tatstcs & Data Aalyss 6 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 of all speculato. (Fracs Baco (5666)
αposso - 3 4 5 6 7 8 9 3 4 5 6 7+ Observed 8 8 56 5 6 46 64 6 3 74 53 3 5 9 5 Expected. 7. 56.5 94.9 3.759.66.955.63.6 99.7 69.7 45. 7. 5. 7.9 7. χ.76.4..7.34.8.5.9.44..7.4.59..57.57 Possoλ.. 3. 4. χ (4).83 8.99 hypothess test X X, X,, X ull hypothessx alteratve hypothessx yp
6. 6.. Example 6. 5mg.5mg5 9 49.5, 5.6, 5.8, 5., 49.3, 5., 5., 5.5, 5. 5.9mg 5.9 5 N(5,.5 ) Example 6..mlcm 4 3 4 5 6 7 8 9 9 56 4 8 6 4 7 9 9 5 3 Posso X Posso Blss C. ad Fsher R. A., (953). Fttg the egatve bomal dstrbuto to bologcal data. Bometrc, 9: 74-
6. 6. 6.. N-P Example 6. 5mg.5mg 9 49.5, 5.6, 5.8, 5., 49.3, 5., 5., 5.5, 5. 5.9mg
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
Example 6. 5mg.5mg 9 49.5, 5.6, 5.8, 5., 49.3, 5., 5., 5.5, 5. 5.9mg 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
.5 X 5.9, 5. Z.96 X 5.9 5..8 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
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
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 )
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
Neyma-Pearso α ββ Jerzy Neyma 894-98 Ego Pearso 895-98
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
6..3 63 H H H H 3α αc C 4 H
6..4 64 (α)%( α Example 6. 5mg.5mg 9 49.5, 5.6, 5.8, 5., 49.3, 5., 5., 5.5, 5. 5.9mg σ.5h : μμ =5.H : μ μ X / Z Z C C C
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
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
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 ( )
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
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 ( ) /
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
α X X t ( ) t ( ) / / t(5) t ( ) acceptace rego rejecto rego α 3
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
. 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 ( )
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 ( ) ~ ( )
χ : ~ ( ) α ( ) ( ) ( ) ( ) ~ ( ) ( ) ( ) rejecto rego acceptace rego rejecto rego
. μh : σ σ H : σ σ χ X ( ) α X X ( ) ( ) X ( ) ( ) X ( ) ( ) ( ) rejecto rego acceptace rego rejecto rego
σ X~N(μ, σ )X,X,,..., X X. μh : σ σ H : σ > σ χ ~ ( ) α ( ) ( ) ~ ( ) ( ) α acceptace rego rejecto rego
. μh : σ σ H : σ < σ χ ~ ( ) α ( ) ( ) ~ ( ) ( ) acceptace rego
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 (, )
~ (, ) 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, ) (
~ (, ) F F F(5, 5) F (, ) acceptace rego rejecto rego 63 6.3 Neyma-Pearso98 Lkelhood rato Jerzy Neyma 894-98 Ego Pearso 895-98
6.3. 63 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
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
6.3. 63 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 ( )]
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 ()
GLR = (logglr) χ (k)k k dm dm 6.3.3 633 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
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
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 64 6.4 6. 6.
6.4. 64 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
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
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
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 (887-974) 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.95.99 ( ).63 D
K- 3 6.4. 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 }
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
Example 6..mlcm 4 3 4 5 6 7 8 9 9 56 4 8 6 4 7 9 9 5 3 Posso X Posso Blss C. ad Fsher R. A., (953). Fttg the egatve bomal dstrbuto to bologcal data. Bometrc, 9: 74- Example 6..mlcm 4 3 4 5 6 7 8 9 9 56 4 8 6 4 7 9 9 5 3 Posso 56 4... 9 X.44 4 3 4 5 6 7 56 4 8 6 4 7 9 34.9 85. 3.8 84.4 5.5 5.. 5. O E E.8 4. 5.5 6..8.4.4 45.
Example 6.6Medel AABB aabb AaBb Example 6.73395 H α. 43 6 5.% 3 34 9.7% 56 83 339 6.5%
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
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
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 6.73395 H α. 43 6 5.% 3 34 9.7% 56 83 339 6.5%