主要介绍了非参数假设检验方法:符号检验法、符号秩和检验法、Fisher置换检验法,简单介绍了两样本问题中的非参数假设检验方法:Wilcoxon两样本秩和检验法、两样本置换检验法。

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1. Lec12: šëêÚO•{ Ü•² May 4, 2011 §1 ˜ ¯K¥ šëêb u 3þ˜Ù·‚?Ø oN©Ùx´ œ/, 'uþŠ ˜ tu •{. ´, ·‚Ãrº@•oN©Ùx• .ž, K7L^Ù§•{5u . e¡0 A«~^ šëê•{, =ÎÒu {!ÎÒ•Úu {ÚFisher˜†u {" ˜!ÎÒu { ~1 •' `¯ü«Ë ` , é N ‡< ¬}. Ó˜‡<¬}ü«Ë , ž¦‚© O‰ü«Ëµ©. ùp, z˜‡¬Ë<é`!¯ü«Ë µ©(J ¤˜‡éf, д˜ ‡¤é' .. ±Xi P1i‡¬Ë<é`Ë µ©, Yi P1i‡¬Ë<é¯Ë µ©. PZi = Xi − Yi , i = 1, · · · , N.XJb½Zi ∼ N (µ, σ 2 ),K`!¯üË´Äk` ¯Kò=z• b H0 : µ = 0 u ¯K, ùÒ´·‚3§5.2?ØL ˜ tu ¯K. Œ´3˜ œ¹e, ·‚Ø„ kŠâ b½Zi Ñl ©Ù. ùžþã•{Ò” . e¡´˜‡O“•{: z˜‡µÒ < µ©‰Ñ˜‡ÎÒ   + eZ i > 0   Si = − eZ i < 0 (1.1)   0 eZ = 0  i =¬Ò<‰±/+0ÒL«¦@•/`Ë`u¯Ë0, ,ü‡ÎÒ ¿Âaí. Xd, ·‚ n‡ÎÒS1 , · · · , Sn . b H0 : `¯ü«Ë˜ Ð (1.2) u Òïá3Á (J ùn‡ÎÒ Ä:þ, ¡•ÎÒu (Sign Test).e¡ò¬w : lÚO . ó, ÎÒu ØL´ ‘©Ùëêu ˜‡A~. ÎÒu äN•{Xe: PN ‡Á (JS1 , · · · , Sn ¥/+0Ò gêkn+ g, Ñy/−0Ò kn− g, Ù{•0. Pn = n+ + n− .XJH0 ¤á, =`¯ü«Ë˜ Ð, K3n‡š0(J¥Ñy/+0½/−0 ŬƒÓ. =z‡š0Á (J¥Ñy/+0Ò VÇp = 1/2;e`!¯üË(k` ƒ©, K z‡š0(J¥Ñy/+0 VÇp = 1/2. ePX = n+ , ˜3ù‡œ¹e, n+ ©ÙÑlb(n, 1/2),e`¯ü«Ë(k` ƒ©, Kz‡(JÑy /+0Ò VÇp = 1/2. K¤J¯K=z•u ¯K: X ‘©Ù b(n, p), 0 ≤ p ≤ 1,‡u 1 1 H0 : p = ←→ H1 : p = . (1.3) 2 2 1

2.˜‡Ü· u • |X − n/2| > c žÄ½ H0 . .Šc‡Šâ‰½ u Y²α, d ‘©Ù5û½(„NL10). •¦α•ý¢Y², 7‡ž ^‘Åzu . ˜‡•( •{´OŽu pŠ(„§5.3,o). 3d, -d S1 , · · · , Sn Ž X = n+ äNŠ•x0 ,Px0 = min{x0 , n − x0 },Ku pŠ• x0 n n n n 1 n 1 p= + (1.4) i=0 i 2 i=n−x0 i 2 en•óê, x0 = n/2,K pŠ•p = 1. pŠ C1, KH0 Œ&. X‰½u Y²α,K p < αžÄ½H0 . 3~1¥,‰½u Y²α,Ku ¯K(1.2) Ľ•• {X = n+ ≥ c, ½ X ≤ d}, Ù¥cÚd Šdeª(½: n n n 1 α ≤ , d = n − c. i=c i 2 2 3~1¥,-N = 13, S1 , · · · , S13 ¥+ÒÚ−Ò ‡ê©O´n+ = 2, n− = 10,Ïdn = n+ + n− = 12. u Y²α = 0.05, NL10/ÎÒu .ŠL0 c = 10, d = n−c = 2. u Ľ•D = {X = n+ ≥ 10, ½ X ≤ 2}.u ÚOþX = n+ = 2, ÏdĽ b . = @•`!¯üËؘ . éù˜u ¯K, •ŒÏLOŽu pŠ5)û. d?, n = 12, x0 = n+ = 2,U(1.4), x0 = min(2, 12 − 2) = 2, ‘©ÙL 2 n 12 n 12 1 12 1 p= + = 0.0384 < 0.05 i=0 i 2 i=10 i 2 30.05wÍ5Y²eAĽH0 . ~2 ó‚ ü‡z ¿, zUÓžló‚ e%Yo , ÿþY¥ ¹Åþ˜g. 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¥xi L«z ¿A ÿþP¹, yi L«z ¿B ÿþP¹. ¯ü‡z ¿ÿ½ (Jƒm kÃwÍ É? α = 0.10. ) ©OPz ¿AÚB ÿþØ •ξ Úη. ξ Úη •ëY.‘ÅCþ, ٩ټê©O •F (x)ÚG(x).u ¯K´ H0 : F (x) = G(x) ←→ H1 : F (x) = G(x). (1.5) 2

