Lec12: ëêÚO{

ܲ

May 4, 2011

§1 ��¯K¥�ëêb�u�

3þÙ·?Ø�oN©Ùx´��/, 'uþ���tu�{. �´, �

·Ãrº@oN©Ùx���., K7L^Ù§{5u�. e¡0�A«~^�

ëê{, =ÎÒu�{!ÎÒÚu�{ÚFisheru�{"

!ÎÒu�{

~1 '�`¯ü«Ë�`�,éN

Ü·�u�

� |X − n/2| > c Ľ H0.

�.câ½�u�Y²α, d�©Ù5û½(NL10). ¦αý¢Y², 7

^Åzu�. (��{´Ou��p(§5.3,o). 3d,-d��S1, · · · , Sn��X = n+�äNx0,Px′0 = min{x0, n− x0},Ku��p

p =

x′0∑i=0

(n

i

)(1

2

)n+

n∑i=n−x0

(n

i

)(1

2

)n(1.4)

enóê, x0 = n/2,K�pp = 1. p��C1, KH0�&. X½u�Y²α,K

�p < αĽH0.

3~1¥,½u�Y²α,Ku�¯K(1.2)�Ľ

{X = n+ ≥ c, ½ X ≤ d},

Ù¥cÚd�deª(½:

n∑i=c

(n

i

)(12

)n≤ α

2, d = n− c.

3~1¥,-N = 13, S1, · · · , S13¥+ÒÚ−Ò�ê©O´n+ = 2, n− = 10,Ïdn =n+ + n− = 12.�u�Y²α = 0.05,�NL10/ÎÒu��.L0�c = 10,�d = n−c = 2.�u��ĽD = {X = n+ ≥ 10, ½ X ≤ 2}.u�ÚOþX = n+ = 2, ÏdĽ�b�. =@`!̄ üËØ�.

éùu�¯K, ÏLOu��p5)û. d?, n = 12, x0 = n+ = 2,U(1.4),

x′0 = min(2, 12− 2) = 2,��©ÙL�

p =

2∑i=0

(12

i

)(1

2

)n+

12∑i=10

(12

i

)(1

2

)n= 0.0384 < 0.05

�30.05wÍ5Y²eAĽH0.

~2 ó�üz�¿, zUÓló�e%Yo��, ÿþY¥�¹Åþg.

e¡´n = 11U�P¹:

i 1 2 3 4 5 6 7 8 9 10 11

xi 1.15 1.86 0.76 1.82 1.14 1.65 1.92 1.01 1.12 0.90 1.40

yi 1.00 1.90 0.90 1.80 1.20 1.70 1.95 1.02 1.23 0.97 1.52

Ù¥xiL«z�¿A�ÿþP¹, yiL«z�¿B�ÿþP¹. ¯üz�¿ÿ½�(Jm

kÃwÍ�É? �α = 0.10.

) ©OPz�¿AÚB�ÿþØ�ξÚη.�ξÚηëY.ÅCþ,٩ټê©O

F (x)ÚG(x).u�¯K´

H0 : F (x) = G(x)←→ H1 : F (x) 6= G(x). (1.5)

2

w,¹Åþ�ÿ½,Øz�¿�ØÓk',�FY¥¹Åþ�õ�k'. ·

±@XiÚYiäkêâ(�:

Xi = µi + ξi, Yi = µi + ηi, i = 1, 2, · · · , n.

Ù¥µi1iUY¥�¹Åþ, ξiÚηi©OL«1iUz�¿A!B�ÿþØ�. w,ξ1, · · · , ξnÚη1, · · · , ηn Ñ´Ø*�ÕáÓ©Ù�ÅCþ. cöξ ∼ F (x)Ó©Ù, �öη ∼G(x)Ó©Ù.

ØÓF�üêâXiYiw,ؽ´Ó©Ù�, XiXj , ±9YiYjؽ

´Ó©Ù�. §m��ÉØ�ÿþØ�k', µiÚµj��Ék'. Ïd

,X1, · · · , XnpÕá, �ØUb½§Ó©Ù, Y1, · · · , Yn´Xd. ¤±ü���ÚO'�{, Xü�����tu�{±9�¡0��ü��ëêu�{ÑØU^uù

aêâ�u�ó. ·3§5.2¥J�L¤éêâ�þãA:.

