The bootstrapping method for everyone

୭Ͱ΋෼͔Δ#PPUTUSBQͷॳา ౷ܭֶษڧձʢԾʣ !LVSDLZ@Z

ࣗݾ঺հ w LVSDLZ !LVSDLZ@Z w 3%BU4BOTBO *OD w μΠϨΫτϦΫϧʔςΟϯάαʔϏεͷݚڀ։ൃ
w స৬ࢢ৔ͷ෼ੳ w ͓ञ͕޷͖ʢνϣϩ͍ʣ 2

͜ͷൃදʹ͍ͭͯ w ର৅ ‣ ϒʔτετϥοϓ๏ʹ͍ͭͯ஌Βͳ͍ਓ ‣ ࢖͏Πϝʔδ͕͍͍ͭͯͳ͍ਓ w ࿩͢͜ͱ ‣
͓ؾ࣋ͪɼ࢖͍Ͳ͜Ζ ‣ جຊͷϊϯύϥϝτϦοΫϒʔτετϥοϓ w ࿩͞ͳ͍͜ͱ ‣ ਺ֶతഎܠ 3

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

͜ͷ࣌఺Ͱ͸ʮ;ʔΜʯͰ0,Ͱ͢

͜͹ͳ͠ ਪଌ౷ܭͬͯͳΜ͚ͩͬ ϒʔτετϥοϓ๏ ໨࣍

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

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

ਪଌ౷ܭͬͯͳΜ͚ͩͬ ͍͍ͩͨ͜ΜͳྲྀΕ 11

ਪଌ౷ܭͬͯͳΜ͚ͩͬ ͍͍ͩͨ͜ΜͳྲྀΕ 12 ͲΕ͙Β͍ྑ͍ਪଌʁ ‣ ෆภੑɿظ଴஋͕ਅͷ஋ʹҰக ‣ Ұகੑɿ ͕େ͖͘ͳΔͱਅͷ஋ʹۙͮ͘ ‣
༗ޮੑɿਪఆྔͷ෼ࢄ͕࠷΋খ͍͞ ‣ FUD n

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

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

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

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)

ͳΜͰʮ෮ݩʯநग़ʁ w ʮಠཱಉ෼෍ͰඪຊΛಘΔʯͱ͍͏͜ͱ͸ɼ෼෍ؔ਺͔Β ཚ਺ΛऔΔͱ͍͏͜ͱ 18 ෼෍ؔ਺ʹҰ༷ཚ਺Λ౰ͯΔͱ ͦͷ෼෍͔Βͷඪຊ͕ग़ͯ͘Δ ܦݧ෼෍ʹҰ༷ཚ਺Λ౰ͯΔͱ ٙࣅඪຊ͕ग़ͯ͘Δ ʢಉ͡஋͕ॏෳ͠ಘΔʣ

͕͜͜ྑ͍ͧ w ಛఆͷ෼෍ΛԾఆ͠ͳ͍ w ண໨͍ͯ͠Δਪఆྔʹґଘͤͣɼશࣗಈ w δϟοΫφΠϑ๏ͷܽ఺͕ͳ͍ ‣ ඪຊू߹ʹؔͯ͠׈Β͔ʹಈ͔ͳ͍ਪఆྔʹऑ͍ʢFHதԝ஋ʣ ‣
ඪ४ޡ্͕ࠩʹภΔ 19

͜͹ͳ͠ ਪଌ౷ܭͬͯͳΜ͚ͩͬ ϒʔτετϥοϓ๏ Ԡ༻ͷ঺հ ໨࣍

#BHHJOH w #PPUTUSBQBOE"HHSFHBUJPOˠ#BHHJOH ‣ ॏෳ͋Γͷ࠶நग़Ͱσʔλ ηοτෳ੡ˠֶश ‣ ͦΕͧΕͷ݁ՌΛΞϯαϯϒϧʢଟ༷ੑɼநग़ʹΑΔ༳Β͗΋൓өʣ w 3BOEPN'PSFTUͳͲʹར༻͞Ε͍ͯΔ
‣ #BHHJOH ‣ ֶशʹ࢖༻͢Δಛ௃ྔ΋ϥϯμϜʹબ୒ ‣ ܾఆ໦Λֶश͠Ξϯαϯϒϧ B 21

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

࣌ܥྻͷϒʔτετϥοϓ 23
ʜ ʜ ॏෳ͋ΓϦαϯϓϦϯά݁߹ ॏෳ͋ΓϒϩοΫԽ ʜ ʜ ʜ ॏෳ͋ΓϦαϯϓϦϯά݁߹ ॏෳͳ͠ϒϩοΫԽ ඇॏෳϒϩοΫϒʔτετϥοϓʢࠨʣͱҠಈϒϩοΫϒʔτετϥοϓʢӈʣ w ߏ଄Λ͋Δఔ౓อ࣋͢Δϒʔτετϥοϓ͕ݚڀ͞Ε͍ͯΔ

