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The bootstrapping method for everyone

9c42c4bc1d91c409d754da88c91cb2ef?s=47 kur0cky
March 28, 2021

The bootstrapping method for everyone

2021/03/28
統計学勉強会(仮)#2 LT資料

9c42c4bc1d91c409d754da88c91cb2ef?s=128

kur0cky

March 28, 2021
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  1. ୭Ͱ΋෼͔Δ#PPUTUSBQͷॳา ౷ܭֶษڧձʢԾʣ !LVSDLZ@Z

  2. ࣗݾ঺հ w LVSDLZ !LVSDLZ@Z  w 3%BU4BOTBO *OD w μΠϨΫτϦΫϧʔςΟϯάαʔϏεͷݚڀ։ൃ

    w స৬ࢢ৔ͷ෼ੳ w ͓ञ͕޷͖ʢνϣϩ͍ʣ 2
  3. ͜ͷൃදʹ͍ͭͯ w ର৅ ‣ ϒʔτετϥοϓ๏ʹ͍ͭͯ஌Βͳ͍ਓ ‣ ࢖͏Πϝʔδ͕͍͍ͭͯͳ͍ਓ w ࿩͢͜ͱ ‣

    ͓ؾ࣋ͪɼ࢖͍Ͳ͜Ζ ‣ جຊͷϊϯύϥϝτϦοΫϒʔτετϥοϓ w ࿩͞ͳ͍͜ͱ ‣ ਺ֶతഎܠ 3
  4. ϒʔτετϥοϓ๏ &GSPOBOE5JCTIJSBOJ  w ݸͷඪຊ ͔Βॏෳ͋ΓͰେ͖͞ ͷແ࡞ҝநग़Λߦ ͏ ʢϒʔτετϥοϓඪຊʣ w

    ͔Βɼ஫໨͢Δ౷ܭྔ Λࢉग़͢Δ w Ҏ্Λ ճ͘Γฦ͠ɼ ΛಘΔ w Λݩʹɼ ͷ࣋ͭภΓ΍෼ࢄɼ৴པ۠ؒͳͲʹ͍ͭͯ ࿦͡Δ n {x1 , x2 , …, xn } n {x* 1 , x* 2 , …, x* n } {x* 1 , x* 2 , …, x* n } ̂ θ* B { ̂ θ* 1 , ̂ θ* 2 , …, ̂ θB *} { ̂ θ* 1 , ̂ θ* 2 , …, ̂ θB *} ̂ θ 4
  5. ͜ͷ࣌఺Ͱ͸ʮ;ʔΜʯͰ0,Ͱ͢

  6. ͜͹ͳ͠ ਪଌ౷ܭͬͯͳΜ͚ͩͬ ϒʔτετϥοϓ๏ ໨࣍

  7. ͜͹ͳ͠ ਪଌ౷ܭͬͯͳΜ͚ͩͬ ϒʔτετϥοϓ๏ ໨࣍

  8. ͜͹ͳ͠ ʮࠓ݄ͷ,1*͸ Ͱͨ͠ɼྑ͍ײ͡Ͱ͢ʂʂʯ ʮͦͷ਺஋ͬͯͲͷ͙Β͍৴༻Ͱ͖Δͷʁʯ ʮɾɾɾ😥ʯ ʮࠐΈೖͬͨ,1*ʹͳ͓ͬͯΓ·ͯ͠ɽɽɽʯ ɹʢ΍΂͐෼͔ΜͶ͐ʣ 8 ৽ถ%4 εςʔΫϗϧμʔ

    ˞͜Ε͸ϑΟΫγϣϯͰ͢
  9. ͜͹ͳ͠ ʮ༧ଌ஋͸ Ͱͨ͠ʂʂʯ ʮͦͷ༧ଌΛ۠ؒͰग़͢ͱ͢Ε͹Ͳͷ͙Β͍ʁʁʯ ʮɾɾɾ😥ʯ ʮઢܗճؼͳΒʢඪ४ޡࠩʣ͕ग़ΔͷͰ͕͢ɽɽɽʯ ɹʢ΍΂͐෼͔ΜͶ͐ʣ 9 ৽ถ%4 εςʔΫϗϧμʔ

