-0/36 & Q; & 2iL)/ & 2 ( -0/ ;MBN?JD /36o:)=)2i:e-0/360/( =);>782i/3-0/32 ( /S;MBN?w/3- ( /3-Q]=)-2i=)L)/36*F7'40/3?K*r*,+.?KL)/3-3OP2CBN-*e7 0F ;>= ( /3?K?K/32wD 8 7*F7K?s7 ( /36060/ & Q 402i7K6o7K2 ( :e-0/ ( 49/)OAge*e7K?8:)/32 Y/36 ( 2i7K2iLp* & ?KL)/36 & Q ( =);M7K2i/3-0/32 ( /;rBN?Wg & -Q]=)-g`*)=)- (i& 2 ?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/32r49/3-h ( D9 u;:H^eX=<^ t?>;@ %'&)( X *N+.-0//3:F49@A=)2i/32e607 & ? Q]=)- ( /3?K6;M/ ( @ & - & ;>/360/3- λ > 0 DA~e=); 7/3:40/3;>@Y/3?WkFDKlS40/3-d*e7 & 6 ( /32i2i/496 & 607840607840:)/;M= ( /3?w/3- ( =);>7K2i/3-0/36 & Q % /3Y/H40L)hi/3;MBN?K/36

S F A Fd 8. R

m @CB R Oi=)L>*e7w: & 2i-0hiL)/S69+.60gi/ ( /32

fλ(x) =

1λ e−x/λ x > 0

0 x ≤ 0,

g`*F7K?K:)/36QBN-0/3-6078?w?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2Lx(λ) =

1λ e−x/λ x > 0

0 x ≤ 0.

?K60/3-02 & 6078*e6w: & 2S*F7 ( =);>7K2i/3-0/;M/ ( m(0,∞) = 1(0,∞)·m O & ?K6949B;>/ (S% /3Y/H40L)hi/3;MBN?K/3694-0/H4060-07K:F607K=)2 607K? ( /325@A=b497K607K*)/og & ?K* & :40/)DC49B Q & ? ( : & 2*F7wi-0hiL)/S69+.60gi/ ( /32gλ(x) =

1

λe−|x|/λ x ∈ R. jJkFD em

/3-Lb+.? ( /3-2i/3;>?K7KL & 6∫

Agλ(x) dm(0,∞)(x) =

∫

Agλ(x) 1(0,∞)(x) dm(x) =

∫

Afλ(x) dm(x)

QR=)- & ?K?8/ A ∈ R D C& :F607840:49/3-*e7 & 6 ( /36/3-78-0-0/3?K/3* & 2e6g`*)=)- (i& 2r*e7 ( /GC2i/3-0/3- gλ @CB ( /322i/3L & 607K*)/g & ?K* & :F40/)D /32 ( /36 & 2iL)7K*F2i/1* & ?8L & Q gλ QN-0/3-607K?w?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2i/32Lx(λ) =

1

λe−|x|/λ x ∈ R.

8 740/3- & 6 Q]=)- x > 0 /3- ( /3->7K2iL)/32 Q]=)-940:)/3?@CB Lx =)L Lx I ( /60=fQ]hi2i:e6078=)2i/3-</3-407K;>@Y/3?K60gi/32p/32C43D /32 Q]=)-2i/3L & 607K*)/ x /3-/3- Lx =)L Lx *F7 ( 6.Q]=)-940:)/3?K?K78L)/)Dn - ( /36 Lx /3?K?K/3- Lx

( /3-/3- ( /32 -078L)607KL)/d?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2 /3-./3-*e7|4940/L)-0hi2 ( /607K? & 6 QR=)-0/360-9+.:F:)/ Lx D /32r7 ( /36.4060=)-0/d=)L gi/3?K/d/3- ( /o?K7KL)/L)= ( /)OF=)L ( /3-/3-4]q0+.? ( /32e62i=)L)/325L)-0hi2 ( 607K? & 640:)/3?82i/;>/3?K?K/3; ( /3;5D //3-:Fhi2hi/32i7KL)/1QR=)-=)C49/3-0* & 607K=)2i/3- ( /3-& ? ( -07KLQ]=)-0/3:)=);>;>/3-hi2 ( /3-;>= ( /3?K?K/32e4 & ;60?87KL)//3:40@A=)2i/32e607 & ? Q]=)- ( /3?K7K2iL)/3-6078? ( /3?K/3-q0=(−∞, 0] 4 & 2 ( 40F2i?K7KL)gi/ ( /32 : D

E*F784*e7 & <</3@i60/3-0/3- & 6 ( /36:ehi2 /3--0/3? & 607K*)/oQR=)-0gi=)? ( ;>/3?K?K/3;?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2i/32C4*,+.- ( 7K/3-17Q]=)-940:)/3?K?K78L)/@ihi2i:F60/3-3O ( /3-Y/36 ( /3-S2i=)L)/363Oc40: & ?*e740/ & 6S* & ?KL)/36 & Q ( =);>7 I2i/3-0/32 ( /r;MBN? 7K2iL)/32 -0/3/3? -0=)?8?K/M40@i7K?8?K/3-3D 6 =)-06949/36SQ]- & Q]=)-S2i=)L)?K/M/3:4060-0/3;>/M@ihi2i:e60/3-7-0/3@i-9+40/32`6 & 607K=)2C49-0hi;>;>/36I'@ihi2i:F60/3- ( /3-7 QN?KL)/.;>= ( /3?8?K/32 49/3?K*T40?K/367K:F:)/.*e78?F=)@i60-9+ ( /40=); *F7K-0:)/3?K7KL)/;rBN?K/3-0/H40hi?K6 & 60/3-I./3- & ?8?K/S?K7K:)/3?87Kgi=e= ( Q]hi2i:e607K=)2i/3-.Y/H4060/3; 6 & Q ( /32 496 & I607840607840:)/;M= ( /3?s40/3?K*YOP7K:F:)/ & Q ge* &)( *e7W*,+.?8L)/3- & 6oA/32ee6060/ 40=);$-0/QR/3-0/32=</3@ihi2i:F612CBN-

: S3 e F Ni ] A*F740: & ?Y/H40:F-07K*)/;M= ( /3?K?8/32wD 61Q]=)-0:e? & -0/ 7g`*)=)- (i& 2 ( /36gC+.2iL)/3-S4 & ;>;>/32wOY:e-9+.*)/3-*F7wh ( *e78:e?K/3-.?K7 ( 660/3=)-07JD8 74078L)/3- & 6T/32 406 & 607849607840:;>= ( /3? P @CB (X , E) /3-Z]\YaH^`_N\YafUsVAX#Z]\c^e_^`aMg`*F784 ( /3-GC2 ( /H4/36 ν0 ∈ P 49B (i& 2 & 6

ν ν0 QR=)- & ?8?K/ ν ∈ P.

s60/3=)-0/3607840:)/d4 & ;>;>/32igC+.2iL)//3- ( /36=NQR60/dA/3:F*)/3; 6 & 6 & -0Y/ q ( /;M/ ( 7K2`60/3-02e6 ( =);>782i/I-0/ ( /<;>= ( /3?K?K/3-3D dLb49Bp7s@i- & :F607840:)/ 607K? Q +.? ( /: & 2 ( /36*,+.-0/ 2 ( *)/32 ( 7KL)63OQ D /3:43DAQR=)- ( 7; & 2 & -0A/ q ( /3-S;>/ ( :)=);>@i?87</3-0/ ( /<-0/3@i-9+40/32`6 & 6078=)2C40-0hi; (X , E)

( /3-TZRuwuP^@CB Q]=)-IgCBN2 ( /3- h ( 406 F-0/36 ;>/ ( /36 L)-0hi2 ( ;MBN?JD~e60=)-06 49/36 & ?K?8/ ( /p;>= ( /3?8?K/3-*e7*F7K? & -0Y/ q ( /;>/ ( /3-782`60/3-02`6 ( =);>7K2i/3-0/ ( /)D9 u;:H^eX=<^ t7> @ %'&)( (νθ)θ∈Θ *N+.-0/ /32@ & - & ;>/360-07840/3-0/36s406 & 607840607849:;>= ( /3?b@CB (X , E) O49B (i& 2 & 6 νθ = fθ · µ Q]=)- & ?K?K/ θ D 2`6 & L & 6

fθ0(x) > 0 Q]=)- & ?K?K/ x ∈ X .

1& /3-.Q]=)- & ?K?K/ θ ∈ Θ

νθ = fθ · µ =fθ

fθ0

fθ0 · µ =fθ

fθ0

· νθ0 .

?K6949B /3-Q & ;>7K?K78/32M7K2e60/3-02`6 ( =);>782i/3-0/363Dn :40/3;>@Y/3?kFD *e7849/3-1Id7Q]=)-0; & Q69+.60gi/ ( /32 jJkFD emI & 6 ( /32i2i/ & -0L)hi;M/32`6 & 607K=)2 : & 2i-07K2iL)/H47 & 2e*)/32 ( /3?840/7FQ]=)-0i7K2 ( /3?840/.;>/ ( /3:40@Y=)2i/32`607 & ? Q]=)- ( /3?K7K2iL)/3-3O,;>/ ( /32<*F7K?K:NBN-0?K7KL& Q.;>= ( /3?K?K/32C4o4 & 2 ( 40F2i?K7KL)gi/ ( 40;MBN?40=); ( =);>7K2i/3-0/32 ( /<;rBN?JDdLL)/32i/3-0/3?K6: & 2f; & 2;>/ ( ?K7 ( 6Q & 2e6 & 4077* & ?KL)/36 & Q ( /361@i-0/3?K7K;M7K2C+.-06 ( =);>7K2i/3-0/32 ( / ;MBN? OP*F7840/ & 6oQ]=)-0i? ,QRIQ]/32 ( /S; & 2iL)/S406 & 607|40607840:)/o;>= ( /3?K?K/3-./3-7K2`60/3-02e6 ( =);>7K2i/3-0/ ( /)D

~e=);$607 ( ?K7KL)/3-0/T2C+.*e2e6d*F7K?'; & 2 7c@i- & :40784L)/3-02i/ & -0Y/ q ( /;>/ ( ?8=)L & -07K60;>/32 & Q?K78:)/3?K7 Igi=F= ( Q]hi2i:F607K=)2i/32wDwE*F784 ( /36S40: & ?sL)78*)/ ;M/32i7K2iLiO;MB & ?K?K/ 69+.60gi/ ( /3-q=*N+.-0/ = *)/3- & ?K6@Y=b407K607K*)/)D dL ( /3-0;M/ ( g & -*F7PQ & :e607|40:>4 & L)6 & 6;>= ( /3?K?K/32r/3-782`60/3-02`6 ( =);>782i/3-0/36 & 2 & I?K=)L)6o;>/ ( /3:F49/3;>@A/3?kFD r40/3-1*e7s2i/3;>?K7KL & 6o/360ge*)/3-06S4 & 2 ( 40F2i?K7KL)gi/ ( 40;rBN?c7s;>= ( /3?8?K/32: & 2i-0hiL)/H440=); ( =);M7K2i/3-0/32 ( /S;MBN?

d; *)/32 ( 63O ge*e784 & ?8?K/ ;MBN?7;>= ( /3?K?K/32 : & 2 i-0hiL)/H4M40=); ( =);M7K2i/3-0/32 ( /;MBN?JO49B /3-& ?K?8/ @ & - & Q;rBN?Q]- & ;>= ( /3?8?K/32 +.:F*e78* & ?K/32e60/)O=)L ( /3-0;>/ ( : & 2 *e7 & 2`6 & L)/ & 6 & ?K?8/

S F A Fd 8. R l69+.60gi/ ( /3- ;>/3?K?8/3; ;>= ( /3?K?K/32C4;rBN?i/3-@A=b497K607K*)/)D`~FBi-0hiL)/H4/36 & QY;>= ( /3?K?K/32C4s;MBN?C49=);( =);>7K2i/3-0/32 ( /.;MBN?JOL)7K*)/3- ( /36s;M/32i7K2iL & 66 & L)/.?K=)L & -07K60;M/32607K?F?K7K:)/3?87Kgi=e= ( Q]hi2i:e607K=)2i/32wD9 u;:H^FX < ^ t >;@ %'&)( νθ *N+.-0/<?K7KL)/Q]=)- ( /3?K7K2iL)/32@CB 7K2`60/3-0* & ?K?K/36 (0, θ) O'g`*)=)- θ > 0/3-Thi:)/32 ( 63D 8 740/3- & 6 νθ = 1(0,θ) · m OW49B ;>= ( /3?8?K/32 (νθ)θ∈(0,∞) /3- ( =);M7K2i/3-0/36 & Q% /3A/H49L)hi/3;MBN?K/363D /32 ;>= ( /3?K?K/32>/3-.Z|uwuA^<7K2e60/3-02`6 ( =);>7K2i/3-0/363D =)-.g`*F784 νθ Y/360/3L)2i/3-?K7KL)/Q]=)- ( /3?K7K2iL)/32p@CB (0, θ) OP4B g & -*e7Q]=)- & ?K?K/ θ & 6

νθ

(

(θ, 2θ))

= 0, ν2θ

(

(θ, 2θ))

=1

2.