?n¤éêâu�¯K,ég,/�XÛrµi�KØK.duézi,XiYim

', eòÓU�üêâ~, lrµi�KØK. -

Zi = Xi − Yi = ξi − ηi, i = 1, 2, · · · , n. (1.6)

w,Zi=z�¿A!B31iF�ÿþØ��k'. PZ = ξ − η, KZ1, · · · , Znw¤5goNZ�Å��,=Z1, · · · , Zn ´ÕáÓ©Ù���. duZ´üÿþØ��,ÏdZ�þ0, y²§´'u�:é¡�.

-n+Z1, · · · , Zn¥���ê, n−Z1, · · · , Zn¥�K�ê, §Ñ´r.v..dub½ξÚη´ëY.ÅCþ, �Z1, · · · , Zn¥�0�ê±VÇ1�0. ÏdPn =n

++ n− .�H0,=(1.5)¤á, K3nÁ�ü�¥Zi�/+0Ú�/−0�U5� 12 . Ïd

u�¯K=z: n+∼ b(n, p), 0 ≤ p ≤ 1,u�

H ′0 : p =1

2←→ H ′1 : p 6=

1

2

ĽD = {n+ ≥ c ½ n+ ≤ d}.

Ïd, 3½wÍ5Y²α�, cÚd�d

n∑k=c

(n

k

)(12

)n≤ α

2, d = n− c

¤(½.

3�~¥n = 11, α = 0.10, ��©ÙL

2∑k=0

(11

k

)(12

)11= 0.0327,

3∑k=0

(11

k

)(12

)11= 0.113,

¤±d = 2, c = 11− 2 = 9 (�NL10�c = 9, d = n− c = 2). �Y²α = 0.10 �ÎÒu��Ľ

{n+ ≤ 2 ½ n+ ≥ 9}

3

�zi = xi − yi,�

0.15, −0.04, −0.14, 0.02, −0.06, −0.05,

−0.03, −0.01, −0.11, −0.07, −0.12,

Ù¥��ê�ên+ = 2, Ïd3Y²α = 0.10eĽH0,=@z�¿A!Bÿ½(J

mkwÍ�É.

ÎÒu��,A^´© ê(AO´¥ ê)u�. we~.

~3 u�,«Z×�nÝ, ÿ�100êâXeL¤« Á¯TZ×nÝ�¥ 

L 1.1

?Ò 1 2 3 4 5 6 7 8 9 10

nÝ 1.26 1.29 1.32 1.35 1.38 1.41 1.44 1.47 1.50 1.53

ªê 1 4 7 22 23 25 10 6 1 1

ême´Ä1.40? (α = 0.05)

) �K3wÍY²α = 0.05e, u�b�

H0 : me = 1.40←→ H1 : me 6= 1.40

e-L¥¤�100êâ�nÝXi, i = 1, · · · , 100, -Yi = Xi − 1.40, i = 1, · · · , 100. OYi���ên+Ú�K�ên−, �0�ê0, Ïdn+ + n− = 100.3H0¤á

�cJe,KzYi�½K�U5�1/2,�100êâ¥n+Ún−A�OØ,ePX =

n+,´X ∼ b(100, 1/2),Ïdu�¯K=z: X ∼ b(100, p), 0 ≤ p ≤ 1,u�

H0 : p =1

2←→ H1 : p 6=

1

2, α = 0.05

ĽD = {X ≥ c2 ½ X ≤ c1}. |^¥%4½n: �H0¤á, n→∞k

X − n/2√n/4

=2X − n√

n

L−→ N(0, 1)

�K¥n = 100, -c1∑i=0

(100

i

)(12

)100≈ Φ

(c1 − 505

)=α

2= 0.025,

�L�(c1 − 50)/5 = −1.96, )�c1 = 40.2

aq/d100∑i=c2

(100

i

)(12

)n≈ 1− Φ

(c2 − 505

)= 0.025

�L�(c2 − 50)/5 = 1.96,)�c2 = 59.8,�Ľ

{X : X ≤ 40.2 ½ X ≥ 59.8}

dL1.1�X = n+ = 43,§0u(40.2, 59.8)m, �Øv±Ä½H0, �@TZ×�n

Ý�¥ ê´1.40.