    ˞͜Ε͸ϑΟΫγϣϯͰ͢
  10. ͜͹ͳ͠ ਪଌ౷ܭͬͯͳΜ͚ͩͬ ϒʔτετϥοϓ๏ ໨࣍

  11. ਪଌ౷ܭͬͯͳΜ͚ͩͬ ͍͍ͩͨ͜ΜͳྲྀΕ 11

  12. ਪଌ౷ܭͬͯͳΜ͚ͩͬ ͍͍ͩͨ͜ΜͳྲྀΕ 12 ͲΕ͙Β͍ྑ͍ਪଌʁ ‣ ෆภੑɿظ଴஋͕ਅͷ஋ʹҰக ‣ Ұகੑɿ ͕େ͖͘ͳΔͱਅͷ஋ʹۙͮ͘ ‣

    ༗ޮੑɿਪఆྔͷ෼ࢄ͕࠷΋খ͍͞ ‣ FUD n
  13. ਪଌ౷ܭͬͯͳΜ͚ͩͬ ͍͍ͩͨ͜ΜͳྲྀΕ 13 ͲΕ͙Β͍ྑ͍ਪଌʁ ‣ ෆภੑɿظ଴஋͕ਅͷ஋ʹҰக ‣ Ұகੑɿ ͕େ͖͘ͳΔͱਅͷ஋ʹۙͮ͘ ‣

    ༗ޮੑɿਪఆྔͷ෼ࢄ͕࠷΋খ͍͞ ‣ FUD n ਪఆྔ͕ෳࡶʹͳΔͱ೉͍͠ʢղੳ΍ల։Ͱۙࣅʣ ฏۉ஋ͳͲͳΒத৺ۃݶఆཧ ਖ਼ن෼෍ΛԾఆ͢Δͱ؆୯͔΋ʢFHඪ४ޡࠩʣ
  14. ͜͹ͳ͠ ਪଌ౷ܭͬͯͳΜ͚ͩͬ ϒʔτετϥοϓ๏ ໨࣍

  15. ϒʔτετϥοϓ๏ʢ࠶ܝʣ w ݸͷඪຊ ͔Βॏෳ͋ΓͰେ͖͞ ͷແ࡞ҝநग़Λߦ ͏ ʢϒʔτετϥοϓඪຊʣ w ͔Βɼ஫໨͢Δ౷ܭྔ Λࢉग़͢Δ

    w Ҏ্Λ ճ͘Γฦ͠ɼ ΛಘΔ w Λݩʹɼ ͷ࣋ͭภΓ΍෼ࢄɼ৴པ۠ؒͳͲʹ͍ͭͯ ࿦͡Δ n {x1 , x2 , …, xn } n {x* 1 , x* 2 , …, x* n } {x* 1 , x* 2 , …, x* n } ̂ θ* B { ̂ θ* 1 , ̂ θ* 2 , …, ̂ θ* B } { ̂ θ* 1 , ̂ θ* 2 , …, ̂ θB *} ̂ θ 15
  16. ϒʔτετϥοϓ๏Ͱ΍͍ͬͯΔ͜ͱ w ඪຊΛٖࣅతͳ฼ूஂͱΈͳ͠ɼ͔ͦ͜ΒԿճ΋ඪຊநग़ Λߦ͏ʢࣗ෼ͰͳΜͱ͔͢Δʂʣ ‣ ະ஌ͷ֬཰෼෍ ˠܦݧ෼෍  ‣ ౷ܭྔ

    ˠ  ‣ ਪఆྔͷ෼෍ ˠ F Fn θ = f(F) θn = f(Fn ) θ ∼ Gn f(F* n ) ∼ G* n 16
  17. ਤͰ੔ཧ 17 ඪຊ  ٙࣅ฼ूஂ ඪຊ ɾɾɾ ඪຊ ඪຊ# ॏෳ͋ΓϦαϯϓϦϯά

    େ͖͞ ɼ ճ n B ̂ θ*(B) ̂ θ*(2) ̂ θ*(1) ɾɾɾ 0 50 100 150 200 250 0.4 0.6 0.8 1.0 1.2 1.4 Variance count ϒʔτετϥοϓ৴པ۠ؒ ཧ࿦஋  ෆภਪఆ஋  ϒʔτετϥοϓฏۉ  ৴པԼݶɾ্ݶ θ ̂ θ ̂ θ* = 1 B B ∑ b=1 ̂ θ*(b) 
  18. ͳΜͰʮ෮ݩʯநग़ʁ w ʮಠཱಉ෼෍ͰඪຊΛಘΔʯͱ͍͏͜ͱ͸ɼ෼෍ؔ਺͔Β ཚ਺ΛऔΔͱ͍͏͜ͱ 18 ෼෍ؔ਺ʹҰ༷ཚ਺Λ౰ͯΔͱ ͦͷ෼෍͔Βͷඪຊ͕ग़ͯ͘Δ ܦݧ෼෍ʹҰ༷ཚ਺Λ౰ͯΔͱ ٙࣅඪຊ͕ग़ͯ͘Δ ʢಉ͡஋͕ॏෳ͠ಘΔʣ