/3-0;M/ ( : & 2 ν2θ 7K:F:)/g & *)/S69+.60gi/ ( ;>/ ( gi/32C40F25607K? νθ D

8 7Y407KL)/3- & 6/36 σ IJ/32 ( /3?87KL)6 ;MBN? µ /3-X#ZR\WZRXt aUsVAX#ZR\W^`_,^e\WUW^TQ]=)- ( /32 406 & 607840607|40:)/

;>= ( /3? P @CB (X , E) OCg`*F784ν µ QR=)- & ?8?K/ ν ∈ P,

=)L<g`*F784 ( /36.Q]=)-/360g`*)/3-06 σ IJ/32 ( /3?K78L)6;MBN? ξ Lb+.? ( /3- & 6ν ξ ∀ν ∈ P ⇒ µ ξ.

E*F784; & 2pg & -.60= ;>7K2i7K; & ?K6 ( =);>7K2i/3-0/32 ( /1;MBN?AQR=)-.;>= ( /3?K?K/32 P 49B ;MB ( /12 ( *)/32I( 7KL)*e7|4*N+.-0/S+.:F*e7K* & ?K/32e60/)D9 u;:H^FX < ^ t > @ Ed*e784;>= ( /3?K?K/32 P /3-d782`60/3-02`6 ( =);M7K2i/3-0/36 & Q ν0 ∈ P O4BM/3- ν0 /36;>7K2i7K; & ?86 ( =);M7K2i/3-0/32 ( /d;rBN?JD =)-.g`*F784 ξ /3-/36*F7K?K:NBN-0?K7KL)6 ( =);>7K2i/3-0/32 ( /o;MBN?JOe49BT;rBν0 @Y/3- ( /GC2i7K607K=)2g & *)/T69+.60gi/ ( ;M/ ( gi/32C40e26078? ξ D /3-0/SL)/32i/3-0/3?863OPg`*F784 P /3- ( =);>7K2i/3-0/36 & Q/32569+.?8?K/3?K7KL :)=)2`*)/3:4:)=); i7K2 & 607K=)2 & Q;rBN?QR- & Q & ;>7K?K78/32wO

µ =∑

n=1

anνn, jJkFD 9emg`*)=)- νn ∈ P Q]=)- & ?K?K/ n ∈ N Oc=)L5g`*)=)- (an)n∈N /3-S/32[QBN?8L)/ & Q.78:e:)/IJ2i/3L & 607K*)/M6 & ?( /3- 49hi;>;>/3- 607K?olNO4B ;rB ( /36060/ µ *,+.-0/ ;>782i7K; & ?K6 ( =);M7K2i/3-0/32 ( /)D =)-<g`*F784 ξ /3-/32 ( 2`h /36 ( =);>7K2i/3-0/ ( / ;MBN?JOY49BM/3- νn = fn · ξ QR=)-1@ & 4940/32 ( / 69+.60gi/ ( /3- fn ∈ M+ O=)L ( /3-0;>/ ( /3-

µ =∞∑

n=1

an (fn · ξ) =

(

∞∑

n=1

anfn

)

· ξ.

S3 e F Ni ] A8 7W*F7K?'2ehfL N-0/-0/ ( / Q]=)- & 6o/32ig`*)/3- ( =);>7K2i/3-0/36o406 & 607840607|40:pQ & ;>78?K7K/1g & -d/36o;>7K2i78; & ?K6( =);>782i/3-0/32 ( /p;rBN?1j]ge* &)(( /3- 7K:F:)/p/3- :F? & -06 @CB QR=)-0gCBN2 ( m9O=)L & 6 ( /36060/ ;>7K2i78; & ?K6( =);>782i/3-0/32 ( /;MBN?wg & -Q]=)-0;>/32 jJkFD 9em9D 8 7w*e7K?i-0hiL)/S2i=)6 & 6078=)2i/32

Bf = x ∈ X | f(x) > 0 Q]=)- f ∈ M+(X , E).

n 2 ;M+.2iL ( / & QcQ]=)-0;>/32C =

∞⋃

n=1

Bfn

Q]=)-/32 QBN?KL)/ (fn)n∈N

& Q M+ I QRhi2i:F607K=)2i/3-: & ? ( /H4W/32>u<Us^CD 8 7F6 & ?8/3-s=);/32 D IJ:,+ ( /)Oge*e784 & ?K?K/ fn /3-02i/p?K7KL)L)/3-T7/32#L)78*)/36;M+.2iL ( / D ⊂ M+ D 6 /3;M+.-0: & 6 /32#69+.?K?8/3?K7KLQ]=)-0/32i7K2iL & Q D IJ:N+ ( /3-7KL)/32 /3-/32 D IJ:,+ ( /)D ^eX X >;@

(X , E, µ) σ ! "$#%'&()* D + ,!- . M+(X , E) /10'324 $5 D 768 95

µ (Bf \ C) = 0. ( :; f ∈ D. jJkFD m

<>=?A@:B 'QN-94060/ =);>L & 2iL & 2`6 & L)/3-*e7 & 6 µ /3-/36S4 & 2 ( 40F2i?K7KL)gi/ ( 40;MBN? D 8 7*e7K?WGC2 ( //32 D IJ:,+ ( /;M/ ( 406$N-9406; hi?K7KL)6 µ IJ;MBN?c~F+.6

α = supµ(C) | C /3-/32 D IJ:,+ ( / .1&

µ /3- /36 4 & 2 ( 49e2i?K78L)gi/ ( 40;MBN?JOw/3- ( /36:F? & -06 & 6 α ≤ 1 D 8 7: & 2#GC2 ( / D IJ:N+ ( /3-C1, C2, . . . 49B

µ(Cn) > α − 1

n.

( /36.QR=)-0/32i782iL)/32 & Q ( 7|4940/ D IJ:N+ ( /3-40/3?K*r/3-/32 D IJ:N+ ( /)OCg & -*F7 & 6α ≥ µ

(

∞⋃

i=1

Ci

)

≥ µ(Cn) > α − 1

n,

=)L<*)/ ( & 6? &)( / n → ∞ Oi40/3-.*e7 & 6 ⋃∞i=1 Ci /3-/32 D IJ:,+ ( /1;>/ ( ;MBN? α O ( /36*e7K?407KL)/;>/ ( ; & :F4078; & ?K6 ;rBN?JD %'&)( =b4.Q]-0/3;>= *)/3-: & ? ( / ( /32i2i/S:N+ ( / C D

Ed*e7|4 f ∈ D OY4B>/3- C ∪ Bf /32 ( 2`hf/32 D IJ:,+ ( /)D7 & :407K; & ?K7K60/360/32 & Q C L N- & 6 ( /322e)/S:,+ ( /78:e:)/1: & 2pg & *)/T406$N-0-0/;MBN?/32 ( C 49/3?K*AD dL ( /3-Q]=)-;MB Bf \ C g & *)/;MBN?: D dL<*F7wg & -49BN?8/ ( /H4.*e784961jJkFD m9D

S F A Fd 8. R kE*F784 µ /3-1/36 σ IJ/32 ( /3?K7KL)61;MBN?JOA;>/32f7K:e:)/ 2 ( *)/32 ( 7KL)*e7|4/36S4 & 2 ( 49e2i?K78L)gi/ ( 40;MBN?JOP4BGC2 ( /H4 ( /3-/364 & 2 ( 49e2i?K78L)gi/ ( 40;MBN? µ O ( /3-/3-+.:F*e78* & ?K/32e6;>/ ( µ D 8 7Ag & - ?K7KL)/QRhi2 ( /36/32 D IJ:N+ ( / C 49B µ(Bf \ C) = 0 Q]=)- & ?K?K/ f ∈ D D /32 (i& µ =)L µ /3-+.:F*e7K* & ?K/32e60/)Og & - ( /T4 & ;>;>/12ehi?K;M+.2iL ( /3-3D dL ( /3-0;>/ ( /3-SjJkFD m =)@QRF? ( 63D

2

a \WZR\ > @t X[V: P^ 9 5 ( , P (X , E) ( :&5

2 $5 ν1, ν2, · · · ∈ P 5 4! ";

λ =

∞∑

n=1

2−n νn jJkFD m

()85 ( : P /

< = ?A@ B 2e6 & L & 6 P /3- ( =);>7K2i/3-0/36 & QA/36 σ IJ/32 ( /3?87KL)6;MBN? µ D =)-g`*)/3-06 ν ∈ P *N+.?KL)/3-*e7w/3269+.60gi/ ( fν ∈ M+ OC49B ν = fν · µ DA~F+.6

D = fν | ν ∈ P,

=)L<? &)( =b440:F-07K*)/ Bν 7c4960/ ( /36.QR=)- BfνDC QN?KL)/1?K/3;>; & kFD GC2 ( /H4 ν1, ν2, · · · ∈ P 49B

µ

(

Bν \∞⋃

n=1

Bνn

)

= 0 Q]=)- & ?K?K/ ν ∈ P .

/GC2i/3- λ h ( Q]- &>( 784949/ νn /3-=)L jJkFD m9DAE*F784 λ(A) = 0 Q]=)-/32;M+.2iL ( / A ∈ E Oi;MB( /3-2 ( *)/32 ( 7KL)*F784.Lb+.? ( / & 6 νn(A) = 0 Q]=)- & ?K?8/ n D ?K6949B & 6∫

Afn dµ = 0,

49B *e7w: & 2 Q & :e607849:M40?Khi6060/ & 6µ(A ∩ Bνn) = 0. =)-/360g`*)/3-06 ν ∈ P g & -*F7w2`h & 6

µ (A ∩ Bν) = µ

(

A ∩ Bν ∩∞⋃

n=1

Bνn

)

+ µ

(

A ∩ Bν \∞⋃

n=1

Bνn

)

≤∞∑

n=1

µ (A ∩ Bνn) + µ

(

Bν \∞⋃

n=1

Bνn

)

= 0 .

