Contents͸͡Ίʹ‫ه‬ड़౷‫ܭ‬ͷʮͳͥʁʯΛ౷‫ܭ‬తਪଌʹΑΓཧղ͢Δʙ࢛෼Ґ਺Λྫͱͯ͠ʙ‫ه‬ड़౷‫ͱܭ‬౷‫ܭ‬తਪଌͷ઀఺࢛෼Ґ਺ͷఆٛञંจ෢ɼૣ࡚‫ొٱ‬2019 ೥ 3 ݄ 2 ೔ඪຊ࢛෼Ґ਺ͷධՁதԝେֶཧ޻ֶ෦਺ֶՊɼதԝେֶେֶӃཧ޻ֶ‫ڀݚ‬Պ਺ֶઐ߈

·ͱΊ

2 / 43ʮ࢛෼Ґ਺ʯ౳ͷѻ͍த౳‫ڭ‬ҭʹ͓͚Δ࢛෼Ґ਺ɼ࢛෼Ґൣғɼശͻ͛ਤͷѻ͍• ‫ߦݱ‬ͷֶशࢦಋཁྖɿߴ౳ֶߍ ਺ֶ I
• ৽ֶशࢦಋཁྖɿதֶ 2 ೥͸͡Ίʹ‫ݱ‬৔Ͱͷࠞཚ• ࢛෼Ґ਺ͷ‫݁ࢉܭ‬Ռ͕ Excel ͱҟͳΔʂ
• ίϯϐϡʔλͷར༻͕ਪਐ͞Ε͍ͯΔͷʹʜͲ͏ͨ͠ΒΑ͍ͷ͔ʁ