ÎÒu��©Ùëêu��'X:

4

b�·a,�¢ëY.ÅCþU , PÙp0© êmq,=

p0 = P (U ≤ mq)

¢S¥· Ø�mq�, =B´½p0�,ù´du·Ø�U�©Ù. é,A

½�m0, P

p = P (U ≤ m0)

dduU�©Ù, �p. duUëY.ÅCþ, �

mq = m0 �=� p = p0

mq ≤ m0 �=� p ≥ p0

mq ≥ m0 �=� p ≤ p0

u´'umq�b��du'up�b�. PU�|��U1, · · · , Un, lÎÒu�ÚOþ

T =∑

I(Ui ≤ m0)

w,T ∼ B(n, p). u´d�©Ù�u�N´��d'uU�© ê�b�u�{K.

�!ÎÒÚu�

4·2£�eÎÒu�, EÒ~1¥¬Ë�¯K5`². 3OZi = Xi − Yi�, ·ïZi�äNê�ÙÎÒSi,¿&E.ù«&E�¿,¦ÎÒu���Ç

k¤ü$. dJÑÎÒÚu�, §´ÎÒu��U?.

~4 Ew~1, �13

½Â6.2.2 �X1, · · · , Xn5güëY.oN���, ½5gõëY.oN�Ü��. KR = (R1, R2, · · · , Rn)¡(X1, · · · , Xn)�ÚOþ, Ù¥RiXi�. dR�Ñ�ÚOþ¡ÚOþ. ÄuÚOþ�u�{¡u�.

y3E£�~4, rL1.2*¿¤eL. ·rÎÒ/+0�@ü(=11Ú12) )å

L 1.3

¬Ë 0;

0 Ù§.

Ri|Zi|3(|Z1|, · · · , |Zn|)¥�,KWilcoxonÎÒÚ (the sum of Wilcoxon signed rank)u�ÚOþ½Â

W+ =

n∑i=1

ViRi. (1.7)

N´n): 3~6.2.5¥,e``u¯,KØ=/+0Ò¬õ,/+0Ò*A�,

 , �o��J´W+A . , e¯`u`, KW+ò �. Ïdu�¯K(1.2), =

H0 : `!̄ üË�Ð

¤á, W+A�ØØ�. u��Ľ´

{W+ ≤ d ½ W+ ≥ c}, (1.8)

d?dÚc�ûun (�~¥n = 12),9½�u�Y²α.=,�½α, c, d©Ode�üª

û½:

P (W+ ≤ d |H0) ≤ α/2, P (W+ ≥ c |H0) ≤ α/2.

H0ýW+�©Ùë©z[4] P246. é,A½�α9Ø�n, cÚd±�L¦�,

Ö"NL11. L¥=��c,d = n(n+ 1)/2− c.

6

dL1.3�K¥n = 12, W+ = 23.�α = 0.05,�L¥α/2@9,3n = 12?�c = 65,

�d = 13,U(1.8)�Ľ

{W+ ≤ 13 ½ W+ ≥ 65}.

13 < W+ = 23 < 65,�A�ÉH0, =¤�*(JØ�¤`!̄ k`�©�¿©yâ.

ùu�¡WilcoxonVýÎÒÚu� (±e{¡VýW+u�) ,¤±�α/2,´

duù/Vý05.

±y²:

E(W+) =n(n+ 1)

4, D(W+) =

1

24n(n+ 1)(2n+ 1)

e!�ÚÚOþWaq, �n→∞, W+�IOzÅCþ

W+∗ =W+ − n(n+ 1)/4√n(n+ 1)(2n+ 1)/24

L−→ N(0, 1) (1.9)

�~6.2.5�Y²Cqα�VýW+u��Ľ{|W+∗ | ≥ uα/2

}�α = 0.05,�|W+∗ | = 1.26 < 1.96 = u0.025 ,��ÉH0, âyk*Øv±Ä½H0.

