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The bootstrapping method for everyone

kur0cky
March 28, 2021

The bootstrapping method for everyone

2021/03/28
統計学勉強会(仮)#2 LT資料

kur0cky

March 28, 2021
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  1. ͜ͷൃදʹ͍ͭͯ w ର৅ ‣ ϒʔτετϥοϓ๏ʹ͍ͭͯ஌Βͳ͍ਓ ‣ ࢖͏Πϝʔδ͕͍͍ͭͯͳ͍ਓ w ࿩͢͜ͱ ‣

    ͓ؾ࣋ͪɼ࢖͍Ͳ͜Ζ ‣ جຊͷϊϯύϥϝτϦοΫϒʔτετϥοϓ w ࿩͞ͳ͍͜ͱ ‣ ਺ֶతഎܠ 3
  2. 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
  3. 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
  4. 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
  5. 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) 
  6. 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
  7. ࣌ܥྻͷϒʔτετϥοϓ 23        

      ʜ                    ʜ                   ॏෳ͋ΓϦαϯϓϦϯά݁߹ ॏෳ͋ΓϒϩοΫԽ           ʜ           ʜ        ʜ ॏෳ͋ΓϦαϯϓϦϯά݁߹ ॏෳͳ͠ϒϩοΫԽ ඇॏෳϒϩοΫϒʔτετϥοϓʢࠨʣͱҠಈϒϩοΫϒʔτετϥοϓʢӈʣ w ߏ଄Λ͋Δఔ౓อ࣋͢Δϒʔτετϥοϓ͕ݚڀ͞Ε͍ͯΔ
  8. 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