4 / 43ʮ࢛෼Ґ਺ʯ౳ͷѻ͍

͞·͟·ͳࢦఠɾҙ‫ݟ‬͍ͣ΋Γɹɹ (2013)
Կ৿த౳‫ڭ‬ҭʹ͓͚Δ࢛෼Ґ਺ɼ࢛෼Ґൣғɼശͻ͛ਤͷѻ͍• ‫ݱ‬৔ͰͷࠞཚΛආ͚ΔͨΊʹ‫ݻ‬ఆͷఆٛΛԡ͠෇͚Δ΂͖Ͱ͸ͳ͍• ‫ߦݱ‬ͷֶशࢦಋཁྖɿߴ౳ֶߍ ਺ֶ I• ೖࢼͳͲʹ࢛෼Ґ਺͸ద͞ͳ͍• ৽ֶशࢦಋཁྖɿதֶ 2 ೥⇒ ࢛෼Ґ਺΍࢛෼ҐൣғΛಡΈऔͬͯͷ൑அ͸ՄೳͰ͋Γɼ͋͘·Ͱ΋࢛෼Ґ਺ͷ‫ࢉܭ‬Λओʹ໰͏΂͖Ͱ͸ͳ͍ͱ͍͏͚ͩ‫ݱ‬৔Ͱͷࠞཚখྛ (2013)• ࢛෼Ґ਺ͷ‫݁ࢉܭ‬Ռ͕ Excel ͱҟͳΔʂ• ࢛෼Ґ਺΍ശͻ͛ਤΛѻΘͳ͍΄͏͕Α͍• ίϯϐϡʔλͷར༻͕ਪਐ͞Ε͍ͯΔͷʹʜͲ͏ͨ͠ΒΑ͍ͷ͔ʁ⇒ ֶशࢦಋཁྖͷҙਤΛཧղ͠Α͏ͱ͠ͳ͍๫࿦
5 / 436 / 43৽ֶशࢦಋཁྖͰ͸৽ֶशࢦಋཁྖͰ͸৽ֶशࢦಋཁྖղઆ (2017)৽ֶशࢦಋཁྖղઆ (2017)• ࢛෼Ґ਺Λ‫ٻ‬ΊΔํ๏ͱͯ͠‫͔ͭز‬ͷํ๏͕ఏҊ͞Ε͍ͯΔ͕ɼ͜͜• ࢛෼Ґ਺Λ‫ٻ‬ΊΔํ๏ͱͯ͠‫͔ͭز‬ͷํ๏͕ఏҊ͞Ε͍ͯΔ͕ɼ͜͜Ͱ͸࢛෼Ґ਺ͷҙຯΛ೺Ѳ͠΍͍͢ํ๏Λ༻͍ΔͰ͸࢛෼Ґ਺ͷҙຯΛ೺Ѳ͠΍͍͢ํ๏Λ༻͍Δ੢஥ (2018)੢஥ (2018)• தֶͰͷಋೖʹࡍͯ͠ɼ࠶౓͜ͷ఺ʹ‫ܯ‬৊Λ໐Βͭͭ͠ɼղઆͷ‫ه‬ड़• தֶͰͷಋೖʹࡍͯ͠ɼ࠶౓͜ͷ఺ʹ‫ܯ‬৊Λ໐Βͭͭ͠ɼղઆͷ‫ه‬ड़ͷෆे෼͞Λࢦఠͷෆे෼͞ΛࢦఠԞଜ (2019)Ԟଜ (2019)• Tukey ͷఆٛɼExcel Ͱ࢖͑Δ 2 ௨Γͷఆٛɼ౷‫ܭ‬ιϑτ R Ͱ࢖͑Δ
9 ௨ΓͷఆٛͷͲΕͰ΋ͳ͍ɼ͓ͦΒ͘จՊলಠࣗͷ΋ͷ• Tukey ͷఆٛɼExcel Ͱ࢖͑Δ 2 ௨Γͷఆٛɼ౷‫ܭ‬ιϑτ R Ͱ࢖͑Δ
9 ௨ΓͷఆٛͷͲΕͰ΋ͳ͍ɼ͓ͦΒ͘จՊলಠࣗͷ΋ͷ⇒ ࠷‫ޙ‬ͷ఺͸‫ޙ‬ड़ͷ௨Γ‫ޡ‬Γ⇒ ࠷‫ޙ‬ͷ఺͸‫ޙ‬ड़ͷ௨Γ‫ޡ‬Γ• ҰͭΛબͿͷͰ͋Ε͹ Tukey ͷఆٛʹ͢΂͖Ͱ͋ͬͨ• ҰͭΛબͿͷͰ͋Ε͹ Tukey ͷఆٛʹ͢΂͖Ͱ͋ͬͨ
7 / 438 / 43ຊൃදͷ໨త‫ݱ‬৔ͷࠞཚ͕‫͍ͳ͖ى‬Α͏ɼ಺༰Λ੔ཧ͓͖͍ͯͨ͠ʂαʔϕΠ• ͳ࢛ͥ෼Ґ਺ͷఆ͕ٛෳ਺͋Δͷ͔• தֶߍɾߴ౳ֶߍͰѻ͍ͬͯΔ΋ͷҎ֎ʹͲͷΑ͏ͳ΋ͷ͕͋Δ͔ɼͲͷΑ͏ͳҧ͍͕͋Δ͔҆৺࢛ͯ͠෼Ґ਺Λ࢖ͬͯ΄͍͠ʂ‫ه‬ड़౷‫ͱܭ‬౷‫ܭ‬తਪଌͷ઀఺‫ڀݚ‬• ଞͷ࢛෼Ґ਺ͱൺ΂ͯ‫͠ࢉܭ‬΍͍͢ϝϦοτ͸͋Δ͕ɼଞʹϝϦοτɼσϝϦοτ͸ͳ͍ͷ͔• ͦͷଞ‫ؾ‬Λ͚ͭΔ΂͖఺͸ͳ͍͔

9 / 43‫ه‬ड़౷‫ͱܭ‬౷‫ܭ‬తਪଌฏ‫ۉ‬ɼ෼ࢄ‫ه‬ड़౷‫ܭ‬• ಘΒΕͨσʔλΛཁ໿͢Δํ๏ฏ‫ۉ‬ʢඪຊฏ‫ۉ‬ʣ X̄ =ex. ฏ‫ۉ‬ɼதԝ஋ɼ࠷ස஋ɼඪ४ภࠩɼ෼ࢄɼώετάϥϜɼࢄ෍ਤɼ૬ؔ܎਺ɼ. . .