3. w,¹Åþ ÿ½Š, Ø †z ¿ ØÓk' , „† FY¥¹Åþ õ k'. · ‚Œ±@•Xi ÚYi äkêâ( : Xi = µi + ξi , Yi = µi + ηi , i = 1, 2, · · · , n. Ù¥µi •1iUY¥ ¹Åþ, ξi Úηi ©OL«1iUz ¿A!B ÿþØ . w,ξ1 , · · · , ξn Úη1 , · · · , ηn Ñ´ØŒ* ÕáÓ©Ù ‘ÅCþ. cö†ξ ∼ F (x)Ó©Ù, ö†η ∼ G(x)Ó©Ù. ØÓF ü‡êâXi †Yi w,ؘ½´Ó©Ù , …Xi †Xj , ±9Yi †Yj •Ø˜½ ´Ó©Ù . §‚ƒm ÉØ †ÿþØ k', …•†µi Úµj Ék'. Ïd• ,X1 , · · · , Xn ƒpÕá, ØUb½§‚Ó©Ù, Y1 , · · · , Yn •´Xd. ¤±ü ÚO' •{, Xü tu •{±9 ¡‡0 ü šëêu •{ÑØU^uù aêâ u óŠ. ·‚3§5.2¥•J L¤éêâ þãA:. ?n¤éêâu ¯K, ég,/Ž XÛrµi K•žØK. duéz‡i,Xi †Yi ƒm Œ', eòÓ˜U ü‡êâƒ~, l rµi K•žØK. - Zi = Xi − Yi = ξi − ηi , i = 1, 2, · · · , n. (1.6) w,Zi =†z ¿A!B31iF ÿþØ ƒ k'. PZ = ξ − η, KZ1 , · · · , Zn Œw¤5g oNZ ‘Å , =Z1 , · · · , Zn ´ÕáÓ©Ù . duZ ´ü‡ÿþØ ƒ , ÏdZ þŠ•0, …Œy²§´'u :é¡ . -n+ •Z1 , · · · , Zn ¥ Š ‡ê, n− •Z1 , · · · , Zn ¥ KŠ ‡ê, §‚Ñ´r.v..du b½ ξ Úη ´ëY.‘ÅCþ, Z1 , · · · , Zn ¥ Š•0 ‡ê±VÇ•1 0. ÏdŒPn = n+ + n− . H0 ,=(1.5)¤áž, K3n‡Á ü ¥Zi /+0Ú /−0 ŒU5 • 21 . Ïd u ¯K=z•: n+ ∼ b(n, p), 0 ≤ p ≤ 1,u 1 1 H0 : p = ←→ H1 : p = 2 2 Ľ•D = {n+ ≥ c ½ n+ ≤ d}. Ïd, 3‰½wÍ5Y²αƒ , cÚd Šd n n n 1 α ≤ , d=n−c k 2 2 k=c ¤(½. 3 ~¥n = 11, α = 0.10, ‘©ÙL• 2 11 11 1 = 0.0327, k 2 k=0 3 11 11 1 = 0.113, k 2 k=0 ¤±d = 2, c = 11 − 2 = 9 (•Œ NL10 c = 9, d = n − c = 2). Y²α = 0.10 ÎÒu Ľ•• {n+ ≤ 2 ½ n+ ≥ 9} 3

4. Š Šzi = 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 ‡ê•n+ = 2, Ïd3Y²α = 0.10eĽH0 ,=@•z ¿A!Bÿ½(Jƒ mkwÍ É. ÎÒu ,˜‡-‡A^´© ê(AO´¥ ê)u . žwe~. ~3 u ,«‘Z× nÝ, ÿ 100‡êâXeL¤« Á¯T‘Z×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 ´Ä•1.40? (α = 0.05) ) K3wÍY²α = 0.05e, u b H0 : me = 1.40 ←→ H1 : me = 1.40 e-L¥¤ 100‡êâ nÝŠ•Xi , i = 1, · · · , 100, -Yi = Xi − 1.40, i = 1, · · · , 