·±w�~1Ú~6.2.5¥�Óu�¯K^ÎÒu�ÚÎÒÚu���ü«Ø

Ó�(Ø. UÎÒu�ĽH0,=@`!̄ üËk`�©, ¯`u`. UÎÒÚu�

����Ú��{, Ñ�ÉH0, =L²Ã¿©yâĽ/`!̄ üË�Ð0. ùp·

w�: Ó¯K, Ó1êâ, ^ØÓ{, u�(JØÓ, ùØv%. �X^Ó1ê

â��O��oN�êÆÏ", ^��þ�O^¥ ê�O, üö(JØÓ. ùÒ�

)¯K: ùü«u�{=«Ð? ù¯KØUVØ, k,��Öö�wë

©z[9] P156¥L9.1¤��(J. ±Ñ�´: ÎÒu��,ØwêwÎÒ; Äu�

�b½�tu�Kwê, W+u�0u�öm: §QØ�Àê, Ø�wê(ê

^uû½, Ø^Ù��) .

n!Fisher�u�∗

~5 '�A!Bü«{Û«`, ÀJ15¬��/, rz¬©¤/G�

��ü�¬, Å/òÙ¥�¬©A,,�¬B. ¼���¬��þXe:

¬Ò 1 2 3 4 5 6 7 8

A 188 96 168 176 153 172 177 163

B 139 163 160 160 147 149 149 122

A−B 49 -67 8 16 6 23 28 41¬Ò 9 10 11 12 13 14 15

A 146 173 186 168 177 184 96

B 132 144 130 144 102 124 144

A−B 14 29 56 24 75 60 -48

Ñ∑

(A−B) = 314,y3u�b�

H0 : A!B ��J�. (1.10)

7

e(1.10)¤á,z¬SA−B(=49,−67, · · ·�)Ø�,¿duA!B�JØÓ,´duÙü�¬��O. �Åz�(J, z�¬kÓ�U©A½B. Ïd, X31¬,

Åz�(JØÓ, A − B±´49, ±´−49,w�Ð�@¬�A´B. ù�5,ùÁ���ÜU�

∑(A−B)k215:

±(49),±(−67),±(8), · · · ,±(60),±(−48),

¢S�Ñ�∑

(A−B) = 314´215¥�. �A!B�Jk��O|∑

(A−B)|A�.é215U(J¥�zÑ

∑(A−B),^xiP, i = 1, 2, · · · , 215.ò§Uì§�ý

él���^Sü�, ØP

x1, x2, · · · , x215 (1.11)

=÷v

|x1| > |x2| > · · · > |x215 |

(1.11)¥�215¥, 3H0¤ácJe, �Uu), =zÑy�VÇÑ´1/215 .u

�¯K(1.10)�Ľ

{|∑

(A−B)| > c}

*ÿ��|∑

(A−B)| = 314, lu��P

P (|∑

(A−B)| > 314|H0) =m

215

Ù¥müS(1.11)¥÷vxm = 314.

äNOp314 < 0.0001 ÏdkndĽH0.

u��":´: 3äN¢Oþ, ¦^å5ØB. �y3kpOÅ,

|^OÅ5¢ØJ¯.

FishergCÚÙ§NõÆö, ÑïÄLù��¯K: �né, Äé�«Cq�

{�¢u�,±{zO?ïÄ(Jy²: 3é�^e,ù«{z�{

Ø=3, Ò´Ï~�tu�ù´ék¿g�(J. Ïm©, tu�´Û3��

�.¥�Ñ�. ÏLùå»uy, =¦32��.e, Á�gêv, tu�

E´·^�, Ïd±`, u��nØlý¡\rtu��/ .