S3 e F Ni ] A /32gi/3- & Q40/3-*e7 & 6

ν(A) =

∫

Afν dµ =

∫

A∩Bν

fν dµ = 0.

n 2 gi/32`*F78402i7K2iL<607K? &)( =)2I 7K:)= ( F;M4.49+.602i7K2iL Q]=)-069+.?K?K/3-2`h & 6 ν g & -69+.60gi/ ( ;>/ (gi/32C40F2607K? λ O ( /3-49BN?K/ ( /H4 ( =);>782i/3-0/3-;>= ( /3?8?K/32 P D

2

%'&)( 2`h P = (νθ)θ∈Θ *N+.-0/>/32 @ & - & ;>/360-07849/3-0/36T406 & 607840607|40:5;>= ( /3?JD'Ed*e7841*F7g & -T/36;>782i7K; & ?K6 ( =);>7K2i/3-0/32 ( /;rBN? µ QR=)-c/32406 & 607849607840:;>= ( /3?JOH49B*e7K?)?87K:)/3?K7Kgi=F= ( Q]hi2i:F607K=)2i/32 & 40/3-0/36@CBd/36c*F7K?K:NBN-0?K7KL)6 ( =);>782i/3-0/32 ( / ;MBN? ξ :ehi2i2i/h ( 60-0F:e:)/H4s*)/ ( ?K7K:)/3?K7Kgi=F= ( QRhi2i:eI607K=)2# & 40/3-0/36 @CB µ

2e6 & L & 6 νθ = fθ · µ Q]=)- & ?8?K/ θ ∈ Θ OW=)L & 6 µ = h · ξ D o&/3-

νθ = fθ · µ = fθ · (h · ξ) = (fθh) · ξ Q]=)- & ?K?K/ θ ∈ Θ,

40/3-*F7 & 6 ξ *e786060/3-0?K7KL)6 ( =);>7K2i/3-0/3-;>= ( /3?K?K/32wOF;>/ ( ?K7K:)/3?87Kgi=e= ( Q]hi2i:e607K=)2Lx(θ) = fθ(x)h(x) = h(x)Lx(θ). jJkFD m

Ed*e7|4 h(x) 6= 0 g & -*F7'49BN?K/ ( /H4.*F78406 & 6 Lx ∝ Lx D~F+.6A = x ∈ X | h(x) 6= 0.

=)- & ?K?K/ x ∈ A /3- Lx ∝ Lx D dL QR=)- & ?8?K/ θ ∈ Θ /3-νθ(A

c) =

∫

Ac

fθ(x)h(x) dξ(x) = 0,

/Q]60/3-940=);7K2e60/3L)- & 2 ( /32T/3-'7 ( /32`607840: : @CB ( /32T;M+.2iL ( / ( /3-'7K2e60/3L)-0/3-0/H4'=*)/3-3D = ( /3?K?8/32Q]=)-0/H40:F-07K*)/3-s4BN?K/ ( /H4 & 6*e7i40?K/36s7K:F:)/: & 2 Q]Bo=)C40/3-0* & 607K=)2i/3-h ( /32Q]=)- A ONh & 2C40/36g`*F7K?K:)/36θ( /3-/3-4 & 2 ( 63D =)7K2`60/32#7jJkFD m/3- & 6T?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/3-02i/49* & -0/32 ( />6078?/36S*e78?K:,BN-0?87KL)6 ( =);>782i/I-0/32 ( /;MBN? ξ /3-Y/H4060/3; 6 & Q?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/3-02i/40* & -0/32 ( /S607K?'/36d;>7K2i7K; & ?86 ( =NI;>782i/3-0/32 ( /T;MBN? µ DA QBN?KL)/E & ?K;M=b4I ~ & * & L)/H4d4+.602i7K2iL>GC2 ( /H4 ( /3- & ?K607 ( /36d;>7K2i7K; & ?K6( =);>782i/3-0/32 ( /5;MBN?JO ( /3->/3:40@i?K7<786T:F2`F6060/3- & 2 607K? ( /32 L)78*e2i/;>= ( /3?JD=); i7K2i/3-0/H4( 78440/-0/H49hi?K6 & 60/3-3OQ]BN-1*F7 & ?K6949B & 6?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/3-02i/S/3-1Y/H4060/3;6 & Q;>= ( /3?8?K/3240/3?8*AO7K:F:)/ & Q ( /36 & -0i7K60-9+.-0/ ( =);>7K2i/3-0/32 ( /;MBN?JD 8 7: & 25=)@C49hi;>;>/3-0/ ( 784940/1=*)/3-0*)/ q I/3?849/3-7w/32549+.602i7K2iL

e e Ni ] A5 )N ] 0 ,

a \WZR\ > @BA P = (νθ)θ∈Θ 7,5 95, 7575$6 !( , % (X , E) &( P ( . ( . ()5$6",:- $#% ! " µ () µ /30' # !6,%( ( . #%6 7 ( $ (Lx)x∈X () (Lx)x∈X ! 5" 7 ( ( : ! " 5

Lx(θ) = h(x) Lx(θ). ( : θ ∈ Θ , x ∈ X ,

. ( 5 5 ! ", . #%6 7 ( h : X → (0,∞) /2

)*+,")$-1../ 032#4 5)$. 41,4 . 57. + 5)$ %7) ,"

%'&)((νθ)θ∈Θ *N+.-0/>/32 @ & - & ;M/360-07840/3-0/36T406 & 607|40607840: ;>= ( /3?Q]=)-S/36S/3:40@Y/3-07K;>/32e6;>/ (-0/3@i-9+40/32`6 & 607K=)2C49-0hi; (X , E) Ob=)L & 2e6 & L & 6 ( /32i2i/;>= ( /3?i/3- ( =);>7K2i/3-0/36 & Q/36L)-0hi2 ( I

;MBN? µ D =)-og`*)/3-06 x ∈ X g & -d*F7'7K2 ( QN-06?K78:)/3?K7Kgi=F= ( Q]hi2i:e607K=)2i/32 Lx : Θ → [0,∞) D Br & L)L)-0hi2 (f& Q/3:F40@Y/3-07K;>/32e60/3694-0/H40hi?86 & 6 x Q]+40602i/3-d*e7W=b4*)/ ( /32A/H4060/3; 6?87K:)/3?K7 Igi=e= ( QRhi2i:F607K=)2 Lx DFEd*e7|4*F7PL)/32`6 & L)/3-./3:40@Y/3-07K;>/32`60/363O`Q]BN-*F7WjRQ]=)-0;>/32`60?K78LmW/32 & 2 ( /32=)C40/3-0* & 6078=)2 x′ ∈ X =)L ( /3-0;>/ ( /32 & 2 ( /32#?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2 Lx′ Dcd/32 *F784oQ]=)-I406 & 2 ( : & 2; & 2 & ?K6949B<407KL)/ & 6?87K:)/3?K7Kgi=F= ( Q]hi2i:F607K=)2i/32>/3-4060=): & 40607849: 8 7: & 2fQR=)-0; & ?K7840/3-0/ ( 784940/ Y/360- & L)602i7K2iL)/3-02i/M*)/ ([& 67K2 ( QBN-0/<-0hi;M;>/36 [0,∞)Θ OwY/I4069BN/32 ( / & Q & ?K?K/dQRhi2i:F607K=)2i/3- Θ → [0,∞) D 8 7wg & -:)=)2C406 & 60/3-0/36 & 6

Lx ∈ [0,∞)Θ Q]=)- & ?K?K/ x ∈ X . /3-0;M/ ( : & 2r*F7P=)@Q & 6060/1gi/3?K/d4 & ;M?K7K2iL)/32 & Q'?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2i/3-49=); /32p60- & 2C4Q]=)-I; & 607K=)2 L : X → [0,∞)Θ Oig`*F7K?K:)/36./3*)/32`60hi/3?K6: & 25782 ( 7K:)/3-0/H4.;>/ ( 40:F-07K*)/3;MB ( /32

L(x)(θ) = Lx(θ) QR=)- & ?8?K/ x ∈ X , θ ∈ Θ. jJkFDKl : mW& -0/32e60/H40/3-02i/>@CBp*)/32C4060-0/<407 ( / & QdjJkFDKl : m/3-14 & 6Q]=)- & 6d; & -0:)/3-0/ & 6 L /3-d/32f60- & 2C4IQR=)-0; & 6078=)2 & Q x D /325-0/H40hi?K6 & 60/36 L(x) /3-40/3?K*r/32 QRhi2i:F607K=)2wOi;>/ (& -0L)hi;M/32`6 θ D /36/3-; hi?K7KL)6 & 6Th ( 406ze-0/ [0,∞)Θ ;>/ ( /32#QR=)-0gi=)? ( 40*e7|42 & 60hi-0?K7KL σ I & ?8L)/3i- & D =)-g`*)/3-06 θ ∈ Θ g & -*F7F/32<2 & 60hi-0?87KL<W_,V ^`uaHZRVA\: ixsZ t U\cZ]\M@i- θ : [0,∞)Θ → [0,∞) OL)7K*)/36*)/ (

@i- θ(f) = f(θ) QR=)- & ?8?K/ f ∈ [0,∞)Θ.

dL <s_,V ^`uaHZRVA\: σ t A^s_ i^F\ P @CB [0,∞)Θ /3- ( /32 ;>7K2 ( 4060/ σ I & ?8L)/3i- &<( /3-L N-& ?K?K/ ( 784940/1@i-0=,q0/3:e607K=)2C4 & QRi7K? ( 2i7K2iL)/3-.;MBN?K/3?K7KL)/)D

9 S3 e F Ni ] A ^eX X >;@BA A

(νθ)θ∈Θ ( : 5 7575$6 !( , % (X , E) / 0 6", ( ( . # 6 7:( L (X , E) − ([0,∞)Θ, P) ! ", . :-/

<>=?A@:B o/32i2i/3;M4960-$N;>2i7K2iLb4949+.602i782iL)/32SQ]=)-069+.?K?K/3- & 6s*e7F40: & ?e*F7840/ & 6s@i- θL Q]=)-sg`*)/3-06Q & 406 θ ∈ Θ /3-/32 E −B ;MBN?K/3?K7KLTQRhi2i:F607K=)2wD /32pge*e7|4.*e7'4078L)/3- & 6 νθ = fθ · µ Oi49B /3-

@i- θ L(x) = L(x)(θ) = fθ(x).

dL (i& fθ /3-/32569+.60gi/ ( OP/3- ( /325@Y/3- ( /GC2i7K607K=)2 E − B ;MBN?K/3?87KLiD2

-0782=<7K@i7K/3?K6: & 2[*F72`h ( 7840:Fhi60/3-0/<Q]=)- ( /3?K7K2iL)/32 & Q?87K:)/3?K7Kgi=F= ( Q]hi2i:F607K=)2i/32 hi2 ( /3- Pθ O( /361*F7K?W4078L)/Ti7K?K?8/ ( ;MBN?K/36 L X(Pθ) D /36Q]=)-0/3:)=);>;>/3-ohiY/3g & L)/3?K7KL)63OCQ]=)- ( /36d/3-1/36;MBN?Y@CB/36Ni ] C0HN [0,∞)Θ Oe=)L *e7Yg & -40?K/367K:e:)/dh ( *e78:e?K/36 -0/ ( 40: & Y/3-02i/o607K? & 66 & ?K/ =);$;MBN?W@CB 49Bp4060=)-0/ ;M+.2iL ( /3-3D; /32= *)/3-0=)- ( 2i/36S49/36d/3- ( /36o7K:e:)/ & 2 ( /3-0?K/ ( /H4/32 (& 6 6 & ?K/ =); QR=)- ( /3?87K2iL)/32 & Q/32g`*F7K?K:)/32 40=);gi/3?8406 & 2 ( /3260- & 2C4Q]=)-0; & 607K=)2 & Q=)C40/3-0* & 607K=)2i/3-02i/)D8 7*e78?7K:e:)/ ?8+.L)L)/<*N+.L)6S@CB Q]=)-0;>/3?K?8/ ( /36 & ? q0/3-=); σ I & ?KL)/3i- & /3-S@CBQRhi2i:F607K=)2C40-0hi;5D /32r7 ( /3/32r=); & 6?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/32<: & 2rY/360- & L)60/H4.40=); /32 60- & 2C4Q]=)-0; & 607K=)2 & Q( /325L q0=)-060/=)C40/3-0* & 6078=)2*e7K?*N+.-0/ & Q4060=)-A/36z ( 2i782iLiD

(Ω, F) (X , E)

X

Θ

θ

Pθ νθ Θ

L

[0,∞)Θ

7KL)hi-MkFD ' % -# + % !; +! % + % % # *+ 0 + -% %-(# % # + ( ( - ! " % ( 2 ( % "# 4