1Xin• ฼ूஂͷฏ‫ۉ‬ʢ฼ฏ‫ۉ‬ʣµ Λਪఆ͢Δ΋ͷʢਪఆྔʣ• ‫͑׵͍ݴ‬Ε͹ɼ฼ฏ‫ۉ‬Λਪఆ͢ΔͨΊʹඪຊฏ‫ۉ‬Λ‫ٻ‬ΊΔ౷‫ܭ‬తਪଌʢਪଌ౷‫ܭ‬ʣ• ඪຊ͔Β฼ूஂΛਪଌ͢Δํ๏

෼ࢄʢඪຊ෼ࢄʣ S 2 =ex. ఺ਪఆɼ۠ؒਪఆɼԾઆ‫ݕ‬ఆ

1(Xi − X̄)2n• ฼ूஂͷ෼ࢄʢ฼෼ࢄʣσ 2 Λਪఆ͢Δ΋ͷʢਪఆྔʣ• ‫͑׵͍ݴ‬Ε͹ɼ฼෼ࢄΛਪఆ͢ΔͨΊʹඪຊ෼ࢄΛ‫ٻ‬ΊΔҰൠతʹ͸ ‫ه‬ड़౷‫ ⇒ ܭ‬౷‫ܭ‬తਪଌ ͱֶͿ͕ɼ‫ه‬ड़౷‫ܭ‬Λਂ͘ཧղ͢ΔͨΊʹ͸౷‫ܭ‬తਪଌͷߟ͔͕͑ͨॏཁʂ
11 / 43

12 / 43฼தԝ஋ͱඪຊதԝ஋฼࠷ස஋ͱඪຊ࠷ස஋

標本中央値0.50.5

標本最頻値

母中央値

母最頻値ͦ΋ͦ΋ώετάϥϜࣗମ΋฼ूஂ෼෍Λਪଌͨ͠΋ͷʂ
13 / 43਺ཧ౷‫Ͱֶܭ‬͸14 / 43Ҏ্Λ౿·͑ͯதԝ஋ɼ࠷ස஋ɼ࢛෼Ґ਺ͳͲ΋฼ूஂͰͷ஋Λਪఆ͢Δ΋ͷ• ͋ΔύϥϝʔλΛਪఆ͢ΔͨΊͷਪఆྔ͸ແ਺ʹ͋Δ• ਪఆྔ͕ෳ਺ଘࡏ͢Δͷ͸ࣗવ• ͦͷ͏ͪͲΕΛ༻͍ΔͱΑ͍͔Λ༷ʑͳࢦඪͰධՁ͢Δ• ͲΕΛ༻͍Δ΂͖͔ɼ਺ཧ౷‫ֶܭ‬తʹ༏ྼΛ൑அͰ͖Δ฼ฏ‫ ۉ‬µ = E(X) ͷਪఆྔͷ୅දྫ• த౳‫ڭ‬ҭͰ͸ͱ͘ʹ࢖͍΍͢͞ͱ͍͏‫఺؍‬΋ॏཁ

n1!
Xin i=1• ඪຊฏ‫ ۉ‬X̄ =• τϦϜฏ‫ۉ‬ෆภੑҰகੑ༗ޮੑ

⇒ தֶ 2 ೥ͷஈ֊Ͱ͸฼ूஂͱඪຊʹ͍ͭͯະֶश• ‫ࠩޡ‬Λ൐͏͜ͱͷཧղ͸೉͍͠ϩόετੑ✞฼෼ࢄ σ 2 = V (X) = E[(X − µ)2 ] ͷਪఆྔͷ୅දྫ• ඪຊ෼ࢄ S 2 =• ෆภ෼ࢄ U 2 =n1!

n i=1(Xi − X̄)2n!