§2 ü��¯K¥�ëêb�u�

3ü���'�¯K¥, ����ÅØ�ØÑl��©Ù, ÒIJÑ�

b�, ¿¦^A�ëêu�{. ù¡�nØÚ{�õ, �Ñé;, ùp

éWilcoxonÚu�Úu�{Ñ0�.

!Úó9½Â

·Äk5wwùu��¢S�µ. ü��u�¯K�J{Xe: �X1, · · · , XmÚY1, · · · , Yn©O´läk©ÙF1ÚF2�oN¥Ä��{ü��,b½Ü��X1, · · · , Xm,Y1, · · · , Yn�NpÕá. u�e�b�

H0 : F1 = F2 ←→ H1 : F1 6= F2. (2.1)

8

3ênÚOÆ¥, S.þ¡ùu�¯K/ü��¯K0. ·5©OÄe�A«¹:

1. �â¯K�¢S�µ,XJ·kndb½F1ÚF2äkÓ����©Ù,=

b½

F1 ∼ N(a, σ2), F2 ∼ N(b, σ2)

Ù¥a!bÚσ2�, −∞ < a, b < +∞, σ2 > 0,ùu�¯K=z

H ′0 : a = b←→ H ′1 : a 6= b. (2.2)

3ùb½e, oN©ÙF1ÚF26unëêa!bÚσ2, u�¯K(2.1)8(u�

ùëê´Ä÷v(2.2). U§5.1¤ãùáu/ëê.b�u�¯K0. ùÒ´§5.2¥?Ø�ü��tu�.

2. XJ·é¯K�¢S�µ¤$�, ·Ð@éF1ÚF2��. 3ù�°

2�b½e, ·2ØU¦^Ï~�ü��tu�. ?nù¯K�«{´/d�â

Å0(Smirnov)u�, ùò3�Ù1Ê!¥?Ø.

3ù/e,oN©ÙF1ÚF2ØU^k¢ëê�x,Ïd¡ëêu�¯K.

3. y3·?Ø«¥m¹. �X´«�¬3½)�ó²e�þI, Y´

T�¬3,)�ó²e�þI. knd@, UC)�ó²ØK�¬þI�V

Ç©Ù, U¦d©Ùu)²£. Ò´`, e±FPX�©Ù, KY©ÙF (x − θ),ùpθ´� ëê. 3ùb½e,/X!YÓ©Ù0�b��/θ = 00, éá

b�/θ 6= 00. Ïdu�(2.1)8(u�

H0 : θ = 0←→ H1 : θ 6= 0. (2.3)

(2.3)´é�b�u�¯K. 3ù�.¥, ·b½F, Ï'���.2.

,ù�.q'/d�âÅu�0¥��.Ä, Ïé�öó, ü©ÙF1ÚF2Î

Ã'X, 3dF1ÚF2mkF2(x) = F1(x− θ).

,L¡þw(2.3)ëêu�¯K: b�¥�9θ,§´¢ëê. Ù¢Ø,,

ÏoN�©ÙFÚθÑk',F�©Ù,ÏdUëêÚO¯K�½Â, (2.3)AÀ

ëêu�¯K.

/, ü��¯K(2.1)käk¢S�µ�¥m¹. ~XF2(x) = F1(x/σ),

dσ > 0�Ýëê, ©ÙF. u�¯K(2.1)3d¹e=z

H∗0 : σ = 1←→ H∗1 : σ 6= 1. (2.4)

Wilcoxonü��Úu�Ò´Ä(2.3)�b�u�¯K.e¡ÄkÑWilcoxonü��

ÚÚOþ�½Â.

½Â6.3.1 �X1, · · · , Xm, Y1, · · · , Ynùn + müüØÓ, r§U�ü�, (J

Z1 < Z2 < · · · < ZN , N = m+ n, (2.5)

w,, zYi7(2.5)¥�,. eYi = ZRi ,KYi3Ü��X1, · · · , Xm,Y1, · · · , Yn¥�Ri. Y1, · · · , Yn�Ú

W = R1 + · · ·+Rn, (2.6)

§¡Wilcoxonü��ÚÚOþ. ù´Wilcoxon31945c�ó¥Ú?�.