8 7W*F7K?c49=);-0/3L)/3?i-0hiL)/ A/360/3L)2i/3?849/32 LX 2CBN-d*F7W69+.2i:)/3-o@CBr?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/3240=); /32#4060=): & 40607849: * & -07 & Y/3?JO'=)L Lx 2CBN-*F769+.2i:)/3- @CB ( /32 :)=)2i:e-0/36 =)C40/3-0*)/3-0/ ( /

e e Ni ] A5 )N ] 0 , b

?K7K:)/3?K7Kgi=F= ( Q]hi2i:F607K=)2wDY~e/3?K*eQBN?8L)/3?K7KL</3-Q]hi2i:F607K=)2i/32 7K:F:)/T/3L)/32e60?K7KL =)C40/3-0*)/3-0/36 g`* &)(( /3-/3-=)C40/3-0*)/3-0/36/3- x ∈ X Oi=)L>@CB & L)L)-0hi2 ( & Q ( /32i2i/T*,+.- ( 7':)=)2C4960-0hi/3-0/3-*e7 Lx D9 u;:H^FX < ^ t>;@BA 7 %'&)( X1, . . . , Xn *,+.-0/rh & QRgC+.2iL)7KL)/p4060=): & 40607840:)/>* & -07 & i?K/)O & ?K?K//3:F40@Y=)2i/32`607 & ? QR=)- ( /3?860/S;>/ ( /32 hi:)/32 ( 6o;>7 (i( /3?K*N+.- ( 7 λ > 0 D 2 & ?K=)L)6d;M/ ( -0/3L)2i7K2IL)/3-02i/M7 /3:40/3;>@Y/3?kFDKl>GC2 ( /3-*F7 & 6T?8=)LNIJ?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/32@CB5 & L)L)-0hi2 ( & Q/32=)C40/3-0* & 6078=)2 (X1, . . . , Xn) = (x1, . . . , xn) /3-

`x1,...,xn(λ) = n log λ +x•

λQ]=)- & ?K?K/ λ ∈ (0,∞)

g`*)=)- x• =∑n

i=1 xi D n 2 ?87 ( 6:)=)-060/3-0/<40:F-07K*)/3;MB ( /)O ( /3-d?K78L)L)/3-;>7K2 ( -0/*,+.L)6o@CBr/32:)=)2i:e-0/36=)C49/3-0* & 607K=)2=)L<;M/3-0/*,+.L)6@CB ( /36d4060=): & 49607840:)/1/3-`X(λ) = n log λ +

X•

λQR=)- & ?8?K/ λ ∈ (0,∞).

E*F784r*e7o:)/32 ( /3- ( /3654 & 2 ( / λ O49B : & 2 *F7407K; hi?K/3-0/ (i& 6 & Q]- & /3:F40@Y/3-07K;>/32e60/363O=)L( /3-0;>/ ( 4078;hi?K/3-0/ ?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/32wD BdGCL)hi-kFD g & -s*e7e=)@i60/3L)2i/36 d407K; hi?K/3-0/ ( /?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/3-QR=)- λ = 2/3 =)L MQR=)- λ = 3/2 D =)-94$NL)/36/3-o: N-06160=rL & 2iL)/)O( /3?84.;>/ ( n = 10 =)L ( /3?84.;>/ ( n = 100 D

0.5 1.0 1.5 2.0 2.5

010

2030

4050

n = 10

λ

`(λ)

0.5 1.0 1.5 2.0 2.5

010

020

030

040

050

0

n = 100

λ

`(λ)

78L)hi-dkFD +*# "+ "!+)8 - ( - # # " 8 +!,*! -+% 0 ) + %- +"4 !, *!+ % 5. %- + 8 0 -+% )02 # ;/ !; +!+- 5 %- + 80 +% 02 # - 4 0=) " (# ".+ % ! % #$# = 0 + 8# ( % 0 # ( λ = 2/3 4 1% 0" ( # " += % ! % #$# - 0 ( # " %= 0 # λ = 1.5 4

6 /3;r+.-0: & 6dQ]=)-94$NL)/36oI.607K?W/32*F784L)- &)( Q]=)- n = 10 O=)L 7Wge*)/3-061Q & ? ( ;>/ ( n = 100I6078?K? &)( /3-=b4 & 6+:HuP^ t \W^r;>/3?K?K/3; ( /60=<@ & - & ;>/360-0/)DPEd*e7|4/32 Q q0/32 ( 60?K7KLb407K2 ( /36@A/3-949=)2i-0hiL)60/p/32 & Q ( / 60=@ & - & ;>/360-0/ 607K? & 6 407K; hi?K/3-0/r2`)/ =)C40/3-0* & 607K=)2i/3-;M/ ( Os49B *F7K?K?K/

S3 e F Ni ] A*F7:ehi2i2i/ 60/3L)2i/ ?K=)LNIJ?K7K:)/3?87Kgi=e= ( Q]hi2i:e607K=)2i/32=)@Q]=)- ( /36060/ 2e)/</3:F49@A/3-07K;M/32`63OP=)L *)/ (& 64 & ;>;M/32igi=)? ( /;M/ ( GCL)hi-SkFD p*F7K?K?K/*F7L & 2C40:)/<407K:F:)/3-0gi/ ( :Fhi2i2i/ & QRL N-0/ ge*e7K?8:)/32@ & - & ;>/360/3- ( /3-* & -Y/32`F6060/363D

~e=); 607 ( ?K7KL)/3-0/.2C+.*e2e6 /3-?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2i/32 ( /32>@i-07K;r+.-0/C+.-0/3- & QY782QR=)-0; & 6078=)2=); 4 & ;M40@i7K?8?K/36d;M/3?K?K/3; =)C40/3-0* & 607K=)2[=)L5;M= ( /3?JD 8 784949/>406 & 607840607K:)/3-0/<g & -6 & L)/36 ( /32( - & 40607|40:)/:)=)2C40/3:F*)/32C4 & 6=)@ig q0/ ( /32i2i/S/3-Q & -07K2iL<607K?/36 ( =)L);>/+ Z|uA^ t ZRvWVwVwU<s_,ZR\bZ <<^`a d dr, iJC F A ] ] 0MpF(X , E,P) ) HN >,W5 N ] x ∈ X i , Np, , F N F e F b Ni ] /36060/o@i-07K2=<7K@M/3-:)=)2`60-0=*)/3-9407K/3?K63OF=)L 7 ( 78440/2i=)60/3-L N-0/H4782`60/36QR=)-94$NL @CB & 6-0/H40@Y/3:`I60/3-0/ ( /363O`60*,+.-0607K;M= ( D -0=)i?K/3;M/36;>/ ( ?K7K:)/3?K78gi=e= ( @i-07K2=<7K@i@Y/36'/3- & 6 ( /3678:e:)/607K?8? &)( /3-=*)/3-0*)/ q/3?849/3-=);QR=)- ( /3?K782iL)/32 & Q LX D o-0=NQR64 & L)6)?K7K:)/3?K7Kgi=F= ( @i-07K2=<7K@i@Y/36w607K?K? &)( /3-s:ehi2& 6.; & 2rhi2 ( /3-94$NL)/3- ( /32r=)C49/3-0*)/3-0/ ( /?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2wON7K:F:)/ & 6.; & 2 ( 7|40:ehi60/3-0/3-ge*e7K?8:)/?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2i/3-; & 2f/3?K?K/3-94d:ehi2i2i/ g & *)/<=)C49/3-0*)/3-0/363D dL & -0L)hi;>/32`6 & I607K=)2 =);M:e-07K2iL 7K:F:)/IJ=)C40/3-0*)/3-0/ ( /r?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/3-*F7K?*N+.-0/<L & 2C40:)/>*F7KL)607KLpQR=)-=b4I=)LrQ]=)- ( / C/H4060/ & 2 ( -0/ 406 & 607849607K:)/3-0/)D /32 QD /3:43D 6.& )/H407 & 2i/3-0/ -0/H49@A/3:F60/3-0/3- & hI60=); & 607|40:M?K78:)/3?K7Kgi=F= ( @i-07K2=<78@i@A/363D% 78:)/3?K7Kgi=F= ( @i-07K2=<78@i@A/36sh ( L N- /36 & ?8*)=)-0?K7KL)6=)L ( Fi6@i-0=)i?K/3; Q]=)- ( /32r60/3=)-0/3607840:)/406 & I607849607K:AD /36*e7849/3-W407KLo2i/3;>?K78L & 6/32 -9+.:e:)/6078?840F2i/3? &)( /32 ( / ;>/3-0/40@i7|40/3?K7KL)/@i-07K2=<7K@i@Y/3-;>/ ( QN-0/3-?K7K:)/3?87Kgi=e= ( @i-07K2=<7K@i@Y/363D d*)/3-0gi=)? ( /3-; & 2 49h <7K/32C40@i-07K2=<7K@i@Y/36=)L<A/3607K2iLNI2i7K2iLb49@i-07K2=<7K@i@Y/363O4B<= *)/3-0gi=)? ( /3-d; & 2=)Lb49B<?87K:)/3?K7Kgi=F= ( @i-0782=<7K@i@A/363O`h & 2C40/36=);; & 2i-0 ( /3-T407KL =); ( /36S/3?8?K/3-1/ qD 8 7*e7K?40/32i/3-0/M:)=);>;>/ 7K2 ( @CB g`* &)( ( 78440/ @i-07K2=<7K@i@Y/3-LbBN-h ( @CBD

o P,

. #)$4 +465%

%'&)((νθ)θ∈Θ *N+.-0/ /32 @ & - & ;>/360-07840/3-0/36s406 & 607840607849:d;>= ( /3?`@CB (X , E) D n 2 uPVAX sZ]\ C\Ya

/3-/32 & QRi7K? ( 2i7K2iL R : Θ ×X → Y Oig`*)=)- Y /3-/32 ( 2eh5/32;M+.2iL ( /)D Q]60/i-0hiL)/H4/32:)=);i7K2 & 2e6S607K? & 6S;rBN?K/ g`*)=)-TL)= ( 6S/32 :)=)2i:F-0/36S=)C40/3-0* & 607K=)2 =)L5/32[:)=)2i:e-0/36@ & - & I;>/360/3- /3-7P:)=)2i:)=)- (i& 2C43D /32r@i-9+<7849/;MB ( / ( /36060/oL N-0/H4@CBOe* & -07K/3-0/3-Q]- & /3:40/3;>@Y/3?607K?/3:40/3;>@Y/3?JD d@i? & L)60/ /3:40/3;>@i?K/3-r@CB :)=);i7K2 & 2e60/3- /3-r?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2i/32=)L?K=)LNIJ?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2i/32wD /32 ( /3-: & 2 GC2 ( /H4; & 2iL)/ & 2 ( -0/)D9 u;:H^FX < ^ t>;@BA > %'&)( X1, . . . , Xn *,+.-0/ -0/3/3?K?K/ 4060=): & 40607840:)/ * & -07 & i?K/)O1=)L ? &)((Pθ)θ∈Θ *,+.-0//32 ;>= ( /3?eQ]=)- ( /3-0/H4407K; hi?K6 & 2i/Q]=)- ( /3?K7K2iLiD 2e6 & L & 6sg`*)/3-06 Xi g & -WQN-I4060/1;>=);>/32`6hi2 ( /3- Pθ Q]=)- & ?K?K/ θ ∈ Θ D 8 7*e78?A2eh=)LQ]-0/3;>=*)/3--0/QR/3-0/3-0/1607K?w49B (i& 2i2i/;>=);>/32`60/3- 40=); EθXi Io;M7 (i( /3?K*N+.- ( 7K/32 & Q Xi g`*F784 θ /3- ( /32 4 & 2 ( / @ & - & ;>/360/3-3D hi2i:e60/36 (EθX1, . . . , EθXn) ∈ R

n /3- ( /36 F1@ihi2i:F63OCg`*F784 θ /3- ( /32 4 & 2 ( /@ & - & ;>/360/3-3D /3-QR=)-.Q]=)-069+.?K?K/3-.:F* &)( - & 6949hi;>;>/32