• ߴ౳ֶߍɾ਺ֶ I ͷԾઆ‫ݕ‬ఆͰͷཧղ͕େࣄ1. ʮ͜ͷ‫࢛͕ࣜࢉܭ‬෼Ґ਺Ͱ͋Δʯͱ͍͏Α͏ͳ‫ݻ‬ఆͷఆٛΛԡ͠෇͚Δ΍Γํ͸ආ͚Δ΂͖Ұகੑ

1(Xi − X̄)2n − 1 i=1☎

2. ʮूஂΛ࢛౳෼͢Δ఺Λ࢛෼Ґ਺ͱ‫Ϳݺ‬ɽͦΕΛσʔλ͔Β‫ࢉܭ‬ෆภੑҰகੑ

UMVU15 / 43

͢Δํ๏͸͍͔ͭ͋͘Δ͕ɼ͜͜Ͱ͸؆қతͳ͜ΕΛ༻͍Δɽ͜Μͳ͜ͱ͕ಡΈऔΕΔΑʯͱ͍͏ࢦಋΛ͢΂͖✝

✆16 / 43฼෼Ґ਺

X: ྦྷੵ෼෍ؔ਺ F (x) = P (X ≤ x) Λ΋ͭ֬཰ม਺p ෼Ґ਺ʢԼଆ 100p ύʔηϯτ఺ʣQ(p) = F −1 (p) = inf{x : F (x) ≥ p},0≤p≤1

࢛෼Ґ਺ͷఆٛ
●

●pp

●

●p●

●

F −1(p )●●F −1(p )

●

●

●

●

F −1(p )ਖ਼੔਺ m ʹର͠ɼi = 1, . . . , m − 1 ʹର͢Δ i/m ෼Ґ਺Λɼm Λ‫Ͱࣈ਺׽‬ද‫ͯ͠ه‬ୈ i m ෼Ґ਺ͱ΋͍͏
18 / 43฼࢛෼Ґ਺ͳ࢛ͥ෼Ґ਺Λߟ͑Δͷ͔࢛෼Ґ਺• ୈ 1 ࢛෼Ґ਺ Q1 = Q(1/4)ύϥϝτϦοΫͳ෼ੳ• ୈ 2 ࢛෼Ґ਺ Q2 = Q(2/4) = Q(1/2) = தԝ஋• ฼ूஂ෼෍ʹਖ਼‫ن‬෼෍ͳͲಛఆͷ෼෍ΛԾఆ• ୈ 3 ࢛෼Ґ਺ Q3 = Q(3/4)• ඪຊ͔Βɼฏ‫ۉ‬ɼ෼ࢄͳͲͷύϥϝʔλΛਪଌ฼ूஂ෼෍ F (x) ʹΑͬͯҰҙʹఆ·Δ• ෼෍ͷ‫ܗ‬΍ಛੑΛ༷ʑͳํ๏Ͱ஌Δ• தԝ஋ɿ෼෍ͷதԝͷҐஔ• ࠷ස஋ɿ෼෍ͷϐʔΫ• ෼Ґ਺ɼ࢛෼Ґ਺ɿ෼෍ͷ‫ܗ‬΍ϘϦϡʔϜκʔϯ• ີ౓ਪఆɿ෼෍ͷ‫ܗ‬

1/4

0.25
Q1• ฼ूஂ෼෍ʹԾఆΛஔ͔ͳ͍

3/4

pnorm(x)0.25ϊϯύϥϝτϦοΫͳ෼ੳ

Q1Q3

Q3
19 / 43ඪຊ෼Ґ਺

20 / 43͞·͟·ͳඪຊ࢛෼Ґ਺• X1 , . . . , Xn : F (x) ͔Βͷແ࡞ҝඪຊLangford (2006): 15 छྨͷਪఆྔ⇒ X(1) , . . . , X(n) : X1 , . . . , Xn ͷॱং౷‫ྔܭ‬(X(1) ≤ · · · ≤ X(n) )• ඪຊ p ෼Ґ਺ͷ p = 1/4, 3/4 ͷ৔߹Λ༻͍Δ͜ͱ͕Մೳྦྷੵ෼෍ؔ਺ʹ‫ํͮ͘ج‬๏ඪຊ p ෼Ґ਺‫ݧܦ‬෼෍ؔ਺Λྦྷੵ෼෍ؔ਺ͱࢥͬͯɼ฼࢛෼Ґ਺ͷཁྖͰඪຊ࢛෼Ґ਺ඪຊ࢛෼Ґ਺ Q̂1 , Q̂3 ΛఆΊΔํ๏ɽn = 4j + r, r = 0, 1, 2, 3 ͷͱ͖• ओʹॱং౷‫ྔܭ‬Λ༻͍ͯ฼෼Ґ਺ Q(p) Λਪఆ• ଟ͘ͷ΋ͷ͸ҎԼͷ‫ܗ‬Q̂(p) = (1 − ε)X(j) + ϵX(j+1)