  19. ͕͜͜ྑ͍ͧ w ಛఆͷ෼෍ΛԾఆ͠ͳ͍ w ண໨͍ͯ͠Δਪఆྔʹґଘͤͣɼશࣗಈ w δϟοΫφΠϑ๏ͷܽ఺͕ͳ͍ ‣ ඪຊू߹ʹؔͯ͠׈Β͔ʹಈ͔ͳ͍ਪఆྔʹऑ͍ʢFHதԝ஋ʣ ‣

    ඪ४ޡ্͕ࠩʹภΔ 19
  20. ͜͹ͳ͠ ਪଌ౷ܭͬͯͳΜ͚ͩͬ ϒʔτετϥοϓ๏ Ԡ༻ͷ঺հ ໨࣍

  21. #BHHJOH w #PPUTUSBQBOE"HHSFHBUJPOˠ#BHHJOH ‣ ॏෳ͋Γͷ࠶நग़Ͱσʔλ ηοτෳ੡ˠֶश ‣ ͦΕͧΕͷ݁ՌΛΞϯαϯϒϧʢଟ༷ੑɼநग़ʹΑΔ༳Β͗΋൓өʣ w 3BOEPN'PSFTUͳͲʹར༻͞Ε͍ͯΔ

    ‣ #BHHJOH ‣ ֶशʹ࢖༻͢Δಛ௃ྔ΋ϥϯμϜʹબ୒ ‣ ܾఆ໦Λֶश͠Ξϯαϯϒϧ B 21
  22. 34° N 34.5° N 35° N 35.5° N 36° N

    36.5° N 84° W 82° W 80° W 78° W 76° W 0 5 10 15 0 100 200 300 t value ߏ଄ͷ͋Δσʔλͷϒʔτετϥοϓ w ֬཰తߏ଄ͷ͋Δσʔλʹରͯ͠JJEʹجͮ͘෮ݩநग़Λ ߦ͏ͱɼͦͷσʔλੜ੒ߏ଄͸่ΕΔ w ࣌ܥྻɼ஍ཧɼάϥϑσʔλͳͲ 22
  23. ࣌ܥྻͷϒʔτετϥοϓ 23        

      ʜ                    ʜ                   ॏෳ͋ΓϦαϯϓϦϯά݁߹ ॏෳ͋ΓϒϩοΫԽ           ʜ           ʜ        ʜ ॏෳ͋ΓϦαϯϓϦϯά݁߹ ॏෳͳ͠ϒϩοΫԽ ඇॏෳϒϩοΫϒʔτετϥοϓʢࠨʣͱҠಈϒϩοΫϒʔτετϥοϓʢӈʣ w ߏ଄Λ͋Δఔ౓อ࣋͢Δϒʔτετϥοϓ͕ݚڀ͞Ε͍ͯΔ
  24. ͍͞͝ʹ w ϒʔτετϥοϓ๏͸͍͍ͧ w ͱ͸͍͑ɼղੳతʹ෼͔͍ͬͯΔʹͨ͜͜͠ͱ͸ͳ͍ w ͋Γ͕ͱ͏͍͟͝·ͨ͠ʂʂ 24

  25. ࢀߟจݙ 1. Efron and R. J. Tibshirani. An Introduction to

    the Bootstrap. CRC Press, 1994. 2. E. Carlstein. The use of subseries values for estimating the variance of a general statistic from a stationary sequence. The Annals of Statistics, 14(3):1171--1179, 09 1986. 3. H. R. Kunsch. The jackknife and the bootstrap for general stationary observations. The Annals of Statistics, pages 1217--1241, 1989. 4. ᔉۚ๕, ా܀ਖ਼ষ, ܭࢉ౷ܭⅠ- ֬཰ܭࢉͷ৽͍͠ख๏, ϒʔτετϥοϓ๏ೖ໳, ؠ೾ ॻళ, 2003. 5. ᔉۚ๕, ࡩҪ༟ਔ, ϒʔτετϥοϓೖ໳, ڞཱग़൛, 2011. 6. Լฏӳण, 21ੈلͷ౷ܭՊֶ3, ୈ8ষ ϒʔτετϥοϓ, ౦ژେֶग़൛ձ, 2008. 25