R(θ;x1, . . . , xn) =n∑

i=1

(

EθXi − xi

)2 jJkFDKl)l,m

2i=)L)/36=); g`*)=)-S? & 2iL)6 ( /32 :)=)2i:F-0/360/<=)C40/3-0* & 607K=)2 (x1, . . . , xn) ?K78L)L)/3-dQ]- & ( /361QR=)-I*)/32`60/ ( /M@ihi2i:F63D /3L)/32 406 & 607840607|40:5;>/360= ( 7K: /3-h ( *F7K:F?K/361@CB & L)L)-0hi2 ([& Q & 2 & ?K40/& Q.:F* &)( - & 6940hi;M49406$N-0-0/3?|40/3-40=); jJkFDKl)l,m9OW=)L ( /36T/3-QN-9406S7 gi/3?K6S;>= ( /3-02i/>607 ( & 6T?K7 I:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2i/32 =)L:)=)2C4060-0hi:F607K=)2i/3-gi/3-0h ( Q]- & g & -= *)/3-06 & L)/365-0=)?8?K/32 40=); ( /*e7KL)6078Lb4060/d:)=); i7K2 & 2e60/3-3DE*)=)-s-0/3?K/3* & 2`6/32<406$N-0-0/3?840/1jJkFDKl)l,m'/3-sQR=)-s:)=)2i:)=)- (i& 2C40*Fhi- ( /3-07K2iL)/3-3O & Q]gC+.2iL)/3-s;M/3L)/36& Q ( /32[:)=)2i:F-0/360/r;>= ( /3?JDcEd*F784 Xi /3-02i/rg & -TQR=)-949:)/3?K?K7KL* & -07 & 2C43Oc4B5;MB R A/360- & LNI60/H4;>/ ( ;>7|4060-0=iOYQR=)-S/32 & QR*F7KL)/3?840/ @CB/32 :)=F=)- ( 7K2 & 6S;M/ ( 4060=)-* & -07 & 2C4>j ( /32#49? & Lb4& QR*F7KL)/3?840/p;MB74 & L)/32C4M2 & 60hi-<QR=)-0*)/32e60/H4mT69+.?K?8/3- ?87KL)/49Bf;>/3L)/36>40=); /32 & Q]*F7KL)/3?840/@CB5/32 :)=e=)- ( 7K2 & 6 ;>/ ( ?K7K?K?K/ * & -07 & 2C4Mj ( /3240? & Lb4 & Q]*e7KL)/3?|40/3-6 ( /3- @CB ( 7840:)=)- (i& 2C43m9D 7K?840* & -0/32 ( /)ONg`*F784 Xi /3-02i//3- & Q]gC+.2iL)7KL)/)O`49B1/3- R /36hC4078:e:)/3-06;MBN?Q]=)-:)=)2i:)=)- (i& 2C43D 7|4940/ ( /QR/3:F60/3-: & 22i=)L)/325L & 2iL)/Q q/3-02i/H4*)/ ( & 6A/360- & L)60/T/32*,+.L)60/36:F* &)( - & 6940hi;

RW (θ;x1, . . . , xn) = (x − EθX)T W (x − EθX) . jJkFDKlHemg`*)=)- W /3-r/32 40F;>;>/360-07|40:AO@Y=b407K607K*F6 ( /GC2i7K6 Pa X aH_,ZD 6 /3;M+.-0: & 6 jJkFDKl)l,m/3-/36T40@Y/<7K/3?K61607K? Q +.? ( / & QjJkFDKlHemd;>/ ( *,+.L)60; & 60-07 W = I D /32 7:)=);>@i?K7</3-0/ ( /;>= ( /3?8?K/3-/3- ( /36=NQR60/?K/36060/3-0/4 & L)6/32 ( L q=)-06 & 6.GC2 ( /S/32 @ & 4940/32 ( /*N+.L)63D

: S3 e F Ni ] A8 7i*e78?Fgi/3- 7K2 ( QBN-0/60=T:)=);i7K2 & 2e60/3-3O ( /3-/3- & C/ ( 60/ & QY?K78:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/32 I ( 784949/:)=); i7K2 & 2`60/3-i?K7K*)/3-*F7KL)607KL)/dQR=)-=b47'40/32i/3-0/ & Q 402i7K63D ^\WZ8aHZRVA\ > @ A

(νθ)θ∈Θ ( >5, 7 5,7 5 6 !( , (X , E) & ( ,!- Θ ,!- .

Rk / 5 (

6,%( ( . #%6 7 ( $ `x C1 % Θ &5 : `VP_,^`xzys\WuaHZ|VY\c^F\ Θ → Rk :

5 ( 5(D`x(θ) =

(

∂`x(θ)∂θ1

∂`x(θ)∂θ2

. . . ∂`x(θ)∂θk

)

.

5 ;() 6", %( ( . # 687:( $ `x C2 % Θ & 5 6 :2 >ZR\Wx VA_ XaHZ|VY\ : xzys\cuaHZRVA\W^e\ Θ → R

k×k 5 (

D2`x(θ) =

∂2`x(θ)∂θ1

2∂2`x(θ)∂θ1∂θ2

. . . ∂2`x(θ)∂θ1∂θk

∂2`x(θ)∂θ2∂θ1

∂2`x(θ)∂θ2

2 . . . ∂2`x(θ)∂θ2∂θk/// /// / / / ///

∂2`x(θ)∂θk∂θ1

∂2`x(θ)∂θk∂θ2

. . . ∂2`x(θ)∂θk

2

.

9 u;:H^eX=<^ t >;@BA %c&)(X1, . . . , Xn *N+.-0/1h & QRgC+.2iL)78L)/-0/3/3?K?K/14060=): & 49607840:)/o* & -07 & i?8/)O& ?K?8/p/3:40@Y=)2i/32`607 & ?KQR=)- ( /3?K60/;M/ ( hi:)/32 ( 6>;>7 (i( /3?8*,+.- ( 7 λ > 0 D /32i2i/5;>= ( /3?g & -

?K=)LNIJ?87K:)/3?K7Kgi=F= ( Q]hi2i:F607K=)2`X(λ) = n log λ +

X•

λ.

/3-0;>/ ( g & -;M= ( /3?K?8/32p4 <=)-0/Q]hi2i:F607K=)2`′X(λ) =

n

λ− X•

λ2,

=)L>782QR=)-0; & 6078=)2C4QRhi2i:F607K=)2`′′X(λ) = − n

λ2+

2X•

λ3.

9 u;:H^eX=<^ t+>;@BA %'&)( =b4Y/360- & L)60/ ; N2`60: & 4060;>= ( /3?K?K/32wOige*)=)-d*F7'g & -oh & QRgC+.2iL)7KL)/-0/3/3?K?8/14960=): & 40607|40:)/1* & -07 & i?8/ X1, . . . , Xn ;>/ (

P (Xi = 1) = p, P (Xi = 0) = 1 − p Q]=)- i = 1, . . . , n, p ∈ (0, 1).

o P, l% 7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/32p/3-

LX(p) =n∏

i=1

pXi(1 − p)1−Xi = pX•(1 − p)n−X•

=)L ( /3-0;>/ ( /3- ?K=)LNIJ?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2i/32wO4 <=)-0/QRhi2i:F607K=)2i/32 =)L 782QR=)-0; & 6078=)2C4QRhi2i:eI607K=)2i/32

`X(p) = −X• log p − (n − X•) log(1 − p),

`′X(p) = −X•

p+

n − X•

1 − p,

`′′X(p) =X•

p2+

n − X•

(1 − p)2.

9 u;:H^FX < ^ t1> @BA %'&)( X1, . . . , Xn *N+.-0/ph & QRgC+.2iL)7K/ -0/3?K?8/r4060=): & 40607840:)/p* & -07 & i?K/)O& ?K?K/ N (ξ, σ2) I Q]=)- ( /3?K60/)D /32i2i/S;>= ( /3?wg & -?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2

LX(ξ, σ2) =n∏

i=1

1√2πσ2

e−(Xi−ξ)2/2σ2,

=)L ( /3-0;>/ ( ?K=)LNIJ?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2`X(ξ, σ2) =

n

2log 2π +

n

2log σ2 +

∑ni=1(Xi − ξ)2

2σ2.

~<=)-0/QRhi2i:F607K=)2i/32wOc/3?8?K/3-T4 <=)-0/3*)/3:F60=)-0/32wO40=);"; & 2[=NQR60/p407KL)/3-T2CBN-@ & - & ;>/360/3-0/32 /3-C/3- ( 7K;>/32C407K=)2 & ? Oe/3-D`X(ξ, σ2) =

(

∂`X

∂ξ

∂`X

∂σ2

)

=

(−∑ni=1(Xi − ξ)

σ2

n

2

1

σ2−∑

i=1(Xi − ξ)2

2σ4

)

6 /3;r+.-0: & 6<@ & - & ;>/360/3-0/32 /3- σ2 O7K:F:)/ σ D /36060/ Q]=)-0gi=)? ( : & 2L)= ( 6 Q]=)-0*e78-0-0/)Os2CBN-; & 2 ( 7 /3-0/32`607K/3-0/3-3D 7K?840* & -0/32 ( /1i?K7K*)/17K2Q]=)-0; & 607K=)2C4Q]hi2`607K=)2i/32wO/3?K?8/3-782QR=)-0; & 6078=)2C40; & 60-07</32wO

D2`X(ξ, σ2) =

nσ2

∑ni=1(Xi−ξ)

σ4

∑ni=1(Xi−ξ)

σ4−n2σ4 +

∑ni=1(Xi−ξ)2

σ6

.

S3 e F Ni ] A ^\WZ8aHZRVA\ > @ A

(νθ)θ∈Θ ( >5, 7 5,7 5 6 !( , (X , E) & ( , Θ ,!- . R

k / YVCaHZR^e\Ya3aH^:Ha :3aP_ _,^ t :H^e\ 68( :

Q(θ, x) =Lx(θ)

supϑ∈Θ Lx(ϑ).

% 78L)/H40=); ?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2i/32 Lx =NQR60//3-9406 & 6060/H4 & Q `x Ob4Bdi?87K*)/3-:e*)=)607K/32e6060/H4069406$N-I-0/3?849/32 Q =NQR60/ /3-9406 & 6060/36 & Q −2 log Q Igi/3-*F7K-0:)/3-160=NIJ6 & ?K?K/361;MB)40:)/ ?K7 ( 6 & -0i7K60-9+.-063O;>/32r/3-6078? QBq0/36 & Qc4 & ;M;>/L)-0hi2 ( 40=); Q]=)-060/3L)2i/36FQR=)- & 6L N-0/140/32i/3-0/dQR=)-0;M?K/3-4078;>@I?K/3-0/)D =)-060=)?8:e2i7K2iL)/32 & Q:e*)=)6078/32`6060/H4069406$N-0-0/3?|40/32 /3-L & 2C40:)/>40hii607K? DEd*)=)-1?K7K:)/3?K78gi=e= ( QRhi2i:eI607K=)2i/32 ;MBN?8/3-L)- &)( /32 & Qs:)=)2i:)=)- (i& 2C4;>/3?K?8/3; =)C40/3-0* & 607K=)2i/32 x =)L @ & - & ;>/360/3-0/32 θ O49B ;rBN?K/3-:e*)=)6078/32`6060/H4069406$N-0-0/3?|40/32 7g`*F7K?K:)/32 L)- &)( ;>= ( /3?8?K/32C4 C\cUs_^ @ & - & ;>/360-0/M/3-7 ^eUW_,^ :)=)2i:)=)- (i& 2C4;>/ ( x /32 ( θ /3-3D'~e/3?8*5=); x =)L θ /3-17sQR=)-060-9+ /3?K7KLp:)=)2i:)=)-I(i& 2C43Oc: & 2 Q(θ, x) 4 & L)60/32C4S*N+.-0/>?K78?K?K/SI ( /36TA/36z ( /3-i?K=)6 & 6 & 2 ( -0/ & Q.;>= ( /3?K?K/32C4@ & - & ;>/360-0/ /3->7/32 ( 2eh Y/ ( -0/5:)=)2i:)=)- (i& 2C4p;>/ ( x D 7K?|40* & -0/32 ( /Y/36 ( /3-M/32 g qQ IJ*N+.- ( 7C7K2iL)/32i?8hi2 ( / & 6 x =)L θ /3-7i:)=)2i:)=)- (i& 2C4 I ( /36 Y/36 ( /3-i?K=)6 & 6782iL)/32 & 2 ( -0/@ & - & ;>/360-0/T/3-7w2C+.*e2i/3*N+.- ( 7KL<Y/ ( -0/S:)=)2i:)=)- (i& 2C4;M/ ( x D8 7W*F7K?s7W/361*F78406d=);<Q & 2iLp & 40/3-0/ 7K2Q]/3-0/32C40/32f=); θ @CBp:e*)=)6078/32`6060/H4069406$N-0-0/3?|40/32wD; /32( /32 Q]-0/3;>L & 2iLb40;MB ( //3-:)=)2`60-0= *)/3-9497K/3? ( /32 QBN-0/3-2i=)L)?K/L & 2iL)/607K?F:)=)2i:e?KhC497K=)2i/3-3O ( /3-: & 2*F7K-0:)/ & C49hi- ( /)O,=)L ( /32/3-s/36cF2 ( /36 & 2iL)-0/3C40@ihi2i:e6sQ]=)-c & )/H407 & 2C40:)/.406 & 6078406078:)/3-0/)D