9

�! Wilcoxonü��Úu�—���{

Wilcoxonü��Úu�Ò´Ä(2.3)�b�u�¯K,=�X1, · · · , Xm i.i.d.∼ F (x), Y1, · · · , Yni.i.d. ∼ F (x− θ),Ü��Õá. u�(2.3), =

H0 : θ = 0←→ H1 : θ 6= 0.

���Y1, · · · , Yn�ÚWd(2.6)Ñ. y3ù�ín: zRiÑ�1, 2, · · · , N.e�b�H0¤á, K�Ü��5gÓoN, zÑØÓAÏ , ج���½��,

W¤�A8¥3²þên(N + 1)/2NC. ���e�u�:

� W ≤ d ½ W ≥ c Ľ H0 (2.7)

XÛ(½cÚd ? §�(3�Kþ±)û: �H0¤á, Ü��ÕáÓ©Ù, ddâé

¡5�Ä, ´(R1, · · · , Rn)�éÜ©Ù

P (R1 = r1, · · · , Rn = rn) =

1

N(N−1)···(N−n+1) �r1, · · · , rn ≤ NpØÓ�g,ê,

0 Ù§.

ddØJ/ª/�ÑW�©Ù. ld

α = P (W ≤ d ½ W ≥ c |H0)

½ÑcÚd. é���m!n®¤L.

XJb�u�´ü>�, =u�

H′

0 : θ ≤ 0←→ H′

1 : θ > 0, (2.8)

duW =n∑i=1

Ri´Y1, · · · , Yn3Ü��¥�Ú.eθ > 0KÏzYi�©ÙXi + θ�©Ù

Ó, YiXi'�u��.=Yi�uXi�/Ŭ0õ,�u§�ŬK�.

ù�5R1, · · · , Rn�θ > 0u�8Ü{1, 2, · · · , N} ¥��(d?N = m + n), Ó�θ < 0, KR1, · · · , Rnu�8Ü{1, 2, · · · , N}¥���. ÏW3θ > 0u���, 3θ < 0u����. �u�¯K(2.8)�u�:

� W ≥ c , Ľ H ′0.

Ó�u�¯K

H′′

0 : θ ≥ 0←→ H′′

1 : θ < 0, (2.9)

�u�:

� W ≤ d , Ľ H ′′0 .

òþãnau��¤eL:

L6.3.1 Wilcoxonü��Úu�(���/)

H0 H1 Ľ

θ = 0 θ 6= 0 W ≤ d ½ W ≥ cθ ≤ 0 θ > 0 W ≥ cθ ≥ 0 θ < 0 W ≤ d

10

XÛ(½�.cÚd ? é���m!n®²¤L, Ö"NL12. 'udL, ±eü:`

²:

(1) ©OP(X1, · · · , Xm)Ú(Y1, · · · , Yn)3Ü��¥�ÚW1ÚW2,K

W1 +W2 = 1 + 2 + · · ·+ (m+ n) =(m+ n)(m+ n+ 1)

2

´~ê. Ïd, 3¦^WilcoxonÚu�{, '�üoN©Ù, ^W1u�ÚOþ

^W2u�ÚOþ´£¯.

(2) 3NL12¥ÑÚu���.: P (W ≥ c) ≤ α¥c�,éP (W ≤ d) ≤ α��.dXÛ|^dL¦Ñ? ±y² P (W ≤ α) = P (W ≥ n(m + n + 1) − d) =ePc = n(m + n + 1) − d, ké½� α !m!n ¦Ñ P (W ≥ c) ≤ α ��. c, ,� d dúª d = n(m+ n+ 1)− c Ñ.

~6 ,«�f3?1,«ó²?ncÚ?n�, ÅÄ���, ÿ�Ù¹

ÇXe

?nc: 0.20, 0.24, 0.66, 0.42, 0.12;

?n�: 0.13, 0.07, 0.21, 0.08, 0.19.

¯?n�¹Ç´Äeü? (α = 0.05)

) �XÚY©OL«?nc!��f�¹Ç,§�©Ù¼ê©OF (x)ÚF (x−θ).Ku�¯K

H0 : θ ≥ 0←→ H1 : θ < 0.