͍͞͝ʹ w ϒʔτετϥοϓ๏͸͍͍ͧ w ͱ͸͍͑ɼղੳతʹ෼͔͍ͬͯΔʹͨ͜͜͠ͱ͸ͳ͍ w ͋Γ͕ͱ͏͍͟͝·ͨ͠ʂʂ 24

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

The bootstrapping method for everyone

The bootstrapping method for everyone

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୭Ͱ΋෼͔Δ#PPUTUSBQͷॳา ౷ܭֶษڧձʢԾʣ !LVSDLZ@Z

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

͜ͷൃදʹ͍ͭͯ w ର৅ ‣ ϒʔτετϥοϓ๏ʹ͍ͭͯ஌Βͳ͍ਓ ‣ ࢖͏Πϝʔδ͕͍͍ͭͯͳ͍ਓ w ࿩͢͜ͱ ‣

ϒʔτετϥοϓ๏ &GSPOBOE5JCTIJSBOJ w ݸͷඪຊ ͔Βॏෳ͋ΓͰେ͖͞ ͷແ࡞ҝநग़Λߦ ͏ ʢϒʔτετϥοϓඪຊʣ w

͜ͷ࣌఺Ͱ͸ʮ;ʔΜʯͰ0,Ͱ͢

͜͹ͳ͠ ਪଌ౷ܭͬͯͳΜ͚ͩͬ ϒʔτετϥοϓ๏ ໨࣍

͜͹ͳ͠ ਪଌ౷ܭͬͯͳΜ͚ͩͬ ϒʔτετϥοϓ๏ ໨࣍

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

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

͜͹ͳ͠ ਪଌ౷ܭͬͯͳΜ͚ͩͬ ϒʔτετϥοϓ๏ ໨࣍

ਪଌ౷ܭͬͯͳΜ͚ͩͬ ͍͍ͩͨ͜ΜͳྲྀΕ 11

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

ਪଌ౷ܭͬͯͳΜ͚ͩͬ ͍͍ͩͨ͜ΜͳྲྀΕ 13 ͲΕ͙Β͍ྑ͍ਪଌʁ ‣ ෆภੑɿظ଴஋͕ਅͷ஋ʹҰக ‣ Ұகੑɿ ͕େ͖͘ͳΔͱਅͷ஋ʹۙͮ͘ ‣

͜͹ͳ͠ ਪଌ౷ܭͬͯͳΜ͚ͩͬ ϒʔτετϥοϓ๏ ໨࣍

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

ϒʔτετϥοϓ๏Ͱ΍͍ͬͯΔ͜ͱ w ඪຊΛٖࣅతͳ฼ूஂͱΈͳ͠ɼ͔ͦ͜ΒԿճ΋ඪຊநग़ Λߦ͏ʢࣗ෼ͰͳΜͱ͔͢Δʂʣ ‣ ະ஌ͷ֬཰෼෍ ˠܦݧ෼෍ ‣ ౷ܭྔ

ਤͰ੔ཧ 17 ඪຊ ٙࣅ฼ूஂ ඪຊ ɾɾɾ ඪຊ ඪຊ# ॏෳ͋ΓϦαϯϓϦϯά

ͳΜͰʮ෮ݩʯநग़ʁ w ʮಠཱಉ෼෍ͰඪຊΛಘΔʯͱ͍͏͜ͱ͸ɼ෼෍ؔ਺͔Β ཚ਺ΛऔΔͱ͍͏͜ͱ 18 ෼෍ؔ਺ʹҰ༷ཚ਺Λ౰ͯΔͱ ͦͷ෼෍͔Βͷඪຊ͕ग़ͯ͘Δ ܦݧ෼෍ʹҰ༷ཚ਺Λ౰ͯΔͱ ٙࣅඪຊ͕ग़ͯ͘Δ ʢಉ͡஋͕ॏෳ͠ಘΔʣ

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

͜͹ͳ͠ ਪଌ౷ܭͬͯͳΜ͚ͩͬ ϒʔτετϥοϓ๏ Ԡ༻ͷ঺հ ໨࣍

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

34° N 34.5° N 35° N 35.5° N 36° N

࣌ܥྻͷϒʔτετϥοϓ 23

͍͞͝ʹ w ϒʔτετϥοϓ๏͸͍͍ͧ w ͱ͸͍͑ɼղੳతʹ෼͔͍ͬͯΔʹͨ͜͜͠ͱ͸ͳ͍ w ͋Γ͕ͱ͏͍͟͝·ͨ͠ʂʂ 24

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