Q̂1 = X(⌈n/4⌉) =ྡ઀͢Δ 2 ͭͷॱং౷‫ྔܭ‬ͷઢ‫ؒิܗ‬"X(j)X(j+1)

r=0r = 1, 2, 3Q̂3 = X(⌈3n/4⌉) = X(3j+r)

Hyndman & Fan (1996): 9 छྨͷਪఆྔΛ 6 ͭͷ‫͔఺؍‬Βൺֱ
21 / 43͞·͟·ͳඪຊ࢛෼Ґ਺

22 / 43͞·͟·ͳඪຊ࢛෼Ґ਺ྦྷੵ෼෍ؔ਺ʹ‫ํͮ͘ج‬๏ͷվྑʢCDF ʹ‫ͮ͘ج‬ਪఆྔʣn = 4j ͷͱ͖ʹதԝʹमਖ਼ɽQ̂1 =Q̂3 =""Mathematica ͷਪఆྔ
1
12 X(j) + 2 X(j+1)

X(j+1)

r=0r = 1, 2, 3

1
12 X(3j) + 2 X(3j+1)

ʲྫʳσʔλ͕ {1, 2, 3, 4, 5} ͷͱ͖ɼ5 × 1/4 + 0.5 = 1.75 ΑΓ

r=0r = 1, 2, 3X(3j+r)●

p = 1/4, 3/4 ʹରͯ͠ʮnp + 0.5 ൪໨ʯͷॱং౷‫͢ͱྔܭ‬Δํ๏ɽQ̂1 =●●●●●●●●

●

●
●

●

●

●

Q1n ≠ 4j●●

●

●

●

●

● ●● ●●●Q1

ʮ(n + 1)p ൪໨ʯͷॱং౷‫͢ͱྔܭ‬Δํ๏ɽ

●●

●●
●

●

●●

●●

●

Excel ͷ QUARTILE.EXC ؔ਺΍ MINITAB ͷਪఆྔ

●

●

●●

Excel ͷ QUARTILE.INC ؔ਺΍ R ͷσϑΥϧτͷਪఆྔ●●ʮ(n − 1)p + 1 ൪໨ʯͷॱং౷‫͢ͱྔܭ‬Δํ๏ɽ

Q1

n = 4j1
3
X(1) + X(2) = 1.75
44

修正した場合23 / 43

24 / 43͞·͟·ͳඪຊ࢛෼Ґ਺

͞·͟·ͳඪຊ࢛෼Ґ਺

Excel ͷ QUARTILE.EXC ؔ਺΍ MINITAB ͷਪఆྔ

Tukey ͷώϯδn = 4j + r, r = 0, 1, 2, 3 ͷͱ͖• ඪຊதԝ஋Λ‫ؚ‬Ήલ൒ͷσʔλͷதԝ஋ɼ‫ޙ‬൒ͷσʔλͷதԝ஋• ฼࢛෼Ґ਺ͷਪఆྔͱͯ͠༻͍Δ͜ͱ΋͋Δʢinclusive ͳਪఆྔʣQ̂1 = (1 − ε)X(⌈(n+1)/4⌉) + εX(⌈(n+1)/4⌉+1)"(1 − ε)X(j) + εX(j+1)
r = 0, 1, 2=
(1 − ε)X(j+1) + εX(j+2) r = 3

X (1) X (2) X (3) X (4) X (5) X (6) X (7) X (8) X (9) X (10) X (11) X (12) X (13)Q̂3 = (1 − δ)X(⌈3(n+1)/4⌉) + δX(⌈3(n+1)/4⌉+1)= (1 − δ)X(3j+r) + δX(3j+r+1)

͜͜Ͱ ε, δ ͸ ⌈ ⌉ ಺ͷ਺ͷখ਺෦:#$n+1
n+1−,44#$3(n + 1)
3(n + 1)δ=−44
ε=25 / 43

͞·͟·ͳඪຊ࢛෼Ґ਺

26 / 43͞·͟·ͳඪຊ࢛෼Ґ਺

Tukey ͷώϯδ• ඪຊதԝ஋Λ‫ؚ‬Ήલ൒ͷσʔλͷதԝ஋ɼ‫ޙ‬൒ͷσʔλͷதԝ஋n = 13n = 12• ฼࢛෼Ґ਺ͷਪఆྔͱͯ͠༻͍Δ͜ͱ΋͋Δʢinclusive ͳਪఆྔʣX (1)