100. O ŽYi Š ‡ên+ Ú KŠ ‡ên− , Š•0 ‡ê•0, Ïdn+ + n− = 100.3H0 ¤á cJe, Kz‡Yi • ½K ŒU5 •1/2, 100‡êâ¥n+ Ún− A OØŒ, ePX = n+ ,´„X ∼ b(100, 1/2),Ïdu ¯K=z•: X ∼ b(100, p), 0 ≤ p ≤ 1,‡u 1 1 H0 : p = ←→ H1 : p = , α = 0.05 2 2 Ľ••D = {X ≥ c2 ½ X ≤ c1 }. |^¥%4•½nŒ•: H0 ¤á, …n → ∞žk X − n/2 2X − n L = √ −→ N (0, 1) n/4 n K¥n = 100, - c1 100 1 100 c1 − 50 α ≈Φ = = 0.025, i=0 i 2 5 2 L (c1 − 50)/5 = −1.96, ) c1 = 40.2 aq/d 100 100 1 n c2 − 50 ≈1−Φ = 0.025 i=c2 i 2 5 L (c2 − 50)/5 = 1.96,) c2 = 59.8, Ľ•• {X : X ≤ 40.2 ½ X ≥ 59.8} dL1.1Ž X = n+ = 43,§0u(40.2, 59.8)ƒm, Øv±Ä½H0 , @•T‘Z× n‘ Ý ¥ ê´1.40. ÎÒu † ‘©Ùëêu 'X: 4

5. b ·‚a, ˜‡¢ŠëY.‘ÅCþU , PÙp0 © ê•mq ,= p0 = P (U ≤ mq ) ¢S¥·‚ Ø• mq Š, =B´•½p0 Š,ù´du·‚Ø• U ©Ù. é,‡A ½ m0 , P p = P (U ≤ m0 ) džduU ©Ù™•, p™•. duU •ëY.‘ÅCþ, mq = m0 …= p = p0 mq ≤ m0 …= p ≥ p0 mq ≥ m0 …= p ≤ p0 u´'umq b du'up b . PU ˜| •U1 , · · · , Un , l ÎÒu Ú Oþ• T = I(Ui ≤ m0 ) w,T ∼ B(n, p). u´d ‘©Ù u N´ dž'uU © ê b u {K. ! ÎÒ•Úu 4·‚2£ ˜eÎÒu , EÒ~1¥¬Ë ¯K5`². 3OŽZi = Xi − Yi ,· ‚˜ïZi äNêŠ ÙÎÒSi ž, ¿” ˜ &E. ù«&E ¿”, ¦ÎÒu Ç k¤ü$. •dJÑ ÎÒ•Úu , §´ÎÒu U?. ~4 Ew~1, Žž 13‡<¬}`!¯ü«Ë, µ©(JXe: L 1.2 ¬Ë< 1 2 3 4 5 6 7 8 9 10 11 12 13 ` (xi ) 55 32 41 50.5 60 48 39 45 48 46 52.2 45 44 ¯ (yi ) 35 37 43.1 55 34 50.3 43 46.1 51 47.3 55 46.5 44 ÎÒ(zi ) + − − − + − − − − − − − 0 d?zi = xi − yi .Á¯`¯ü«Ë´Ä˜ Ð? ˜ 12‡š0ÎÒ¥, kü‡/+0Ò, w« õê¬Ë<@•¯ËÐ. 3ÎÒu ¥·‚Ò•UŠâ/+0! /−0Ò ê8 e(Ø. [ w˜e(J, ·‚uy, 3@•/¯Ë'`Ë`0 10<¥, ¯Ë ©'`Ëp Øõ, 3@•/`Ë`u¯Ë0 2<¥, ` © pu¯. ù‡¯¢‰2 : 10ù‡L¡(J, ‹ ˜‡òž, §é«·‚: Ø •ÄÎÒ , „A rù˜:•Ä?5. ÎÒ• VgJø ˜«Š{. ½Â6.2.1 X1 , · · · , Xn •üü؃ ˜| , òÙŒ ü •X(1) < · · · < X(n) , eXi = X(Ri ) , K¡Xi 3 (X1 , · · · , Xn )¥ ••Ri . w,, eX1 , · · · , Xn •5gëY.©ÙF (x) , K±VÇ•1 yX1 , · · · , Xn ¥üü p؃ . 5

