第４回 確率・統計の基礎勉強会

研究室のM1向け勉強会のスライド

Transcript

1. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ ాத ޺ণ (@setten QB) ಸྑઌ୺Պֶٕज़େֶӃେֶ ৘ใՊֶݚڀՊɹɹ஌ೳίϛϡχέʔγϣϯݚڀࣨ Ϗοάσʔλάϧʔϓ March 25, 2017 ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 1 / 33
2. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ 1 Introduction 2 Ծઆݕఆ 3 2 छྨͷաޡͱݕग़ྗؔ਺ 4 ࠷ڧྗݕఆͷߏ੒ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 2 / 33
3. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ ౷ܭతݕఆ • ౷ܭతਪఆཧ࿦ͱڞʹ౷ܭతਪଌཧ࿦ͷ 2 େபΛ੒͢ • “ͳΜͪΌͬͯ౷ܭֶ”ͷߨٛͰ͸ݕఆͷ΍Γํ͚ͩڭ͑Δ৔ॴ΋͋Δ Β͍͠ • ඪຊ͕ै͏֬཰෼෍ʹؔ͢Δ໋୊ (͜ΕΛԾઆͱݺͿ) ͕ਖ਼͍͠ͷ͔Λ σʔλʹج͍ͮͯ൑அ͢Δํ๏ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 3 / 33
4. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ 1 Introduction 2 Ծઆݕఆ 3 2 छྨͷաޡͱݕग़ྗؔ਺ 4 ࠷ڧྗݕఆͷߏ੒ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 4 / 33
5. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ Ծઆݕఆ • લճͷ౷ܭతਪఆཧ࿦ͱಉ༷ͷԾఆΛ͓͘ ౷ܭతਪଌͷجຊతઃఆ ඪຊ X1, . . . , Xn iid ∼ f(x; θ) ʹج͖ͮɼਪఆྔ θ = θ(X1, . . . , Xn) Ͱ θ Λਪఆ͢Δɽ • f(x, θ) ͷະ஌ύϥϝʔλ θ ʹؔ͢ΔԾઆΛɼσʔλʹج͍ͮͯغ٫͢ Δ͔൱͔Λܾఆ͢Δ͜ͱΛ౷ܭతݕఆͱ͍͏ • ύϥϝʔλ θ ؚ͕·ΕΔۭؒΛ Θ Ͱද͢ • Θ0, Θ1 ⊂ Θ ͕ Θ0, Θ1 ̸= ∅, Θ1 ∩ Θ1 = ∅, Θ0 ∪ Θ1 = Θ ͱ͢Δ • ҰൠతʹԾઆ͸ • ؼແԾઆ (null hypothesis) H0 : θ ∈ Θ0 • ରཱԾઆ (alternative hypothesis) H1 : θ1 ∈ Θ1 Ͱߏ੒͞ΕΔ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 5 / 33
6. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ Ծઆݕఆ • null hypothesis ʹ͸غ٫͍ͨ͠ԾઆΛ͓͘ͷ͕ී௨ • null hypothesis ͷඪຊͷΑ͏ͳσʔλ͕ಘΒΕΔ֬཰Λߟ͑ɼͦͷΑ ͏ͳ͜ͱ͕ “ى͜Γʹ͍͘”ͱߟ͑ͨ࣌ɼnull hypothesis Λ reject ͢Δ • null hypothesis Λ reject ग़དྷͨͱ͖ɼͦΕͱରཱ͢Δ alternative hypothesis Λ accept ͢Δ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 6 / 33
7. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ ౷ܭతԾઆݕఆͷྫ ⃝ ൪͘͡ ςϥγϚ܅͸ɼ๭ίϯϏχͰߦΘΕ͍ͯΔ͘͡Ҿ͖ʹߦͬͨɽ൴ͷૂ ͍͸ A ৆ͷ๭σϏϧʔΫ੕ͷԦঁͷϑΟΪϡΞͰɼͦΕҎ֎ʹڵຯ ͸ͳ͍ɽެࣜʹΑΔͱ 60 ຕͷΫδͷதʹ 2 ຕͷׂ߹Ͱ A ৆͕ೖͬ ͍ͯΔͱ͍͏ɽͨͩ͠ɼ͜ͷίϯϏχ͸ಛघͰɼ౰ͨΓͱϋζϨͷׂ ߹͕ৗʹҰఆʹͳΔΑ͏ʹ٬͕͘͡ΛҾͨ͘ͼʹ৽͍͠ΫδΛ௥Ճ ͢ΔɽςϥγϚ͘Μ͸ 1 ϲ݄෼ͷ঑ֶۚΛͭ͗ࠐΜͰ 200 ຕͷΫδ ΛҾ͍ͨɽ͔͠͠ɼA ৆͸ 1 ຕ͔͠Ҿ͚ͳ͔ͬͨɽ͜ͷίϯϏχ͸ ΠϯνΩΛ͍ͯ͠Δͱݴ͑Δ͔ʁ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 7 / 33
8. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ ౷ܭతԾઆݕఆͷྫ • A ৆ΛҾ֬͘཰Λ p ͱ͢Δ • ΠϯνΩΛ͍ͯ͠Δͱओு͍ͨ͠ςϥγϚ܅ཱ͕ͯΔԾઆ͸ H0 : p = 1/30 vs H1 : p ̸= 1/30 • θ = p, Θ = (0, 1), Θ0 = { 1/30 } , Θ1 = (0, 1) − { 1/30 } • ςϥγϚ܅͕Ҿ͍ͨ͘͡͸ X1, . . . , X200 iid ∼ B(1, p) ͷ࣮ݱ஋ͱߟ͑Β ΕΔ • ͨͩ͠ɼA ৆Λ 1ɼͦΕҎ֎Λ 0 ͱ͢Δ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 8 / 33
9. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ ౷ܭతԾઆݕఆͷྫ • A ৆ΛҾ͘ຕ਺Λ Y ͱ͢Δͱɼ Y = 200 ∑ i=1 ∼ B(200, p) Ͱ͋Δ • Αͬͯɼ1 ճ͔͠ A ৆ΛҾ͔ͳ͍֬཰͸ P(Y = 1) = ( 200 1 ) p1(1 − p)199 ͱॻ͚Δ • H0 ͕ਖ਼͍͠ͱ͍͏Ծఆͷ΋ͱͰ͸ɼP(Y = 1) = 0.00783 ͱͳΔ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 9 / 33
10. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ ౷ܭతԾઆݕఆͷྫ • ͜͜·Ͱͷٞ࿦ͰɼH0 ͕ਅͰ͋ΔͱԾఆ͢Δͱʮ200 ຕҾ͍ͯ A ৆͕ 1 ճ͔͠౰ͨΒͳ͍ʯͱ͍͏ঢ়گ͕ى͜Δ֬཰͸ 1% ʹ΋ຬͨͳ͍͜ͱ ʹͳΔ • ͜ͷΑ͏ͳح੻͕ى͜ΔΑΓ΋ɼH0 ͕͓͔͍͠ͱ͢Δ΄͏͕ࣗવͰ ͋Δ • ͕ͨͬͯ͠ɼH0 Λ reject ͠ H1 Λ accept ͢Δ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 10 / 33
11. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ ౷ܭతԾઆݕఆͷྫ • ͜͜·Ͱͷٞ࿦ͰɼH0 ͕ਅͰ͋ΔͱԾఆ͢Δͱʮ200 ຕҾ͍ͯ A ৆͕ 1 ճ͔͠౰ͨΒͳ͍ʯͱ͍͏ঢ়گ͕ى͜Δ֬཰͸ 1% ʹ΋ຬͨͳ͍͜ͱ ʹͳΔ • ͜ͷΑ͏ͳح੻͕ى͜ΔΑΓ΋ɼH0 ͕͓͔͍͠ͱ͢Δ΄͏͕ࣗવͰ ͋Δ • ͕ͨͬͯ͠ɼH0 Λ reject ͠ H1 Λ accept ͢Δ ͭ·Γ౷ܭֶతʹߟ͑ͯɼ͜ͷίϯϏχ͸ΠϯνΩΛ͍ͯ͠Δ! ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 10 / 33
12. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ غ٫Ҭ • Ծઆݕఆ͸ · · · • ฼ूஂ෼෍͔Βͷແ࡞ҝඪຊ X = [X1 , . . . , Xn ] ͷऔΓ͏Δ஋ͷू߹ͷ ෦෼ू߹ C ΛఆΊΔ • X ͷ࣮ݱ஋͕ C ʹଐ͢Δ࣌ʹ H0 Λغ٫͢Δ ͱ͍͏ϧʔϧͰߏ੒͞Ε͍ͯΔ • ʮX ͕ͲͷΑ͏ͳ஋ΛऔΕ͹ H0 Λغ٫͢Δ͔ʁʯͱ͍͏ϧʔϧΛܾ ΊΔ͜ͱ͕ݕఆΛߏ੒͢Δͱ͍͏͜ͱ • ্هͷ C Λݕఆͷغ٫ҬͱݺͿ • ʮ“ྑ͍” غ٫ҬΛͲͷΑ͏ʹܾΊΔͷ͔ʁ͕໰୊ʯͱͳΔ • ͳ͓ɼH0 Λغ٫ग़དྷͳ͔͔ͬͨΒͱݴͬͯɼH0 Λੵۃతʹࢧ࣋͢Δ Θ͚Ͱ͸ͳ͍͜ͱʹ஫ҙ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 11 / 33
13. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ 1 Introduction 2 Ծઆݕఆ 3 2 छྨͷաޡͱݕग़ྗؔ਺ 4 ࠷ڧྗݕఆͷߏ੒ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 12 / 33
14. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ 2 छྨͷաޡ • ʮओு A ͕ਖ਼͍͠ͷ͔ʁʯͱ͍͏ࣄΛ൑அ͢Δঢ়گΛߟ͑Δ • ͜ͷΑ͏ͳঢ়گʹ͓͍ͯɼඞͣද 1 ʹࣔ͢Α͏ͳޡΓ͕ى͜ΓಘΔ Table: 2 छྨͷաޡ ਅ࣮ H0 H1 ൑ H0 ⃝ Type II Error ஈ H1 Type I Error ⃝ • θ ∈ Θ0 ͭ·Γ H0 ͕੒ཱ͍ͯ͠ΔԼͰͷࣄ৅ A ͷ֬཰Λ Pθ(A), Pθ(A|θ ∈ Θ0), Pθ(A|H0) ͳͲͰද͢ 1 • Type I Error, Type II Error ͷ֬཰͸ͦΕͧΕ Pθ(X / ∈ C|H0), Pθ(X ∈ C|H1) ͱॻ͚Δ 1৚݅෇͖֬཰Ͱ͸ͳ͍͜ͱʹ஫ҙ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 13 / 33
15. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ ݕग़ྗ ݕग़ྗؔ਺ ؔ਺ β : θ → [0, 1] Λ β(θ) = Pθ (X ∈ C) (1) ͰఆΊɼݕఆͷݕग़ྗؔ਺ͱ͍͏ɽಛʹɼθ1 ∈ Θ1 ͷͱ͖ β(θ1) Λ θ1 ʹର͢Δݕఆͷݕग़ྗ (Power) ͱ͍͏ɽ • ݕग़ྗ͸ H1 ͕ਅͷ࣌ʹɼH0 Λਖ਼͘͠غ٫Ͱ͖Δ֬཰Λද͍ͯ͠Δ • Type I Error ͷ֬཰͸ β(θ), θ ∈ Θ0 • Type II Error ͷ֬཰͸ 1 − β(θ), θ ∈ Θ1 ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 14 / 33
16. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ ݕఆؔ਺ ݕఆؔ਺ X = [X1, . . . , Xn] ͷ࣮ݱ஋ x = [x1, . . . , xn] ͷؔ਺ φ (x) = 1{ X∈C } = { 1 (x ∈ C) 0 (x / ∈ C) Λݕఆؔ਺ͱ͍͏ɽ • ݕఆؔ਺ͷ͜ͱΛݕఆͱ͍͏͜ͱ΋͋Δ • ݕఆؔ਺Λ໌ࣔ͢Δ͜ͱΛݕఆΛߏ੒͢Δͱ͍͏ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 15 / 33
17. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ ݕఆͷߏ੒ྫ ݕఆͷߏ੒ͱաޡͷ֬཰ ֬཰ม਺ X ͷ෼෍ؔ਺͕ Fθ(x) = 1 − exp(−θx) Ͱ͋Δͱ͖ɼ H0 : θ = 1 vs H1 : θ > 1 ͷݕఆΛߟ͑Δɽݕఆؔ਺Λ φ(x) = { 1 (x > 2) 0 (x ≤ 2) ͱ͢ΔɽҎԼͷ໰ʹ౴͑Αɽ (1) Tyepe II Error ͷ֬཰ΛٻΊɼθ ʹ͍ͭͯͷάϥϑΛඳ͚ɽ (2) Type I Error ͷ֬཰ΛٻΊΑɽ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 16 / 33
18. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ ݕఆͷߏ੒ྫ (1). ·ͣɼغ٫Ҭ͸ C = { x | x > 2 }ɽΑͬͯɼType II Error ͷ֬཰͸ P(X / ∈ C|H1) =P(X ≤ 2|θ > 1) =Fθ(2) =1 − exp(−2θ) (θ > 1). (2). Type I Error ͷ֬཰͸ P(X ∈ C|H0) =P(X > 2|θ = 1) =1 − F1(2) = exp(2). ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 17 / 33
19. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ 1 Introduction 2 Ծઆݕఆ 3 2 छྨͷաޡͱݕग़ྗؔ਺ 4 ࠷ڧྗݕఆͷߏ੒ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 18 / 33
20. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ 2 छྨͷ Error ֬཰ • Type I Error ͷ֬཰ͱ Type II Error ͷ֬཰Λখ͍ͨ͘͞͠ • ͔͠͠ɼ2 ͭͷ֬཰Λಉ࣌ʹ࠷খԽ͸ग़དྷͳ͍ • Ͱ͸Ͳ͏͢Δͷ͔ʁ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 19 / 33
21. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ 2 छྨͷ Error ֬཰ • Type I Error ͷ֬཰ͱ Type II Error ͷ֬཰Λখ͍ͨ͘͞͠ • ͔͠͠ɼ2 ͭͷ֬཰Λಉ࣌ʹ࠷খԽ͸ग़དྷͳ͍ • Ͱ͸Ͳ͏͢Δͷ͔ʁ P (X ∈ C|θ ∈ Θ1) ≤ α ͷԼͰ Pθ (X ∈ C|θ ∈ Θ1) Λ࠷େԽ͢ΔΑ͏ͳݕ ఆΛߏ੒͢Δ! • ఆ਺ α ͸લ΋ܾͬͯΊ͓ͯ͘ఆ਺Ͱɼ“༗ҙਫ४” ͱ͍͏ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 19 / 33
22. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ ࠷ڧྗݕఆ ࠷ڧྗݕఆ • Φα = { φ | Eθ [φ (X)] , θ ∈ Θ0 } • θ1 ∈ Θ1 β(θ) Λ࠷େʹ͢Δ φ ∈ Φα Λɼ༗ҙਫ४ α ͷ θ = θ1 ʹର͢Δ࠷ڧ ྗݕఆ (Most Powerful Test; MP ݕఆ) ͱ͍͏ɽ∀θ ∈ Θ1 Ͱ β(θ) Λ ࠷େʹ͢Δ φ ∈ Φα ΛҰ༷࠷ڧྗݕఆ (Uniformly MP ݕఆ) ͱ͍͏ɽ • Ұ༷࠷ڧྗݕఆ͕ଘࡏ͢Ε͹ɼଞͷݕఆΛߏ੒͢Δඞཁ͸ͳ͍ • Ͱ͸ɼҰ༷࠷ڧྗݕఆ͸ͲͷΑ͏ʹߏ੒͞ΕΔͷ͔? ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 20 / 33
23. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ Neyman-Pearson ͷجຊఆཧ ωΠϚϯɾϐΞιϯͷجຊఆཧ X1, . . . Xn iid ∼ f(x; θ) ͱ͢Δɽݕఆ໰୊ H0 : θ = θ0 vs H1 : θ = θ1 Λߟ͑Δɽf0(x) = ∏ n i=1 f(x; θ0), f1(x) = ∏ n i=1 f(x; θ1), ͱ͢Δ ͱɼ༗ҙਫ४ α (0 < α < 1) ͷ MP ݕఆ͸ ϕ (x) =      1 (f1(x) > kf0(x)) γ (f1(x) = kf0(x)) 0 (f1(x) ≤ kf0(x)) Ͱ༩͑ΒΕΔɽ͜͜Ͱ k, γ ͸ Eθ0 [φ (X)] = α Λຬͨ͢ఆ਺ (k ≥ 0, 0 ≤ γ ≤ 1) Ͱ͋Δɽ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 21 / 33
24. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ ໬౓ൺ ໬౓ൺ H0, H1 ͷԼͰͷ p.d.f. (p.m.f.) ͷൺ Λ (x) = f1 (x) f0 (x) Λ໬౓ൺͱ͍͏ɽ • Neyman-Peason ͷجຊఆཧ͸ɼ໬౓ൺʹج͍ͮͯ MP ݕఆ͕ߏ੒͞Ε Δ͜ͱΛ͍ࣔͯ͠Δ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 22 / 33
25. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ MP ݕఆͷߏ੒ྫ MP ݕఆͷߏ੒ྫ –ਖ਼ن෼෍Λྫʹ– X1, . . . , Xn iid ∼ N ( µ, σ2 ) ɼσ2ɿط஌ɼµɿະ஌ͱ͍͏ঢ়گͰɼݕఆ ໰୊ H0 : µ = µ0 vs H1 : µ = µ1 > µ0 Λߟ͑Δɽ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 23 / 33
26. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ MP ݕఆͷߏ੒ྫ H0 ͷԼͰͷ j.p.d.f. ͸ f0 (x) = ( 1 √ 2πσ2 )n exp [ − 1 2σ2 n ∑ i=1 (xi − µ0 )2 ] H1 ͷԼͰͷ j.p.d.f. ͸ f1 (x) = ( 1 √ 2πσ2 )n exp [ − 1 2σ2 n ∑ i=1 (xi − µ1 )2 ] Ͱ͋Δ͔Βɼ໬౓ൺ͸ Λ (x) = exp [ − 1 2σ2 n ∑ i=1 (xi − µ1 )2 + 1 2σ2 n ∑ i=1 (xi − µ1 )2 ] = exp [ n (µ1 − µ0 ) σ2 x − n ( µ2 1 − µ2 0 )] ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 24 / 33
27. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ MP ݕఆͷߏ੒ྫ X1, . . . , Xn ͸ C-type ͳͷͰ φ (x) = { 1 Λ (x) > k 0 Λ (x) < k ͕࠷ڧྗݕఆͰ͋ΔɽΛ (x) ͕୯ௐ૿Ճؔ਺Ͱ͋Δ͜ͱ͔Β ∃k′ ∈ R s.t. Λ (x) > k ⇐⇒ x > k′. ༗ҙਫ४Λ α > 0 ͱ͢Δͱ α = Eµ0 [φ (X)] = ∫ Λ(x)>0 dFµ0 (x) =Pµ0 (Λ(X) > k|H0) =Pµ0 ( X > k′|H0 ) ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 25 / 33
28. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ MP ݕఆͷߏ੒ྫ √ n ( X − µ0 ) /σ ∼ N (0, 1) Ͱ͋Δ͜ͱΛ༻͍Δͱɼ α = P ( √ n ( X − µ0 ) σ > √ n (k′ − µ0 ) σ µ0 ) = 1 − Φ (√ n (k′ − µ0 ) σ ) ͱͳΔɽzα Λ N (0, 1) ͷ্ଆ 100α% ఺ͱ͢Δͱɼ zα = √ n (k′ − µ0) σ =⇒ k′ = µ0 + σ √ n zα Ͱ͋Δ͔Βɼ࠷ڧྗݕఆͷغ٫Ҭͱͯ͠ C = { (x1, . . . , xn) x > µ0 + σ √ n zα } ͕ٻ·Δɽ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 26 / 33
29. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ MP ݕఆͷߏ੒ྫ • ͜ͷغ٫Ҭ C ͸ɼµ1 ͷ஋ʹ͸ґଘ͍ͯ͠ͳ͍ͷͰɼµ1 > µ0 Λຬͨ͢ ೚ҙͷ µ1 ʹରͯ͠΋ಉ༷ͷغ٫Ҭ͕ಋ͔ΕΔ • ͕ͨͬͯ͠ɼ͜ͷݕఆ͸ UMP ݕఆͰ͋Δ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 27 / 33
30. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ MP ݕఆͷߏ੒ྫ –Type II Error ͷ֬཰– ࣍ʹɼType II Error ͷ֬཰ΛٻΊΔɽµ = µ1 ͷͱ͖ʹ X / ∈ C ͱͳΔ֬཰ ͳͷͰɼ P ( X / ∈ C µ1 ) =P ( X < µ0 + σ √ n zα ) =P (√ n ( X − µ1 ) σ < zα − √ n (µ0 − µ1) σ ) =Φ ( zα − √ n (µ0 − µ1) σ ) . α Λখ͘͢͞ΔͱɼఆٛΑΓ zα ͸େ͖͘ͳΔͷͰɼType II Error ͷ֬཰΋ େ͖͘ͳΔɽ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 28 / 33
31. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ MP ݕఆͷߏ੒ྫ – ݕग़ྗ – ݕग़ྗ͸ β(µ1) =P ( X ∈ C µ1 ) =1 − P ( X / ∈ C µ1 ) =1 − Φ ( zα − √ n (µ0 − µ1) σ ) ͱٻ·Δɽ • ݕग़ྗ β(µ1) ͸ɼµ0 − µ1 ͕େ͖͍΄Ͳେ͖͘ͳΔ • 2 ܈ͷ฼ฏۉ͕େ͖͘཭Ε͍ͯΔ΄Ͳ H0 ͕੒ཱ͍ͯ͠Δ͔ H1 ͕੒ཱ ͍ͯ͠Δͷ͔Λݟ෼͚Δͷ͸؆୯ͱ͍͏௚ײʹ߹க͍ͯ͠Δ • n ͕େ͖͍΄Ͳݕग़ྗ͕େ͖͘ͳΔ • ͜Ε΋௚ײʹ߹க͍ͯ͠Δ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 29 / 33
32. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ MP ݕఆͷߏ੒ྫ • ઌఔߟ͑ͨྫ͸ H1 : µ = µ1 > µ0 Ͱ͕͋ͬͨɼରཱԾઆͱͯ͠ H1 : µ = µ1 < µ0 ΋ߟ͑Δ͜ͱ͕ग़དྷΔ • ͜ͷ৔߹ɼಉ༷ʹͯ͠࠷ڧྗݕఆͷغ٫Ҭ C = { (x1, . . . , xn) x < µ0 + σ √ n zα } Λߏ੒Ͱ͖Δ • ͜ͷΑ͏ʹɼରཱԾઆ H1 ͷؼແԾઆ H0 ͔ΒͷζϨͷํ޲ʹରԠ͠ ͯغ٫Ҭ͕ H0 ͷ஋ (ࠓճ͸ µ0) ยଆʹ࡞ΒΕΔݕఆΛยଆݕఆͱ͍͏ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 30 / 33
33. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ MP ݕఆͷߏ੒ྫ –྆ଆݕఆ΁ͷ֦ு– • ରཱԾઆ͕ H1 : µ ̸= µ0 ͷ৔߹ʹ͸༗ҙਫ४ α ͷ UMP ݕఆ͸ଘࡏ͠ ͳ͍ • ͔͠͠ɼ͜Ε·Ͱͷٞ࿦͔Βغ٫Ҭ C = { (x1, . . . , xn) |x − µ0| > σ √ n zα } Λߏ੒Ͱ͖Δ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 31 / 33
34. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ ৴པ۠ؒͱݕఆ • ਖ਼ن฼ूஂʹ͓͚Δ෼ࢄط஌Ͱͷ฼ฏۉͷ۠ؒਪఆʹ͓͍ͯɼ৴པ܎ ਺ 100(1 − α)% ͷ µ ͷ৴པ۠ؒ͸ Iµ = [ X − zα/2 σ0 √ n , X + zα/2 σ0 √ n ] • Iµ ͕ µ0 ΛؚΉͱ͢ΔͱɼؼແԾઆ H0 ͸ઌఔͷ྆ଆݕఆʹ͓͍ͯ༗ ҙਫ४ α Ͱغ٫͞Εͳ͍ • ͢ͳΘͪɼx / ∈ C =⇒ µ0 ∈ Iµ • ͭ·ΓɼؼແԾઆ͕غ٫͞Εͳ͍͜ͱͱɼ৴པ͕۠ؒؼແԾઆͷ஋Λ ؚΉ͜ͱ͸ಉ஋ɿ ؼແԾઆ͕غ٫͞ΕΔ ⇐⇒ ৴པ͕۠ؒؼແԾઆͷ஋Λؚ·ͳ͍ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 32 / 33
35. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ ͦͷଞͷݕఆߏ੒ํ๏ • ࠶ڠྗݕఆ͸ωΠϚϯɾϐΞιϯͷجຊఆཧʹΑͬͯߏ੒͞ΕΔ͕ɼ ෼෍࿦͕ෳࡶʹͳͬͯਖ਼֬ͳغ٫ҬΛ࡞Εͳ͍͜ͱ΋͋Δ • ͦΜͳͱ͖͸ɼ࠷໬ਪఆྔʹج͍ͯݕఆΛߏ੒͢Δ͜ͱ͕͋Δ • ଞʹ΋ɼefficient score Λ༻͍ͨݕఆͷߏ੒ํ๏΋͋Δ ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 33 / 33
36. Introduction Ծઆݕఆ 2 छྨͷաޡͱݕग़ྗؔ਺ ࠷ڧྗݕఆͷߏ੒ Reference • ੺ฏণจ, ౷ܭղੳೖ໳, ৿๺ग़൛גࣜձࣾ, 2007 • ಺౻؏ଠ, ਺ཧ౷ܭֶ II ߨٛϊʔτ, 2014 • ౦ژେֶڭཆֶ෦౷ܭֶڭࣨɼجૅ౷ܭֶ III ࣗવՊֶͷ౷ܭֶɼ2013 ాத ޺ণ (@setten QB) (AHC-Lab) ୈ 4 ճ ֬཰ɾ౷ܭͷجૅษڧձ ౷ܭతݕఆ March 25, 2017 34 / 33