X (6) X (7)X (2)X (12)↑ X (8)中央値

X (5)X (1) X (2) X (3) X (4) X (5) X (6) X (7) X (8) X (9) X (10) X (11) X (12) X (13)X (3) X (4)

X (7)
X (2)

X (6)
X (3)

X (5)

↑中央値

X (4)

↑下ヒンジ

X (5)

X (6)

X (3)

X (10)

X (5)

X (4)

↑下ヒンジ

↑
上ヒンジ

↑上ヒンジ

X (13)X (10)X (12)X (11)

↑上ヒンジ

X (1)

X (8)

X (2)

X (7)

X (3)

X (15)X (9)↑X (6) 中央値 X (10)X (4)X (5)↑下ヒンジ

27 / 43͞·͟·ͳඪຊ࢛෼Ґ਺

X (11)X (10)↑下ヒンジ

X (14)X (9)↑中央値X (12)n = 15

X (7)X (8)X (2)X (11)X (13)
X (8)↑中央値 X (9)X (4)n = 14X (1)

X (12)
X (9)

X (6)
X (3)

↑上ヒンジ

X (13)X (8)

X (7)X (2)X (11)
X (9) X (10)

↑
下ヒンジ

X (1)

X (1)

X (14)X (13)X (11X) (12)↑上ヒンジ

28 / 43͞·͟·ͳඪຊ࢛෼Ґ਺ߴ౳ֶߍͷ࢛෼Ґ਺• ඪຊதԝ஋Λ‫͍ͳ·ؚ‬લ൒ͷσʔλͷதԝ஋ɼ‫ޙ‬൒ͷσʔλͷதԝ஋• exclusive ͳਪఆྔ֤σʔληοτͷ (Q̂1 , Q̂3 )ਪఆྔ

n = 4j + r, r = 0, 1, 2, 3 ʹରͯ͠Q̂1 =Q̂3 =""1
12 X(j) + 2 X(j+1)
X(j+1)

inclusive
exclusive
CDF
Mathematica
QUARTILE.EXC
R ͷσϑΥϧτr = 0, 1r = 2, 31
12 X(3j+r) + 2 X(3j+r+1)
X(3j+r)r = 0, 1r = 2, 329 / 43

{1, 2, 3, 4}(1.5, 3.5)
(1.5, 3.5)
(1.5, 3.5)
(1.5, 3.5)
(1.25, 3.75)(1.75, 3.25){1, 2, 3, 4, 5}(2, 4)(1.5, 4.5)
(2, 4)(1.75, 4.25)(1.5, 4.5)
(2, 4){1, 2, . . . , 6}
(2, 5)(2, 5)(2, 5)(2, 5)(1.75, 5.25)(2.25, 4.75){1, 2, . . . , 7}
(2.5, 5.5)
(2, 6)(2, 6)(2.25, 5.75)(2, 6)(2.5, 5.5)

30 / 43਺஋࣮‫ ݧ‬1• ਖ਼‫ن‬෼෍ N (50, 102 ) ͔Βେ͖͞ n = 8, 9, . . . , 20, 30, 40 ͷσʔληοτΛൃੜʢ1,000,000 ճʣඪຊ࢛෼Ґ਺ͷධՁ• ฼࢛෼Ґ਺ Q1 ≈ 43.2551, Q3 ≈ 56.7449 ͔Βͷඪຊ࢛෼Ґ਺ͷฏ‫ۉ‬తͳζϨʢόΠΞεʣΛ‫ࢉܭ‬