=ò/?n��f¹Çvkeü0�b�.

dL6.3.1: �W ≤ dĽ�b�. dc¡'uNL12�¦^`²(2), klNL¥dP (W ≥ c) ≤ α�Ñc, Kd = n(m+ n+ 1)− c.

�K¥m = n = 5, α = 0.05,dNL12, �Ñc = 36,�k

d = n(m+ n+ 1)− c = 5× 11− 36 = 19.

ùL²

P (W ≤ 19) = P (W ≥ 36) ≤ 0.05.

Ïdþãu�¯K�Ľ:

D = {(X,Y) : W ≤ 19}.

yòü|��*Ul��ü¤�¤eL

0.07 0.08 0.12 0.13 0.19 0.20 0.21 0.24 0.42 0.66

1¯

2¯

3 4¯

5¯

6 7¯

8 9 10

ey�ê´?n��f¹Ç(Y )�*�. �Y�*�Ú

W = 1 + 2 + 4 + 5 + 7 = 19

òÙĽ¥�.'�19 ≤ d (d = 19),ÏdĽH0,=@?n��f¹Çeü.

z

11

b�:

½ü|Õá�Å��X1, X2, · · · , Xn ÚY1, Y2, · · · , Ym, ¦

1. �^SºÝ

2. a,��Cþ´ëY.�

3. F (x)ÚG(x)©OL«XÚY�©Ù¼ê

Kéb�

(A) H0 : F (x) = G(x) é¤kx ⇐⇒ H1 : F (x) 6= G(x) é,x

(B) H0 : F (x) ≤ G(x) é¤kx ⇐⇒ H1 : F (x) > G(x) é,x

(C) H0 : F (x) ≥ G(x) é¤kx ⇐⇒ H1 : F (x) < G(x) é,x

b�(B)Ú(C)¥�H1©OL«”Xu'Y�” Ú”Xu'Y”. u´PWL

«X��3Ü��¥�Ú, Ku��{K©O

(A) éV>u�H1 : F (x) 6= G(x): eW < c1 ½W > c2, áýH0; ÄKØv±áýH0.

(B) éü>b�H1 : F (x) > G(x): eW < c,áýH0 ; ÄKØv±áýH0.

(C) éü>b�H1 : F (x) < G(x): eW > c, áýH0; ÄKØv±áýH0.

n!Wilcoxonü��Úu�—��{

c¡?Ø�m!n��Wilcoxonü��Úu�����{. �m!n�u�

ÚOþW�©Ù�OéE,, é½�αvky¤�L±��Ľ��.. Ïd�

4½n. {Xe: N´¦�

E(W ) = n(m+ n+ 1)/2 = n(N + 1)/2,

D(W ) = mn(n+m+ 1)/12 = mn(N + 1)/12.

d?Wd(2.6)ªÑ, N = m+ n.P

W ∗ =W − E(W )√

D(W )=W − n(N + 1)/2√mn(N + 1)/12

±y²3�b�H0 : θ = 0¤áe, �m, n→∞k

W ∗ =W − n(N + 1)/2√mn(N + 1)/12

L−→ N(0, 1)

Ïd�u�¯K(2.3)�Y²Cqα�Ľ:

D = {(X,Y) : |W ∗| ≥ uα/2}

aq�u�¯K(2.8)Ú(2.9)�Y²Cqα�u��Ľ, eL:

12

L6.3.2 Wilcoxonü��Úu�(��/)

H0 H1 Ľ

θ = 0 θ 6= 0 |W ∗| > uα2

θ ≤ 0 θ > 0 W ∗ > uαθ ≥ 0 θ < 0 W ∗ < −uα

~7 k`¯ü�ÅK\óÓ���¬. lùü�ÅK\ó��¬¥Å/Ä�eZ�

¬, ÿ��¬»(ü :mm)