?K?K/ :)=); i7K2 & 2`60/3-: & 2 Idhi2 ( /3-T@ & 4940/32 ( /M;MBN?K/3?K7KL)gi/ ( 40QR=)-0h ( 49+.602i782iL)/3-dId=)@Q & 6060/H440=); 4960=): & 40607|40:)/)O@CB54 & ;>;>/ ;MB ( />40=); ?87K:)/3?K7Kgi=F= ( Q]hi2i:F607K=)2i/32wD 8 7*e78?49:e-07K*)/ RXge*e784T*e7 69+.2i:)/3-T@CB5:)=); i7K2 & 2e60/32 R @CB ( /32i2i/r;MB ( /)D /36T/3- & ?K6949B /36 4060=): & 49607840:* & ?KLj] & 40/3-0/36@CB =)C49/3-0* & 607K=)2 X = x m & Q/32Q]hi2i:e607K=)2 Θ → Y Dn 2f;MB ( / & 6o60-9+.:e:)/ 7K2Q]=)-0; & 607K=)2 h (& Q /32:)=);i782 & 2`6 R : X × Θ → R

m @CBOY/3-& 6-0/3L)2i/d:)=); i7K2 & 2e60/32rh ( 7P/32rQ & 496 @ & - & ;>/360/3-0*N+.- ( 7 θ1 ∈ Θ D B ( /32r;rB ( /i?K7K*)/3-:)=); i7K2 & 2`60/32pi?K=)6/32r60- & 2C4Q]=)-0; & 607K=)2 & Q (i& 6 & OF=)L ( /3-0;>/ ( /32r4060=): & 40607849:* & -07 & Y/3?JD /322`h : & 2 ; & 2-0/3L)2i/;>7 (i( /3?8*,+.- ( 7K/32 h (& Q ( /32@CBNLb+.? ( /32 ( /5* & -07 & A/3? OWge*e784; & 2 :)/32 ( /3- ( /324 & 2 ( /1@ & - & ;>/360/3-3D 8 7A*F7K? & ?K6949B ( 7840:Fhi60/3-0/d/32 Q]hi2i:F607K=)2 & QcQR=)-0;>/32

(θ1, θ2) 7→ Eθ2R(X, θ1),

o P, k=)L>=)@Q & 6060/ ( /32i2i/QRhi2i:F607K=)2540=); /36h ( 60-0e:rQ]=)-;>= ( /3?K?K/32C4/3L)/32C40: & A/3-3D 78440/1-0/3LNI2i7K2iL)/3-i?K7K*)/3-.;>/H406.7K2e60/3-0/H494 & 2e60/Tg`*F784 θ1 = θ2 DP49B Q & ? ( 4060h ( /3-0/3-*F7

θ 7→ EθR(X, θ).8 7KL)6078Lb406 Q]=)-=b4i?K78*)/3- ( /32 x VP_ A^F\YaH^`Us^ Z]\cxzVP_NX aHZRVA\

i(θ) = EθD2`X(θ),

40=); 2i/360=)@ /3- ( /32i2i/ L)/32i/3-0/3?K?8/5:)=)2C4060-0hi:F607K=)2 & 2`*)/32 ( 6p@CB 782QR=)-0; & 6078=)2C4QRhi2i:F607K=NI2i/32wD /32 QR=)-0*)/32e60/ ( /#782QR=)-0; & 6078=)2 /3- 2 & 60hi-0?K7KL)*F784 :ehi2 ( /GC2i/3-0/36g`*F784 & ?8?K/f?K=)LNI?K7K:)/3?K7Kgi=F= ( QRhi2i:F607K=)2i/3-02i/ `x /3- C2 @CB Θ O=)L g`*F784 ( /32 & 2 ( /32 & C/ ( / ( /<g & ->lNDY;M=NI;>/32`63D /32 40? & Lb4.60/3:e2i7849:)/ ( /36 & ? q/3-@i?K/ q0/3-; & 2L)/3-02i/ & 6hi2 ( /3-Q]=)-94069BD9 u;:H^FX < ^ t >;@BA %'&)( 40=); 7c/3:F40/3;M@A/3?WkFDKl X1, . . . , Xn *,+.-0/h & QRgC+.2iL)78L)/)OP/3:4I@A=)2i/32e607 & ? Q]=)- ( /3?K60/1;>/ ( hi:)/32 ( 6;>7 (i( /3?8*,+.- ( 7 λ D 1& /3-

EλX• = nλ,

=)L ( /3-0;>/ ( /3- ( /32Q]=)-0*)/32`60/ ( /7K2QR=)-0; & 607K=)2i(λ) = Eλ`′′X(λ) = Eλ

(

− n

λ2+

2X•

λ3

)

=n

λ2.

9 u;:H^FX < ^ t=> @7 %'&)( 49=); 7M/3:40/3;>@Y/3?rkFDKl X1, . . . , Xn *,+.-0/ h & QRgC+.2iL)78L)/N (ξ, σ2) I QR=)- ( /3?860/-0/3/3?K?K/* & -07 & i?K/)D 1& /3-

E(Xi − ξ) = 0 , E(Xi − ξ)2 = σ2 ,

=)L ( /3-0;>/ ( /3- ( /32Q]=)-0*)/32`60/ ( /7K2QR=)-0; & 607K=)2C40; & 60-07i(ξ, σ2) =

( nσ2 0

0 n2σ4

)

.

/3-0/ L)/32i/3-0/3?K61/32 ( :ehi2 & 617K2`60/3-0/H440/3-0/<407KLrQR=)-o;>7 (i( /3?K*N+.- ( 7K/32wOA: & 2f; & 27K2`60/3-0/H40I40/3-0/407KL QR=)- gi/3?K/pQ]=)- ( /3?K7K2iL)/32 & Q R(X, θ1) hi2 ( /3- Q]=)-0h ( 49+.602i7K2iL & Q & 6 θ2 /3- ( /324 & 2 ( />@ & - & ;>/360/3-3D L)/32[/3- ( /36T;M/H406S7K2e60/3-0/H494 & 2`6g`*F784 θ1 = θ2 = θ D 8 76 & ?K/3- (i&=);uPVYX WZ]\ C\YaH^e\:rxzVP_,UW^ t Z]\ f7 θ D /36d/3- & ?K6949B Q]=)- ( /3?K7K2iL)/32 & Q ( /324960=): & 40607|40:)/* & -07 & Y/3? R(X, θ) g`*F784 θ /3- ( /32 4 & 2 ( /@ & - & ;>/360/3-3D*e7|4940/fj 4Rq0+.? ( 2i/Nm16078? Q]+.? ( / & Q]gC+.2iL)/3- ( /32i2i/ QR=)- ( /3?K782iL78:e:)/ & Q θ D 8 7.497KL)/3- (i&& 6:)=);i7K2 & 2e60/325/3-/32 <sZ YVCa)OC/3?8?K/3- & 6 ( /325/3-x VA_Us^ t Z]\ :HuPVY\ :3a C\Ya)D

S3 e F Ni ] A . ,465 % , 2#" 5 5% 03. % +. )$4 +465%

2`6 & L#7 ( /36>QN?KL)/32 ( / & 6 (νθ)θ∈Θ /3->/32 406 & 607840607849: ;>= ( /3?@CB (X , E) D 2`6 & L & 6@ & - & ;>/360/3-0;r+.2iL ( /32 Θ /3-1/32 BNA/32 ( /3?K;M+.2iL ( / & Q Rk O=)L & 2`6 & L & 61;>= ( /3?K?K/32 /3-( =);>782i/3-0/36 & Qc/36 σ IJ/32 ( /3?87KL)6 ;MBN? µ @CB (X , E) Oi49B (i& 2 & 6 69+.60gi/ ( /3-02i/ fθ(x) > 0 Q]=)-& ?K?8/ θ ∈ Θ =)L & ?K?K/ x ∈ X D

a \cZ]\ >;@7;A 7 57 46 6",- 7# $: :5 . (,# 57 :--

Eθ(D`X(θ)) = 0. ( θ ∈ Θ , jJkFDKlHkem

()Vθ(D`X(θ)) = i(θ)

. ( θ ∈ Θ. jJkFDKl m

<>=?A@:B /325QRhi2 (i& ;>/32e6 & ?K/-0/3? & 607K=)2/3-∫

fθ(x) dµ(x) = 1 QR=)- & ?K?8/ θ ∈ Θ. jJkFDKlem

Ed*e7|4.; & 2 : & 2 ( 7 Y/3-0/32e607K/3-0/hi2 ( /3-7K2e60/3L)- & ?K60/3L)2i/363OP40/3-*F7QR=)- i = 1, . . . , k& 6

0 =∂

∂θi

∫

fθ(x) dµ(x) =

∫

∂

∂θifθ(x) dµ(x) =

∫ ∂∂θi

fθ(x)

fθ(x)fθ(x) dµ(x)

=

∫ (

∂

∂θilog fθ(x)

)

fθ(x) dµ(x) = −Eθ

(

∂

∂θi`X(θ)

)

.

~ & ;>?8/H4 ( 784949/ k @CB)406 & 2 ( /o7P/32p*)/3:F60=)-07 ( /32`607K60/363OFg & -; & 2M2i/360=)@fjJkFDKlHkemIW@CB2C+.-/367K-0-0/3?8/3* & 2e6.QR=)-060/3L)2wDEd*e7|4.; & 2: & 2 ( 7 Y/3-0/32e607K/3-0//325L & 2iL>607K?whi2 ( /3-7K2`60/3L)- & ?K60/3L)2i/367 jJkFDKlem9OCg & -*F7QR=)-& ?K?8/ i, j = 1, . . . , k

& 6

0 =∂

∂θj

∫

∂

∂θifθ(x) dµ(x) =

∫

∂2

∂θi ∂θjfθ(x) dµ(x).

f H Je P,

n 2 ( *e7 ( /3-0/S40/3-*e7 & 6∂2

∂θi ∂θj`X(θ) = −

∂2

∂θi ∂θjfθ(x)

fθ(x)+

∂∂θi

fθ(x)

fθ(x)

∂∂θj

fθ(x)

fθ(x)

= −∂2

∂θi ∂θjfθ(x)

fθ(x)+

∂

∂θi`X(θ)

∂

∂θj`X(θ).