6. ½Â6.2.2 X1 , · · · , Xn •5gü‡ëY.oN , ½5gõ‡ëY.oN Ü . KR = (R1 , R2 , · · · , Rn )¡•(X1 , · · · , Xn ) •ÚOþ, Ù¥Ri •Xi •. dR Ñ Ú Oþ•¡••ÚOþ. Äu•ÚOþ u •{¡••u . y3E£ ~4, rL1.2*¿¤eL. ·‚rÎÒ•/+0 @ü‡•(=11Ú12) )å L 1.3 ¬Ë<(i) `(xi ) ¯(yi ) ÎÒ(zi ) |Zi | = |xi − yi | • 1 55 35 + 20 [11] 2 32 37 − 5 10 3 41 43.1 − 2.1 4 4 50.5 55 − 4.5 9 5 60 34 + 26 [12] 6 48 50.3 − 2.3 5 7 39 43 − 4 8 8 45 46.1 − 1.1 1 9 48 51 − 3 7 10 46 47.3 − 1.3 2 11 52.2 55 − 2.8 6 12 45 46.5 − 1.5 3 13 44 44 0 0 ؽ• 5, §‚ Ú´W + = 11 + 12 = 23, ‰/ÎÒ•Ú0. ˜„§Œ±^e •ª5½Â: PZi = Xi − Yi , - 1 eZi > 0; Vi = 0 Ù§. Ri •|Zi |3(|Z1 |, · · · , |Zn |)¥ •, KWilcoxon ÎÒ•Ú (the sum of Wilcoxon signed rank)u ÚOþ½Â• n W+ = V i Ri . (1.7) i=1 N´n): 3~6.2.5¥, e``u¯, KØ=/+0Ò¬õ, …/+0Ò* ƒA •, ˜„ • Œ, o J´W + A Œ. ‡ƒ, e¯`u`, KW + ò . Ïdu ¯K(1.2), = H0 : `!¯ü˘ Ð ¤áž, W + A ØŒØ . u Ľ•´ {W + ≤ d ½ W + ≥ c}, (1.8) d?dÚc ûun ( ~¥n = 12), 9•½ u Y²α. =, ‰½αž, c, d©Ode üª û½: P (W + ≤ d |H0 ) ≤ α/2, P (W + ≥ c |H0 ) ≤ α/2. H0 •ýžW +©Ù„ë•©z[4] P246 . é, A½ α9ØŒ n, cÚdŒ± L¦ ,„ Ö"NL11. L¥=Œ c, d = n(n + 1)/2 − c. 6

7. dL1.3Œ• K¥n = 12, W + = 23. α = 0.05, L¥α/2@˜9, 3n = 12? c = 65, Ž d = 13,U(1.8) Ľ•• {W + ≤ 13 ½ W + ≥ 65}. 13 < W + = 23 < 65, A ÉH0 , =¤ * (JØ ¤`!¯k` ƒ© ¿©yâ. ù‡u ¡•WilcoxonVýÎÒ•Úu (±e{¡VýW u + ) , ƒ¤± α/2, •´ duù‡/Vý0 5. Œ±y²: n(n + 1) 1 E(W + ) =, D(W + ) = n(n + 1)(2n + 1) 4 24 †e! •ÚÚOþW aq, n → ∞ž, W + IOz‘ÅCþ W + − n(n + 1)/4 L W∗+ = −→ N (0, 1) (1.9) n(n + 1)(2n + 1)/24 ~6.2.5 Y²Cq•α VýW + u Ľ•• |W∗+ | ≥ uα/2 α = 0.05,Ž |W∗+ | = 1.26 < 1.96 = u0.025 , ÉH0 , Šâyk* ŠØv±Ä½H0 . ·‚Œ±w ~1Ú~6.2.5¥ Ó˜‡u ¯K^ÎÒu ÚÎÒ•Úu ü«Ø Ó (Ø. UÎÒu ĽH0 ,=@•`!¯üËk` ƒ©, …¯`u`. UÎÒ•Úu ÚŒ •{, Ñ ÉH0 , =L²Ã¿©yâĽ/`!¯ü˘ Ð0. ùp·‚ w : Ó˜‡¯K, Ó˜1êâ, ^ØÓ•{, u (JØÓ, ùØv•%. X^Ó˜1ê â O oN êÆÏ"Š, ^ þŠ O†^¥ ê O, üö(JØÓ. ùÒ ) ˜‡¯K: ùü«u {=˜«Ð? ù‡¯KØU˜V Ø, k, ÖöŒ wë• ©z[9] P156 ¥L9.1¤ (J. Œ±•Ñ ´: ÎÒu ,ØwêŠ •wÎÒ; Äu b½ tu K‡wêŠ, W u 0u öƒm: §QØ ÀêŠ, •Ø wêŠ(ꊕ + ^uû½•, Ø^Ù Š) . n! Fisher ˜†u ∗ ~5 •' A!Bü«–••{Û«•`, ÀJ15¬˜ Œ /, rz¬©¤/GŒ ˜ ü ¬, ‘Å/òÙ¥ ˜¬©‰A,,˜ ¬‰B. ¼ž ˆ ¬ 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 ŽÑ (A − B) = 314,y3‡u b H0 : A!B J˜ . (1.10) 7

