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1 tatstcs & Data Aalyss 6 Zhu Uversty The strogest argumets prove othg so log as the coclusos are ot verfed by experece. Expermetal scece s the quee of sceces ad the goal of all speculato. (Fracs Baco (5666)

2 αposso Observed Expected χ Possoλ χ (4) hypothess test X X, X,, X ull hypothessx alteratve hypothessx yp

3 Example 6. 5mg.5mg , 5.6, 5.8, 5., 49.3, 5., 5., 5.5, mg N(5,.5 ) Example 6..mlcm Posso X Posso Blss C. ad Fsher R. A., (953). Fttg the egatve bomal dstrbuto to bologcal data. Bometrc, 9: 74-

4 N-P Example 6. 5mg.5mg , 5.6, 5.8, 5., 49.3, 5., 5., 5.5, mg

5 N(μ, σ )μ μ =5. N(μ, σ ): Gve σ μμ =5. No Yes (ull hypothess) H : μμ (alteratve hypothess) H :μ μ H ad H : whch ca we accept based o the samplg evdece? Jury system

6 Example 6. 5mg.5mg , 5.6, 5.8, 5., 49.3, 5., 5., 5.5, mg σ.5h : μμ =5.H : μ μ H E ( X ) X μ H X / ~ N (, ) α α =.,.,.5, 5. X P Z / X / Z Z

7 .5 X 5.9, 5. Z.96 X Z.96 /.5 / 9 X / Z Z H Uor X Z ~ N (, ) / test tstatstc t t α=.,.,.5, 5. P X Z / sgfcace level test statstc sgfcace level

8 X / Z Z rejecto rego acceptace rego rejecto rego C C C Type I error H H P{H H }α Type II error H H P{H H }β P{H H } β

9 α (β)power of test (α)cofdece Level α β Example 6.3X~N(μ, σ )σ μμ μ μ <μ XX, X,, X α H : μ=μ H : μμ (>μ )

10 Accept H Reject H H s true Correct Decso α: Cofdece Level Type I Error α : gfcace level Type II Error Correct Decso H s false (H s true) β β : Power of a test (-β) U U -ββ Power curve μ μ

11 Neyma-Pearso α ββ Jerzy Neyma Ego Pearso

12 mple hypothess X X~N(μ, σ )σ H : μμ Composte hypothess X X~N(μ, σ )μ σ XPossoλ 3X~B(, p) p>.5 oe-sded alteratve hypothess two-sded alteratve hypothess

13 H H H H 3α αc C 4 H

14 (α)%( α Example 6. 5mg.5mg , 5.6, 5.8, 5., 49.3, 5., 5., 5.5, mg σ.5h : μμ =5.H : μ μ X / Z Z C C C

15 ΘXf(X θ)θ θ Θ αh : θθ A(θ )C(X) = {XA(θ)}θ(α)% ΘXf(X θ)c(x) θ(α)% P C ( X ) αh : θθ A( ) X C( X ) 6. ample: X, X,, X Radom samplg Populato dstrbuto

16 6.. μ μ. σ HH : μμ HH : μ μ X~N(μ, σ )σ X, X,..., X X U U X ~ N (,) / X / Z Z rejecto rego acceptace rego rejecto rego

17 gfcace level. gfcace level.5. σ H : μμ H : μ μ X~N(μ, σ )σ XX, X,..., X X T T X / ~ t( ) α X t ( ) X t ( )

18 t(5) X T ~ t( ) / t ( ) t ( ) rejecto rego acceptace rego rejecto rego μ. σ H : μμ H : μ < μ X~N(μ, σ )σ XX, X,..., X X U X U ~ N (,) / H : μμ H : μ < μ μ

19 α X X Z / / X / Z Z rejecto rego acceptace rego. σ H : μμ H : μ < μ X~N(μ, σ )σ X, X,..., X X Tαα X / t( ) X t ( ) /

20 t(5) ( ) T X / ~ t ( ) t ( ) rejecto rego acceptace rego 3. σ H : μ μ H : μ > μ X~N(μ, σ )σ X, X,..., X X T μμ μ μ μ < μ

21 α X X t ( ) t ( ) / / t(5) t ( ) acceptace rego rejecto rego α 3

22 6.. X~N(μ, σ )Y~N(μ, σ )X, X,..., X XY, Y,..., Y Y. σ σ H :μ μμ H :μ μ E X Y X Y N, X Y U N, UU U X Y N, α X Y X Y Z Z

23 . σ σ σ σ H : μ μ H : μ μ E X Y ( X Y ) ( ) ~ ( ) T t w ( ) ( ) w ( X X ), Y Y ( ) TT ( X Y ) T ~ t( ) w α w X Y t ( ) w X Y t ( )

24 3. σ σ H : μ μ H : μ > μ UUU U X Y N, α X Y Z X Y Z 6..3 σ σ X~N(μ, σ )X, X,..., X X. μhh :σ H :σ σ H σ X X ( ) ~ ( )

25 χ : ~ ( ) α ( ) ( ) ( ) ( ) ~ ( ) ( ) ( ) rejecto rego acceptace rego rejecto rego

26 . μh : σ σ H : σ σ χ X ( ) α X X ( ) ( ) X ( ) ( ) X ( ) ( ) ( ) rejecto rego acceptace rego rejecto rego

27 σ X~N(μ, σ )X,X,,..., X X. μh : σ σ H : σ > σ χ ~ ( ) α ( ) ( ) ~ ( ) ( ) α acceptace rego rejecto rego