=); i7K2i/3-0/H4 ( 784949/o=)@i?8F402i782iL)/3-Q BN-*F7 & 6

Eθ

(

∂2

∂θi ∂θj`X(θ)

)

= −Eθ

∂2

∂θi ∂θjfθ(x)

fθ(x)

+ Eθ

(

∂

∂θi`X(θ)

∂

∂θj`X(θ)

)

= 0 + =*(

∂

∂θi`X(θ),

∂

∂θj`X(θ)

)

,

/QR60/3-940=); ;M7 (i( /3?K*N+.- ( 7K/32 & Q ( /60=S4060=): & 49607840:)/.* & -07 & i?K//3-2ehi?JD /32 ( /36060//3-2i/360=)@; & 60-07?K7KL)2i7K2iL)/32jJkFDKl m ?8+406:)=);>@Y=)2i/32`6QR=)-:)=);>@Y=)2i/32`63D

2

/T2 ( *)/32 ( 7KL)/SQR=)-0h ( 49+.602i782iL)/3-QR=)- & 6 ( 784940/-0/3L)2i782iL)/3-gi=)? ( /3-3OC/3-d4060=)-0640/36 & 6 ( /7K2`*)=)?K*)/3-0/ ( /p406$N-0-0/3?840/3-TL)7K*)/3-S;>/32i7K2iLiD % =)LNIJ?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2i/32 40: & ?*N+.-0/ ( 7 /I-0/32`607 & Y/3?JO)=)L D`X(θ) 40: & ?i*N+.-0/7K2e60/3L)- & A/3?P;>/ ( gi/32C40F2>607K? νθ Q]=)- & ?K?K/ θ Ob49B (i& 2 & 6*)/32C4060-0/H407 ( /32 & QojJkFDKlHkemL)78*)/3-d;>/32i782iLiD dL49B 40: & ?; & 2 & ?K6949Br:ehi2i2i/ ( 7 /3-0/32`6078/3-0/7K2 ( hi2 ( /3-/367K2e60/3L)- & ?K60/3L)2wD dL<607K?849* & -0/32 ( /oQ]=)-TjJkFDKl m9DE*F784 Θ ⊂ R /3-S/32 6078?84060-9+.:F:)/3?K7KLrI=);>/32 ( & ?K61Q]=)-S4069+.-0: IY/3607K2iL)/3?849/)O ( /3-407K:F-0/3-& 6T49+.602i7K2iLfkFDl>/3-T=)@Q]F? ( 63O & 6 ( /3-GC2 ( /H4T/32 QRhi2i:F607K=)2 h ∈ M+(X ) O ( /3-T/3- µ I7K2`60/3L)- & Y/3?JOi=)L>40=);=)@Q]F? ( /3- & 6

∣

∣

∣

∣

∂

∂θfθ(x)

∣

∣

∣

∣

≤ h(x),

∣

∣

∣

∣

∂2

∂2θfθ(x)

∣

∣

∣

∣

≤ h(x) QR=)- & ?K?8/ θ ∈ Θ, x ∈ X .

/32i2i/A/3607K2iL)/3?|40/./3-7K:F:)/49+.-0?K7KL)67K2e60hi7K607K*YD /32>/3- gi/3?K?K/3-7K:F:)/=)@QRF? ( 67i-0/36; & 2iL)/;>= ( /3?8?K/3-3D & 2r: & 2>:)=);M;>/L & 2C40:)/d? & 2iL)6;>/ (>& 6i? ( L N-0/A/360782iL)/3?840/32wOb49B (i& 2 & 6; & 2<: & 2>49/ & 6 49+.602i7K2iLTkFDl^e_o=)@QRF? ( 6sQR=)- ( / C/H4060/;>= ( /3?K?K/3-3ON;>/32 ( /3-*)/ ( i?87K*)/3-& 6*,+.-0/S2i=)L)/36dhi7K2`60hi78607K*e6.=)L<60/3:F2i7840:p= *)/3-Y/3607K2iL)/3?849/3-02i/)Oi=)L>*e7w*F7K? ( /3-Q]=)-7K:F:)/LbB7 ( /36 & ? q/3-;M/ (( /3;5Dn 2*F7KL)607KLd?8+.-0/ & 6h (i( - & L)/ & QA49+.602i782iLokFDlNON/3- & 6 ( /32QR=)-0*)/32e60/ ( /7K2Q]=)-0; & 607K=)2 i(θ)7Cg`*)/3-06@ihi2i:e6: & 2>=)@Q & 6060/H449=); /32>* & -07 & 2C43O)=)L ( /3-Q]=)- /3- ( /32M@A=b4078607K*e640/3;M7 ( /GC2i7K6I 40=); -0/3L)/3?'/32 (i(i& @Y=b407K607K*F6 ( /GC2i7K63D

9 S3 e F Ni ] AEd*e7|4 Θ ⊂ R : & 2>*e7PL)7K*)//32M4 & ;M?K/3660=)?K:F2i7K2iL & QjJkFDKlHkemW=)LrjJkFDKl m %'&)( θ0 *N+.-0/ ( /324 & 2 ( /o@ & - & ;>/360/3-3D ( /36;>7 (i( /3?8*,+.- ( 7K/32 & Q `′X(θ0) /3-2`hi?JOF*F7K? ( /32p6 F@i7840:)/*N+.- ( 7 & Q`′X(θ0) 7s/36d:)=)2i:F-0/361/3:F49@A/3-07K;M/32`6*N+.-0/ 69+.6d*)/ ( 2`hi? D dL ( /3-0;>/ ( *e7K? `′X(θ) *N+.-0/69+.6*)/ ( 2ehi?iQ]=)- θ 7i/32<?K7K?K?K/=);>/3L)2 & Q θ0 Db ( /36;>7 (i( /3?K*N+.- ( 7K/32 & Q `′′X(θ0) /3-@A=b497K607K*YO*F7K? ( /32#6 F@i7840:)/>*N+.- ( 7 & Q `′′X(θ0) =)Lb49B5*N+.-0/M@Y=b407K6078*AOw=)L ( /3-0;>/ ( *F7K? `′′X(θ) *N+.-0/@Y=b407K607K*>QR=)- θ 7'/32?K78?K?K/=);M/3L)2 & Q θ0 D/32?K7K?K?8/=);>/3L)2 & Q θ0 : & 2*F7wi-0hiL)/ c& e?K=)- & @i@i-0=):F4078; & 607K=)2i/32

`′X(θ) ≈ `′X(θ0) + `′′X(θ0)(θ − θ0). jJkFDKl9em~F+.6060/3-*F7

θ = θ0 −`′X(θ0)

`′′X(θ0), jJkFDKl m

49B L)78*)/3-SjJkFDKl9em & 6`′X(θ) ≈ 0.

( /36*e7LbBN-h ( Q]- & & 6 θ ?K78L)L)/3-49B 69+.6@CB θ0& 6 `′′X(θ) > 0 OC40?Khi6060/3-.*e7 & 6 `X g & -/36?K=): & ?K6.;>7K2i78;hi;7 θ j]/3?K?K/3-.7wg`*)/3-06Q & ? ( ;M/3L)/3669+.6*)/ ( m9D

8 7wg & - & -0L)hi;>/32e60/3-0/36=b4.Q]-0/3;607K? & 6 ( /3256ze@i7|40:)/?K=)L)?K7K:)/3?87Kgi=e= ( Q]hi2i:e607K=)2 `X(θ) *F7K?g & *)//36j]?K=): & ?K6m';>782i7K;hi; 7C2C+.-0gi/ ( /32 & Q ( /32r4 & 2 ( /@ & - & ;>/360/3-0*N+.- ( 7 θ0 D N?KL)/3?K78L*F7K? ( /326 F@i7840:)/ ?K7K:)/3?K78gi=e= ( QRhi2i:F607K=)2 g & *)//365j]?K=): & ?K6mo; & :F4078;hi; 7.2C+.-0gi/ ( /32 & Q( /32 4 & 2 ( /S@ & - & ;>/360/3-0*N+.- ( 7JD & 2#: & 2 & -0L)hi;>/32`60/3-0/ @CB ?K7KL)2i/32 ( /M*F784Tg`*F784 Θ ⊂ R

k O40: N2e6 2i=)6 & 6078=)2i/32#i?K78*)/3-;>/3-0/d7K2 ( *F7K:e?8/363D %c&)( =b4 hi2 ( /3-94060-0/3L)/ & 6 ( 784949/-9+40=)2i2i/3;>/32`60/3-.7K:F:)//3-4060-0/32iL)6 ; & I60/3; & 607|40:)/)Oc;>/32#/3- & QQ]=)-060=)?K:F2i7K2iLb40;M+4407KL : & - & :F60/3-3DE*F784S; & 2[LbBN- ( /3; /Q]60/3-7( /36 & ? q0/3-3O/3- ( /<g`hi?K?K/ ( / 40=); /32 407JD 61;>7 (i( /3?K*N+.- ( 7K/32 & Q `′X(θ0) /3-12`hi? OYY/36 ( /3-7K:F:)/ & ?K607 (p& 6 ( /32 6ze@i7849:)/o*N+.- ( 7/3-.?K7K?K?K/)O ( /36Y/36 ( /3-i?K=)6 & 6 ( /12i/3L & 607K*)/1*,+.- ( 7K/3-:)=);>@Y/32C40/3-0/H4 & Q ( /o@Y=b407K6078*)/)D ?K?K/d*,+.- ( 7K/3-.:Fhi2i2i/oQR=)- ( /3254 & Lb440:FF? ( *N+.-0/12`hi;>/I-07849:14960=)-0/)D 7K?|40* & -0/32 ( /7K2 ( *)/32 ( 7K2iL)/3-W: & 2 -0/ q049/H4c;>= ( -0/H4960/32 & QC-9+40=)2i2i/3;M/32`60/3-02i/)D /32#; & 2#*F7K?Q]=)-0*)/32`60/ & 6 & -0L)hi;>/32`60:N+ ( /32 Q & :F607840: u C\ L N-0/H4 @i-9+<7847/32#? & 2iL-9+.:F:)/r;M= ( /3?K?8/3-Io60/3:F2i7840:fg & 2 ( ?K/3- ( /36 =); & 6 :Fhi2i2i/rY/3L)-9+.2C40/ ( /32#aH_,^`UsZR^ & Q|I?K/ ( / ( / & Q.?K=)L)?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2i/32wOP49B (i& 2 & 6; & 2 g & -:)=)2e60-0=)? = *)/3-Tg`*)=)-S;>/3L)/36( />60= QN-94060/ & C/ ( / ( /p+.2 ( -0/3-T407KLiD /32i2i/ & -0L)hi;>/32e6 & 607K=)2 L)7K*)/3-*N+.L)6607K? ( /32 782I60hi7K6078=)2wOP*F7'Q BN-Q]- & :)=)2i:)=)- (i& 2C4Q]=)-060=)?K:F2i7K2iL)/32 & Q?87K:)/3?K7Kgi=F= ( Q]hi2i:F607K=)2i/32 ( /324 & 2 ( /@ & - & ;>/360/3-M & L /32L)7K*)/36<=)C40/3-0* & 607K=)27N->4NL)/H4 i? & 2 ( 6 ( /@ & - & ;>/360-0/)O ( /3-<L N-?K78:)/3?K7Kgi=F= ( Q]hi2i:e6078=)2i/32r4060=)-3D

f H Je P, b a \WZR\ > @77 _ CXe_ iV > Θ ⊂ R / 5$5 #- , :5 . (,# 5 7 :--9 . ( 8 ! ", . - t : X → R

Vθ(t X) ≥ (Eθ′(t X))2

i(θ)

. ( θ ∈ Θ. jJkFDKl m2

< = ?A@ B %'&)( =b47K2 ( QBN-0/1@ & - & ;>/360/3-Q]hi2i:F607K=)2i/32 g : Θ → R *)/ (

g(θ) = Eθ(t(X)).