8. e(1.10)¤á, z¬SA − B Š(=49, −67, · · · )ؘ , ¿šduA!B JØÓ, ´d uÙü ¬ O. ‘Åz (J, z˜ ¬kÓ ŒU©‰A½B. Ïd, X31˜¬, • ‘Åz (JØÓ, A − B Œ±´49, •Œ±´−49,‡w Ð @¬ ‰A„´B. ù ˜5, ù‡Á ÜŒU (A − B)Šk215 ‡: ±(49), ±(−67), ±(8), · · · , ±(60), ±(−48), ¢S Ñ (A − B) = 314´215 ¥ ˜‡. A!B Jk Œ Ož| (A − B)|A ŒŠ. é215 ‡ŒU(J¥ z˜‡ŽÑ (A − B),^xi Pƒ, i = 1, 2, · · · , 215 . ò§‚U짂 ý éŠlŒ ^Sü , Ø”P• x1 , x2 , · · · , x215 (1.11) =÷v |x1 | > |x2 | > · · · > |x215 | (1.11)¥ 215 ‡Š¥, 3H0 ¤ácJe, • ŒUu), =z‡Ñy VÇÑ´1/215 .u ¯K(1.10) Ľ•• {| (A − B)| > c} *ÿ | (A − B)| = 314, l u PŠ• m P (| (A − B)| > 314|H0 ) = 215 Ù¥m•üS(1.11)¥÷vxm = 314. äNOŽŒ•p314 < 0.0001 ÏdkndĽH0 . ˜†u ":´: 3äN¢–žOŽþŒ, ¦^å5Ø•B. y3k p„OŽÅ, |^OŽÅ5¢–•ØŽJ¯ . FishergCÚÙ§NõÆö, ÑïÄLù ¯K: n錞, ŒÄé ˜«Cq • { ¢–˜†u , ±ŒŒ{zOŽ? ïÄ(Jy² : 3阄 ^‡e, ù«{z •{ Ø=•3, …Ò´Ï~ tu œù´˜‡ék¿g (J. Ï•˜m©, tu ´Û•3 .¥ Ñ . ÏLù‡å»uy, =¦3••2• .e, •‡Á gêv Œ, tu E´·^ , ÏdŒ±`, ˜†u nØl˜‡ý¡\r tu / . §2 ü ¯K¥ šëêb u 3ü ' ¯K¥, ‘ÅØ ØÑl ©Ùž, ÒI‡JÑ•˜„ b , ¿¦^ƒA šëêu •{. ù•¡ nØÚ•{ õ, ŒÑé;€, ùp• éWilcoxon•Úu Ú˜†u Š˜{Ñ0 . ˜! Úó9½Â ·‚Äk5w˜wù˜u ¢S µ. ü u ¯K ˜„J{Xe: X1 , · · · , Xm ÚY1 , · · · , Yn ©O´läk©Ù•F1 ÚF2 ˜‘oN¥Ä {ü , …b½Ü X1 , · · · , Xm , Y1 , · · · , Yn NƒpÕá. ‡u e b H0 : F1 = F2 ←→ H1 : F1 = F2 . (2.1) 8

9.3ênÚOÆ¥, S.þ¡ù‡u ¯K•/ü ¯K0. ·‚5©O•Äe A«ž¹: 1. Šâ¯K ¢S µ, XJ·‚kndb½F1 ÚF2 •äkƒÓ• ©Ù, = b½ F1 ∼ N (a, σ 2 ), F2 ∼ N (b, σ 2 ) Ù¥a!bÚσ 2 ™•, −∞ < a, b < +∞, σ 2 > 0,ùžu ¯K=z• H0 : a = b ←→ H1 : a = b. (2.2) 3ù‡b½e, oN©ÙF1 ÚF2 ••6un‡™•ëêa!bÚσ 2 , u ¯K(2.1)8(•u ù ™•ëê´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@•éF1 ÚF2 ™•. 3ù ° 2 b½e, ·‚2ØU¦^Ï~ ü tu . ?nù‡¯K ˜«•{´/d’ â Å0(Smirnov)u , ùò3 Ù1Ê!¥?Ø. 3ù˜œ/e, oN©ÙF1 ÚF2 ØU^k•‡¢ëê •x, Ïd¡•šëêu ¯K. 3. y3·‚?ؘ«¥mœ¹. X ´˜« ¬3˜½) ó²e Ÿþ•I, Y´ T ¬3,˜) ó²e Ÿþ•I. knd@•, UC) ó²ØK• ¬Ÿþ•I V Ç©Ù, •U¦d©Ùu)˜ ²£. •Ò´`, e±F PX ©Ù, KY ©Ù•F (x − θ), ùpθ´˜‡™• ˜ëê. 3ù‡b½e, /X !Y Ó©Ù0 b ƒ /θ = 00, éá b •/θ = 00. Ïdu (2.1)8(•u H0 : θ = 0 ←→ H1 : θ = 0. (2.3) (2.3)´˜‡é-‡ b u ¯K. 3ù˜ .¥, ·‚b½F ™•, Ï ' .•2. , ù˜ .q'/d’ âÅu 0¥ .