`: 18.1, 17.7, 17.2, 19.1, 17.0, 17.5, 17.8, 18.7

¯: 18.3, 19.0, 18.9, 17.3, 16.9, 18.4, 17.6, 18.6, 18.0

�ü�ÅK�°ÝÓ, Á¯`!̄ ü�ÅK\ó�¬�»kÃwÍ�É? (α = 0.10)

) �XÚY©OL«`¯ü�K\ó�¬�»,§�©Ù¼ê©OF (x)ÚF (x−θ).Ku�¯K:

H0 : θ = 0←→ H1 : θ 6= 0

d~¥m = 8, n = 9.òü|êâYUl��ü¤�, �1�|êâ�g: 1, 4,

6, 9, 11, 12, 13, 15, 16, ÙÚ´

W = 1 + 4 + 6 + 9 + 11 + 12 + 13 + 15 + 16 = 87.

éu�Y²α = 0.10,dL6.3.2��Ľ

D = {(X,Y) : |W ∗| ≥ u0.05 = 1.96}

Ù¥

|W ∗| =∣∣∣W − n(n+m+ 1)/2√

mn(m+ n+ 1)/12

∣∣∣ = ∣∣∣87− 81√108

∣∣∣ = 0.58 < 1.96��ÉH0,=â®kêâØv±Ä½ùý�ÅK\ó�»ÃwÍ�É�b½.

k

o!ü��u�{∗

3ênÚOÆ¥, /?n0c�¹¿42. §±L««ó²6§, «f¬

«, «£�{��. ü��u��g~6.2.6 q. �ü?n��ÜÁ�(J

X1, · · · , Xm, Y1, · · · , Yn. òÙü�3å, UZ1, · · · , Zn+m. XJ`!¯ü?nÃ�O,KZ1, · · · , Zn+mm��OØ´du?nØÓ5, ´duùn + mÁ�ü��©�{5. 3n+m ü�¥©�m?n`, ØÓ�{k

(n+mm

)«. e3z«©�{

eÑO±e�þ:

g =1

m(`?nÁ�Ú)− 1

n(¯?nÁ�Ú).

§�uZ1, · · · , Zn+m¥�m²þê~�e�n�²þê. @mKwÁ�ü�´XÛ©��. ù�, �U��N =

(n+mm

): g1, g2, · · · , gN ,òÙýéU�ü�, Ã-

Ù:

|g1| ≤ |g2| ≤ · · · ≤ |gN |, (2.10)

u�¯K:

H0 :`¯ü?n�J� (2.11)

�H0¤á(2.10)¥@N¥�z,kÓ�Ñy�Ŭ1/N,Ò¢S��Ñg,P

g∗ = X − Y . XJH0ؤá, |g∗|u��. Ïd, ½u�Y²α�, éNα, ¦�

α = P (|gi| > |g[Nα]|, i ∈ {1, · · · , N} |H0) (2.12)

dug1, g2, · · · , gN¥�zu)´�U1/N, Ïd(2.12)ª�duéNα,¦�

N − [Nα]N

= α

ÏdĽ

D = { |g[Nα]+1|, |g[Nα]+2|, · · · , |gN | },

Ïd½��X = (X1, · · · , Xm)ÚY = (Y1, · · · , Yn),Ñg∗ = X − Y, e|g∗| = |X − Y | >|g[Nα]|, K|g∗|73ĽD¥, �DL

D = { (X,Y) : |g∗| = |X − Y | > |g[Nα]| }.

d?[Nα]L«Nα��êÜ©.

eU�b�

H′

0 :?n`Ø`u¯,

Kòg1, g2, · · · , gNU�ü�h1 ≤ h2 ≤ · · · ≤ hN

é½�α,éNα,¦�

α = P (hi > h[Nα], i ∈ {1, 2, · · · ., N} |H′

0).

Ïd, �k��X = (X1, · · · , Xm)ÚY = (Y1, · · · , Yn) Ñg∗ = X − Y,ĽL

D = {(X,Y) : g∗ = X − Y > h[Nα]}.

�m!nÑé, þãu��Cuü��tu�. XÓ3��/, ù5l

ý¡\rü��tu��/ .

14