8 7*F7K?7K2e60/3-0/H4940/3-0/>=b4SQR=)-T4 & ;M40@i7K?K?8/36;>/3?K?8/3; ( /r60= 4060=): & 40607840:)/<* & -07 & i?K/ t(X) =)Lfη(X)fθ(X) hi2 ( /3- & 2`6 & L)/3?|40/ & Q & 6 θ /3- ( /32 4 & 2 ( /T@ & - & ;>/360/3-3DPEd/3-/3- η i?K=)6/32 ( 2`h /32@ & - & ;>/360/3-3O ( /3-7wQN-94060/=);>L & 2iL<Y/360- & L)60/H440=); Q & 4063D 8 7w:)=)2C406 & 60/3-0/3- & 6

Eθ

(

fη(X)

fθ(X)

)

=

∫

fη(x)

fθ(x)fθ(x) dµ(x) =

∫

fη(x) dµ(x) = 1.

/3-0;M/ ( /3-Vθ

(

fη(X)

fθ(X)

)

=

∫ (

fη(x)

fθ(x)− 1

)2

fθ(x) dµ(x).

7K?840* & -0/32 ( /1/3-=* θ

(

t(X),fη(X)

fθ(X)

)

=

∫

t(x)fη(x)

fθ(x)fθ(x) dµ(x) − g(θ) · 1 = g(η) − g(θ).

& h=<9g`eI ~<9g & -<hi?K78L)gi/ ( ;M/ ( QBN-0/3-2eh & 6

(g(η) − g(θ))2 ≤ Vθ(t(X))

∫ (

fη(x) − fθ(x)

fθ(x)

)2

fθ(x) dµ(x).

( /32i2i/hi?K7KL)gi/ (5( 7K*F7 ( /3-0/3-*F7';>/ ( (η − θ)2 =)LMLbBN-6078?'L)-9+.2C40/32 η → θ D 1& Q BN-*e7L)-9+.2C40/3hi?K7KL)gi/ ( /32

g′(θ)2 ≤ Vθ(t(X))

∫

(

∂∂θfθ(x)

fθ(x)

)2

fθ(x) dµ(x) = Vθ(t(X))Eθ

(

`′X(θ)2)

.

=); i7K2i/3-0/H4 jJkFDKlHkem =)L jJkFDKl mQ]BN-r*e72i/360=)@ & 6r4 <=)-0/Q]hi2i:F607K=)2i/32C4rFD;>=);>/32`6M/3-7 ( /32`607849:M;>/ ( 7K2Q]=)-0; & 607K=)2i/32wO=)L ( /36 N2C49:)/ ( /S/3-*e7|4063D

2

S3 e F Ni ] A- & ; 3-I & =b4ohi?K7KL)gi/ ( g & -o/32*e7KL)6078L<QR=)-060=)?8:e2i7K2iLr7cQ]=)-0i7K2 ( /3?849/S;>/ ( ( /361@ & - & ;>/I60/3-Q)40=);>gi/ ( 40Y/3L)-0/3wO.*e7 ( 7849:ehi60/3-0/ ( /f7 & Q]402i786 FD FDEd*e784r;>= ( /3?K?K/32 h ( *e7849/3-p4060=)-@ & - & ;>/360/3-Q)40=);>gi/ ( Os*F7K?L)/32`6 & L)/3?|40/3- & Q/3:40@Y/3-07K;>/32e60/36Shi2 ( /3-60= Q]=)-940:)/3?K?K7KL)/ @ & I- & ;>/360-0/L)7K*)/ & 2i?K/ ( 2i7K2iL 607K?60= *)/3?840/3@ & -0/3-0/ ( /f@ihi2i:F6940:F)/3-3O40/ GCL)hi-5FD D /32i2i/40/3@ & - & 6078=)2[: & 2[; & 2[QR=)-0;>/32e60?K7KLQ & 2iL)/M*)/ ( g,q0+.?8@ & Q/32 & Q]i7K? ( 2i782iL t : X → R O49B (i& 2 & 6 t IJ*,+.- ( 7K/3-02i/ & Q ( /d60=T@ihi2i:F6940:F)/3- ?K7KL)L)/3-*)/3?|40/3@ & -0/3-0/36 @CB ( /32M-0/3/3?K?K/ & :40/)DEd*e7|4*e7W7K2`*)=)?8*)/3-0/3-;>/3-0//32 ( 60=r@ & - & ;>/360/3-0*,+.- ( 78/3-7 ( 7|40:ehC4407K=)2i/32wOC/3-d*e7W7K2`60/3-0/H4I40/3-0/367 & 6QR=)- ( /3?87K2iL)/32 & Q t X Ce6060/3-407KLS; & -0: & 2`6;>/ ( θ D n 2p;MB ( / & 6hi2 ( /3-94$NL)/( /32 N2C49:)/ ( /d40/3@ & - & 607K=)2C49/3L)/32C40: & wO`:ehi2i2i/*N+.-0/ & 6GC2 ( /dh (>& Qg`*)=)- ;M/3L)/36X[ZRUWa<yW\WuaH^ba & QQ]=)- ( /3?K7K2iL)/32 CF6060/3-o497KLM;>/ ( θ D /32i2i/6 & 2i:)/3L & 2iLpQN-0/3-d2 & 60hi-0?87KL)6607K?& 6; & 2h ( -0/3L)2i/3-

Eθ′(t X) ,( /3-gi/3?849649: & ?*N+.-0/2`hi;>/3-07849:4060=)-3D /32 - & ; 3-I & =b4hi?87KL)gi/ ( A/36z ( /3- & 6/32>4060=)-

Eθ′(tX) IJ*,+.- ( 7F2 ( *)/32 ( 78L)*e784*F7K?N;>/ ( QN-0//32T4060=)-w* & -07 & 2C4 & Q tX D,~ & L)6'7b=)- ( Hge*e7|4;>7 ( 60@ihi2i:e60/36 & Q.Q]=)- ( /3?K7K2iL)/32 CF6060/3-T407KL g`hi-06078L)6S;>/ ( θ Ow49B;rBpQR=)- ( /3?K782iL)/32 *,+.-0/*N+.-0/i-0/ ( dL ( /3-QR=)-*F7K?AQR=)- ( /3?K782iL)/32 & Q t X hi2 ( /3-60=<2C+.-060?87KL)L)/32 ( /1@ & - & ;>/360-0/& ?K?87KL)/3*)/3?7K:e:)/1*N+.-0/T49+.-0?K78LL)= ( 640/3@ & -0/3-0/363D /32 406$N-0-0/3?840/)O ( /3-rQ]=)-069+.?K?K/3-rv YVP_fi-0/ ( QR=)- ( /3?K782iL)/32 & Q t X ;MB[*,+.-0/ Q]=)- & 660=)@i@ihi2i:F60/36<: & 2 CF6060/ 497KL ;M+.-0:F & -06 ;>/ ( @ & - & ;M/360/3-0/32wO/3- 7840gi/3-94TQ]=)-0*)/32`60/ ( /7K2Q]=)-0; & 607K=)2 i(θ) D n 2 4060=)-*N+.- ( 7 & Q i(θ) Q]=)- & ?K?K/ θ L N- ( /36o@Y=)60/32`6078/3?K6;hi?87KL)6 & 6*N+.?KL)/ t I Q]hi2i:F607K=)2i/3-3OC49B Q]=)- ( /3?K7K2iL)/32 & Q t X /3-QR=)-0gi=)? ( 49*e784Q]=):ehC49/3-0/36.QR=)-ge*)/3-06 θ O=)L & ?K?K7KL)/3*)/3? Ce6060/3-1407KLr2CBN-d; & 2+.2 ( -0/3-1@CB θ D7 / (& 2 ( -0/=)- ( Y/32f4060=)-d*N+.- ( 7 & Qi(θ) L N- ( /36; hi?K7KL)6 & 640/dQ]=)-940:)/3?A@CB@ & - & ;>/360-0/32i/)D /3-QR=)-./3-=)- ( /36 7K2QR=)-0; & 607K=)2 ( /3-9406o*)/3?K* & ?8L)6607K? i(θ) ( /361Y/360/3L)2i/3-1/32 /3L)/32C40: & f*)/ ( ;M= ( /3?K?8/32wO ( /3-1A/H40:F-07K*)/3-ge*)=)-s;M/3L)/36W*F7e*F7K?`Q]B & 6s*F7 ( /=); ( /32 4 & 2 ( /.@ & - & ;>/360/3-s*)/ ( & 6sL N-0/./32=)C49/3-0* & 607K=)2wD /3640: & ?hi2 ( /3-94060-0/3L)/H4 & 6 - & ; 3-I & =b4.hi?K7KL)gi/ ( /3-/32L)/32i/3-0/3?'hi?K78L)gi/ ( OF=)L ( /3-/3-7K:F:)/ & 440=<78/3-0/362i=)L)/32f@CB)406 & 2 ( =); & 6o; & 2 & ?K607 ( : & 2 *N+.?KL)//36d=)@i607K; & ?K640/3@ & -0/I-0/32 ( / t O'49B hi?K7KL)gi/ ( /32fQ]=)-0* & 2 ( ?K/H4607K?/32 7 ( /32e607K60/363D =)-S2i=)L)?8/<;>= ( /3?8?K/3-d: & 2 ; & 2GC2 ( /p49B (i& 2 /36S=)@i607K; & ?K640/3@ & -0/3-0/32 ( / t OQR=)- & 2 ( -0/M;>= ( /3?K?K/3-1: & 2 ; & 2 7K:F:)/)D =)-*F784940/T;>= ( /3?8?K/3-/3- ( /3-o/32 (i(i& /36oL & f;>/3?K?K/3; ( /32Y/ ( 406d=)@i2CBN/3?87KL)/40/3@ & - & 6078=)2=)L( /32 49/3@ & - & 607K=)2wO - & ; 3-I & =b4hi?K7KL)gi/ ( & 2`6z ( /3-d/3-;hi78LiD /3-GC2 ( /H4.* & -07 & 2`60/3- & Q - & ; 3-I & =b4.hi?K7KL)gi/ ( O ( /3-.Lb+.? ( /3-Q]=)-;M= ( /3?K?8/3- g`*)=)-.@ & I- & ;>/360/3-0;r+.2iL ( /32 /3- C/3- ( 7K;M/32C407K=)2 & ?JD 784940/* & -07 & 2e60/3-=)@i2CB)4*)/ ( & 6=)@C40hi;>;M/3-0/7K2Q]=)-0; & 607K=)2i/32 QR- &p( / /36 ( 7K;>/32C407K=)2 & ?8/Shi?K7KL)gi/ ( /3-3O ( /3-d: & 2f@i-0/H4940/h (& Q & ?K?K/S/36 ( 7 I;>/32C497K=)2 & ?K/ ( /3?K;>= ( /3?K?K/3-3D /36/3-L & 2C40:)/40*N+.-06 & 6QR=)-0; hi?K/3-0/S/36o49B (i& 2`6o-0/H40hi?K6 & 6I

f H Je P,

( /36/3- 2i=)L)/36;>/ (>& 6/32r*e784; & 60-07 /3-@A=b497K607K*F649/3;>7 ( /GC2i7K6Ic=)L*e7P*e7K?i78:e:)/L)7K*)//3260/3:e2i7849:rL)/32i2i/3;>L & 2iLiD /32 7K2 ( gi=)? ( /36 & Q - & ; 3-I & =b4dhi?K7KL)gi/ ( 7 C/3-0/ ( 7K;M/32C407K=NI2i/3-3Ow/3- & 6 ( /32 QR=)-0*)/32e60/ ( />7K2Q]=)-0; & 607K=)2C40; & 60-07 j]/3?8?K/3-1;>/3-0/ @i-9+<78406 ( /32 7K2`*)/3-949/QR=)-0*)/32e60/7K2Q]=)-0; & 607K=)2C40; & 60-07Amw49+.6060/3-/32ML)-9+.2C40/Q]=)- ge*)=)- L)= ( 6; & 2>: & 2r49/3@ & -0/3-0/@ & - & ;>/360-0/)D

l :+: S3 e F Ni ] A