Ę , Ï•é ö ó, ü©ÙF1 ÚF2 Î Ã'X, 3dF1 ÚF2 ƒmkF2 (x) = F1 (x − θ). •,L¡þw(2.3)–˜‡ëêu ¯K: b ¥• 9θ, §´˜‡¢ëê. Ù¢Ø,, Ï•oN ©Ù†F ÚθÑk', F ©Ù™•, ÏdUšëêÚO¯K ½Â, (2.3)AÀ• šëêu ¯K. ˜„/, ü ¯K(2.1)„k˜ äk¢S µ ¥mœ¹. ~XF2 (x) = F1 (x/σ), dσ > 0•™• •Ýëê, ©ÙF •™•. u ¯K(2.1)3dœ¹e=z• H0∗ : σ = 1 ←→ H1∗ : σ = 1. (2.4) Wilcoxonü •Úu Ò´•Ä(2.3) b u ¯K. e¡Äk‰ÑWilcoxon ü •ÚÚOþ ½Â. ½Â6.3.1 X1 , · · · , Xm , Y1 , · · · , Yn ùn + m‡Šüü؃Ó, r§‚UŒ ü ,( J• Z1 < Z2 < · · · < ZN , N = m + n, (2.5) w,, z‡Yi 7•(2.5)¥ ,˜‡. eYi = ZRi ,KYi 3Ü X1 , · · · , Xm , Y1 , · · · , Yn ¥ ••Ri . Y1 , · · · , Yn •Ú• W = R1 + · · · + Rn , (2.6) §¡•Wilcoxonü •ÚÚOþ. ù´Wilcoxon31945c ˜‘óŠ¥Ú? . 9

10. ! Wilcoxonü •Úu — •{ Wilcoxonü •Úu Ò´•Ä(2.3) b u ¯K, = X1 , · · · , Xm i.i.d. ∼ F (x), Y1 , · · · , Yn i.i.d. ∼ F (x − θ),…Ü Õá. ‡u (2.3), = H0 : θ = 0 ←→ H1 : θ = 0. •ÚW d(2.6)‰Ñ. y3ù ín: z‡Ri ÑŒ 1, 2, · · · , N ƒ˜•Š. Y1 , · · · , Yn e b H0 ¤á, K Ü 5gÓ˜oN, z‡ÑØÓAÏ ˜, ج ½ Œ Š, W¤ ƒŠA8¥3²þên(N + 1)/2NC. e u : W ≤ d ½ W ≥ c Ľ H0 (2.7) XÛ(½cÚd ? §‚ (3 KþŒ±)û: H0 ¤áž, Ü ÕáÓ©Ù, ddŠâé ¡5 •Ä, ´•(R1 , · · · , Rn ) éÜ©Ù•   1  N (N −1)···(N −n+1)  r1 , · · · , rn ≤ N • P (R1 = r1 , · · · , Rn = rn ) = pØƒÓ g,ê,  Ù§.   0 ddØJ/ª/ ÑW ©Ù. l d α = P (W ≤ d ½ W ≥ c |H0 ) ½ÑcÚd. é m! n®›¤L. XJb u ´ü> , =‡u H0 : θ ≤ 0 ←→ H1 : θ > 0, (2.8) n duW = Ri ´Y1 , · · · , Yn 3Ü ¥ •Ú. eθ > 0KÏz‡Yi ©Ù†Xi + θ ©Ùƒ i=1 Ó, Yi †Xi ƒ' –•u •Œ Š. =Yi ŠŒuXi /Ŭ0•õ, u§ ŬK . ù ˜5R1 , · · · , Rn θ > 0ž–•u 8Ü{1, 2, · · · , N } ¥ Œ Š(d?N = m + n), Ó θ < 0, KR1 , · · · , Rn –•u 8Ü{1, 2, · · · , N }¥ Š. ÏW3θ > 0ž–•u Œ Š, 3θ < 0ž–•u Š. u ¯K(2.8) u •: W ≥ c ž, Ľ H0 . Ó u ¯K H0 : θ ≥ 0 ←→ H1 : θ < 0, (2.9) u •: W ≤ d ž, Ľ H0 . òþãnau ¤eL: L6.3.1 Wilcoxonü •Úu ( œ/) H0 H1 Ľ• θ=0 θ=0 W ≤d½W ≥c θ≤0 θ>0 W ≥c θ≥0 θ<0 W ≤d 10