28 . μh : σ σ H : σ < σ χ ~ ( ) α ( ) ( ) ~ ( ) ( ) acceptace rego

29 6..4 X~N(μ, σ )Y~N(μ, σ )X, X,..., X XY, Y,..., Y Y. μ μ H : σ σ H : σ σ H : σ σ H : σ σ / F ~ F (, ) / X X Y Y ( ), ( ) F F ~ F(, ) α F (, ) F (, ) F (, ) F (, )

30 ~ (, ) F F F(5, 5) F (, ) F (, ) rejecto rego acceptace rego rejecto rego μ μ H : σ σ H : σ > σ F ~ F (, ) F F α F (, ) F, ) (

31 ~ (, ) F F F(5, 5) F (, ) acceptace rego rejecto rego Neyma-Pearso98 Lkelhood rato Jerzy Neyma Ego Pearso

32 N-P NP N-PNeyma-Pearso Lemma H H Xf (x θ )f (x θ ) X,X,..., X X LR, Lkelhood Rato LR lk ( f ) lk ( f ) f X f X LR testα LR < cch α* α (β*) (β) β*β

33 N-P α β 3 Example 6.4X~N(μ, σ )σ μ μ μ μ <μ X, X,, X X α H :μ=μh :μμ LR LR

34 GLR test, Geeralzed Lkelhood Rato test X, X,..., X X Θ f X, H θω ΘHH θω Θω ω =Θ =ω ω GLR, Geeralzed Lkelhood Rato GLR max[ lk ( )], max[ lk ( )] lk( ) f X GLR*GLR* max[ lk( )] GLR max[ lk ( )]

35 lk ( ) H GLR max[ lk( )] GLR m(glr*, ) Example 6.5X~N(μ, σ )σ X, X,, X X αh :μ=μh :μ μ GLR e X X X X GLR log GLR ()

36 GLR = (logglr) χ (k)k k dm dm XX, X,, X m m X X... Xm p,p,,p m pp p p p... p m H p=p(θ)ω θ p p... p m H pθ

37 GLR max[ lk ( )] max[ lk ( )] m! m!! X! MLE max p ( ) ( ) p m m p X! X! X MLE m m! X X p, X p! GLR max[ lk ( )] max[ lk ( )] χ m m p ( ) O log GLR p log O log p E O p, E p ( ) dm dm ( m ) k

38 H H GLR m m p O log GLR p log O log p ( ) E Pearso χ X m X p ( ) m O E p ( ) E χ dm dm

39 K- K Kolmogorov mrov test X, X,, X Xx, x,, x x () x () x (), x x () k F( x), x( k) x x( k),( k,,..., ), x x( ) F ()X (x)xemprcal ldtbt dstrbuto fuctoxf(x) theoretcal dstrbuto fucto Theoretcal dstrbuto fucto

40 x, x,, x F (x) F (x) F (x)b(, () ( F(x)) XF(x)F (x) Plm max F ( ) ( ) x F x x X X

41 K- K XF(x)F F (x) H : F(x) = F (x) F (x)xf(x)h () () F (x)f (x) Kolmogorov ( ) D F x F x x max ( ) ( ) ( ) D F ( x ) F ( x ) XF(x)X, X,, X X H : F(x) = F (x) ( ) lm P D K ( ) K ( ) k, k k ( ) e, K Kolmogorov

42 K-mrov939 XF(x)HH :F(x)=F(x) F (x) α P D D D ( ) ( ) ( ).9.3 (.36 D ).95 ( ).63 D.99 Vladmr mrov ( ) Russa mathematca K- x, x,, x x () x (), x (), x x() k F( x), x( k ) x x( k ), ( k,,..., ), x x( ) KolmogorovD () ( ) D max F( x) F( x) max F ( x( ) ), F ( x( ) ) x ( ) ( ) 3 D D H ( ).3 ( ).36 D.9 D ( ).63 D

43 K χ K. Pearso9 Ch-square test XF(x)X, X,, X X H : F(x) = F (x)h : F(x) F (x) XΩmA, A,, A m m A A Aj, j H p P X A,,,..., m /{XA }

44 Pearso χ X m m p O E p Pearso χ χ (mr)r E χ Ω 5 3p >5A 4F (x)k-pearso χ

45 Example 6..mlcm Posso X Posso Blss C. ad Fsher R. A., (953). Fttg the egatve bomal dstrbuto to bologcal data. Bometrc, 9: 74- Example 6..mlcm Posso X O E E

46 Example 6.6Medel AABB aabb AaBb Example H α % % %

47 6.4.3 (X, Y)x, xymk A, A,, A m B, B,, B k (x,y ), (x,y ),, (x, y ) j xa yb j =,,, m; j=,,.., k k m, j j j j H : XY

48 p P X A, Y B j j j p P X A p P Y B j k p p, p p, j j j j m m k m k p p p j j j j H p p p,,,..., m; j,,..., k j j p j m k m k j pj Oj Ej p j j j E j ( mk ) α m k j pj mk j pj

49 p j p j, j p p j j j p j pp j m k m k p O E j j j j p j E j j j χ dm dm ( m)( k) α m k j pj ( m )( k ) p j j Example H α % % %

Microsoft PowerPoint - Lect06 HypothesesTest

Microsoft PowerPoint - Lect06 HypothesesTest 统计与数据分析 tatstcs & Data Aalyss 6 推断统计学的假设检验 Zhu Huaqu @Pekg Uversty The strogest argumets prove othg so log as the coclusos are ot verfed by experece. Expermetal scece s the quee of sceces ad the goal