11.XÛ(½ .ŠcÚd ? é m! n®²›¤L, „Ö"NL12. 'udL, Š±eü:` ²: (1) ©OP(X1 , · · · , Xm )Ú(Y1 , · · · , Yn )3Ü ¥ •Ú•W1 ÚW2 ,K (m + n)(m + n + 1) W1 + W2 = 1 + 2 + · · · + (m + n) = 2 ´˜‡~ê. Ïd, 3¦^Wilcoxon•Úu {, ' üoN©Ùž, ^W1 Š•u ÚOþ †^W2 Š•u ÚOþ´˜£¯. (2) 3NL12¥•‰Ñ •Úu .Š: P (W ≥ c) ≤ α¥c Š, éP (W ≤ d) ≤ α .ŠdXÛ|^dL¦Ñ? Œ±y² P (W ≤ α) = P (W ≥ n(m + n + 1) − d) =eP c = n(m + n + 1) − d, k鉽 α ! m! n ¦Ñ P (W ≥ c) ≤ α .Š c, , d dú ª d = n(m + n + 1) − c ŽÑ. ~6 ,« f3?1,«ó²?nƒcÚ?nƒ , ˆ‘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 ¹•Ç´Äeü? (α = 0.05) ) X ÚY ©OL«?nc! f ¹•Ç, §‚ ©Ù¼ê©O•F (x)ÚF (x−θ). Ku ¯K H0 : θ ≥ 0 ←→ H1 : θ < 0. =ò/?n f¹•Çvkeü0Š• b . dL6.3.1Œ•: W ≤ džÄ½ 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| * ŠUl Œü¤˜ ¤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 f¹•Ç(Y ) * Š •. Y * Š •Ú• W = 1 + 2 + 4 + 5 + 7 = 19 òنĽ•¥ .Š' 19 ≤ d (d = 19),ÏdĽH0 ,=@•?n f¹•Çeü . ˜„z 11

12. b : ‰½ü|Õá ‘Å X1 , X2 , · · · , Xn ÚY1 , Y2 , · · · , Ym , ‡¦ 1. – •^SºÝ 2. a, Cþ´ëY. 3. F (x)ÚG(x)©OL« X ÚY ©Ù¼ê Kéb (A) H0 : F (x) = G(x) é¤kx ⇐⇒ H1 : F (x) = 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« ”X –•u'Y ” Ú”X –•u'Y Œ”. u´PW L «X 3Ü ¥ •Ú, Ku {K©O• (A) éV>u H1 : F (x) = 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 —Œ •{ c¡?Ø m!n žWilcoxonü •Úu •{. m!n Œžu ÚOþW ©Ù OŽéE,, 鉽 αvky¤ LŒ± Ľ• .Š. Ïd •‚4•½n. •{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?W d(2.6)ª‰Ñ, N = m + n.P W − E(W ) W − n(N + 1)/2 W∗ = = D(W ) mn(N + 1)/12 Œ±y²3 b H0 : θ = 0¤áƒe, m, n → ∞žk W − n(N + 1)/2 L W∗ = −→ N (0, 1) mn(N + 1)/12 ÏdŒ u ¯K(2.3) Y²Cq•α Ľ•: D = {(X, Y) : |W ∗ | ≥ uα/2 } aqŒ u ¯K(2.8)Ú(2.9) Y²Cq•α u Ľ•, •„eL: 12

13. L6.3.2 Wilcoxonü •Úu (Œ œ/) H0 H1 Ľ• θ=0 θ=0 |W ∗ | > u α2 θ≤0 θ>0 W ∗ > uα θ≥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 °ÝƒÓ, Á¯`!¯ü ÅK\ó ¬ †»kÃwÍ É? (α = 0.10) ) X ÚY ©OL«`¯ü •K\ó ¬ †», §‚ ©Ù¼ê©O•F (x) ÚF (x− θ).Ku ¯K•: H0 : θ = 0 ←→ H1 : θ = 0 d~¥m = 8, n = 9.òü|êâYUl Œü¤˜ ,Ž 1 |êâ ••g•: 1, 4, 6, 9, 11, 12, 13, 15, 16, Ù•Ú´ W = 1 + 4 + 6 + 9 + 11 + 12 + 13 + 15 + 16 = 87. éu Y²α = 0.10,dL6.3.2Œ•Œ Ľ•• D = {(X, Y) : |W ∗ | ≥ u0.05 = 1.96} Ù¥ W − n(n + m + 1)/2 87 − 81 |W ∗ | = = √ = 0.58 < 1.96 mn(m + n + 1)/12 108 ÉH0 ,=Šâ®kêâØv±Ä½ùý ÅK\ó †»ÃwÍ É b½. k<ŒU¬ú , ù«•Úu Çجp, Ï•§•|^ ¥ Œ 'X Ñ ÙäNêŠ, Ù¢Ø,. C“'u•u Œ nØy² , ˜„•u – 3 Œ Œž, †DÚ ëêu ƒ'¿ØÖÚ. <Wilcoxonü •Úu †ü tu ƒ' , =¦3‘ÅØ ©Ù• œ/, Wilcoxonu ǃéutu •ˆ 3/π ≈ 0.95 ( Nþ Œž) . éØ ©Ùš ž, ù‡ƒé ÇŒ?¿ C1, …oج$u0.864. ±þ·‚b½ Ü X1 , · · · , Xm , Y1 , · · · , Yn *dØÓ, Ï Yi •Ri Œ±•˜(½. eb½F ??ëY, ù˜¯¢ò±VÇ•1¤á, ùžØ•3¯K. F ØëYž, Ü ¥ ŒUÑyƒÓ , =¤¢/(0 ¯K. ~X` x2 < x1 = x4 < x5 < x3 , S.þrƒÓ n‡Cþ¡•˜‡/(0. ( xi ••˜(½. Xd?x2 , x5 Úx3 •© O•1, 4 Ú5. (S xi •Òؘ٠. dž ù ƒU•ê ²þŠŠ•(Sˆ‡xi •. Xd?x1 Úx4 Ók•2Ú3, 2.5©OŠ•x1 Úx4 •. é¤k /(0Š ?nƒ , U c¡¤ã •{?Øü •Úu ¯K. 13

14. o! ü ˜†u {∗ 3ênÚOÆ¥, /?n0˜c ¹¿42. §Œ±L«˜«ó²6§, ˜‡«f¬ «, ˜«£ •{ . ü ˜†u gŽ†~6.2.6 ƒq. ü?n ÜÁ (J •X1 , · · · , Xm , Y1 , · · · , Yn . òÙü 3˜å, U•Z1 , · · · , Zn+m . XJ`!¯ü?nà O, KZ1 , · · · , Zn+m ƒm OØ´du?nØÓ 5, ´duùn + m‡Á ü © • n+m { 5. 3n + m ‡ü ¥© m‡‰?n`, ØÓ •{k m «. e3z«© •{ƒ eÑOŽ±e þ: 1 1 g = (`?nÁ ŠƒÚ) − (¯?nÁ ŠƒÚ). m n § uZ1 , · · · , Zn+m ¥ m‡²þê~ •e n‡ ²þê. @m‡K‡wÁ ü ´X Û© . ù ,˜ U N = n+m m ‡Š: g1 , g2 , · · · , gN ,òÙýéŠUŒ ü , Ô- Ù•: |g1 | ≤ |g2 | ≤ · · · ≤ |gN |, (2.10) u ¯K•: H0 : `¯ü?n J˜ (2.11) H0 ¤áž(2.10)¥@N ˆŠ¥ z˜‡, kÓ Ñy Ŭ1/N, Ò¢S ŽÑg ƒŠ, P •g = X − Y . XJH0 ؤá, |g |–•u ∗ ŒƒŠ. Ïd, ‰½u Y²α , éNα , ¦ ∗ α = P (|gi | > |g[Nα ] |, i ∈ {1, · · · , N } |H0 ) (2.12) dug1 , g2 , · · · , gN ¥ z˜‡u)´ ŒU1/N, Ïd(2.12)ª duéNα ,¦ N − [Nα ] =α N ÏdĽ•• D = { |g[Nα ]+1 |, |g[Nα ]+2 |, · · · , |gN | }, Ïd‰½ X = (X1 , · · · , Xm )ÚY = (Y1 , · · · , Yn ),ŽÑg ∗ = X − Y, e|g ∗ | = |X − Y | > |g[Nα ] |, K|g ∗ |73Ľ•D¥, D•ŒL• D = { (X, Y) : |g ∗ | = |X − Y | > |g[Nα ] | }. d?[Nα ]L«Nα êÜ©. eU b • H0 : ?n`Ø`u¯, Kòg1 , g2 , · · · , gN UŒ ü • h1 ≤ h2 ≤ · · · ≤ hN 鉽 α,éNα ,¦ α = P (hi > h[Nα ] , i ∈ {1, 2, · · · ., N } |H0 ). Ïd, ˜ k X = (X1 , · · · , Xm )ÚY = (Y1 , · · · , Yn ) ŽÑg ∗ = X − Y,Ľ•ŒL• D = {(X, Y) : g ∗ = X − Y > h[Nα ] }. m! nÑ錞, þ㘆u Cuü tu . XÓ3˜ œ/, ù‡5Ÿl˜‡ ý¡\r ü tu / . 14