Bias(Q̂j ) = E(Q̂j ) − Qj• ʢόΠΞε͕ͳ͍ਪఆྔΛෆภਪఆྔͱ͍͏ʣ

32 / 43਺஋࣮‫ ݧ‬1 ͷ݁Ռʢൈਮʣ

਺஋࣮‫ ݧ‬1 ͷ݁Ռʢൈਮʣ

ୈ 1 ࢛෼Ґ਺
n8910111213
141516inclusive
0.12 1.02 0.19 0.80 0.10 0.72 0.13 0.60 0.08
exclusive
0.12 −0.78 0.19 −0.54 0.10 −0.52 0.13 −0.41 0.08
CDF
0.12 1.03 0.19 −0.54 0.10 0.72 0.13 −0.41 0.08
Mathematica
0.12 0.12 0.19 0.13 0.10 0.13 0.09 0.08 0.05
QUARTILE.EXC −0.83 −0.78 −0.68 −0.54 −0.54 −0.52 −0.47 −0.41 −0.41
R
1.07 1.03 0.88 0.80 0.74 0.72 0.64 0.60 0.56

• n ͕େ͖͘ͳΔʹैͬͯόΠΞε͸‫ݮ‬গ܏޲

ୈ 3 ࢛෼Ґ਺n
8
910111213
141516inclusive−0.12 −1.02 −0.18 −0.79 −0.09 −0.71 −0.12 −0.59 −0.07
exclusive−0.12 0.78 −0.18 0.55 −0.09 0.52 −0.12 0.41 −0.07
CDF−0.12 −1.02 −0.18 0.55 −0.09 −0.71 −0.12 0.41 −0.07
Mathematica−0.12 −0.12 −0.18 −0.12 −0.09 −0.10 −0.12 −0.09 −0.07
QUARTILE.EXC 0.83 0.78 0.68 0.55 0.55 0.52 0.47 0.41 0.41
Rɹ
−1.07 −1.02 −0.88 −0.79 −0.73 −0.71 −0.64 −0.59 −0.56• inclusiveɼMathematicaɼR ͸ୈ 1 ࢛෼Ґ਺Λաେਪఆ• QUARTILE.EXC ͸ɼୈ 1 ࢛෼Ґ਺Λաখਪఆ• શମతͳ܏޲͸ୈ 1 ࢛෼Ґ਺ͱ΄΅ಉ͡

• exclusiveɼCDF ͸ n ͷ஋ʹΑͬͯόΠΞεͷํ޲͕൓స͢Δ• όΠΞεͷූ߸͕͢΂ͯ൓స• inclusiveɼexclusive ͸ n ͕‫਺ح‬ͷͱ͖ͷόΠΞε͕େ͖͍• n ͕େ͖͘ͳΔʹैͬͯόΠΞε͸‫ݮ‬গ܏޲• Mathematica ͸όΠΞε͕͔ͳΓখ͍͕͞ɼn = 4j + 2 ͷͱ͖ʹόΠΞε͕ए‫ׯ‬େ͖͍33 / 43਺஋࣮‫ ݧ‬234 / 43઴ۙతͳධՁ
Mudholkar and Huston (1997): QUARTILE.EXC ͷ઴ۙόΠΞε౳ͷಋग़• ਖ਼‫ن‬෼෍ N (50, 102 ) ͓ΑͼҰ༷෼෍ U (0, 100) ͔Βେ͖͞ n = 10ͷσʔληοτΛൃੜʢ1,000,000 ճʣ઴ۙόΠΞε• ฼࢛෼Ґ਺ Q1 ≈ 43.2551 ͔Βͷඪຊ࢛෼Ґ਺ͷόΠΞεɼඪ४‫ޡ‬3Q′′1B1 (ε)++ O(n−3 ),32n2048n23Q′′3B3 (δ)
Bias(Q̂3 ) =++ O(n−3 )32n2048n2
Bias(Q̂1 ) =ࠩɼฏํࠜฏ‫ۉ‬ೋ৐‫ࠩޡ‬Λ‫ࢉܭ‬

inclusive
exclusive
CDF
Mathematica
QUARTILE.EXC
RN (50, 102 )Bias
SE RMSE0.188 4.185
4.189
0.188 4.185
4.189
0.188 4.185
4.189
0.188 4.185
4.189−0.677 4.085
4.141
0.889 3.982
4.080

U (0, 100)
BiasSE
2.258 12.847
2.258 12.847
2.258 12.847
2.258 12.847−0.015 11.906
4.534 12.605

RMSE
13.04013.04013.04013.04011.906
13.395͜͜Ͱ(4)

B1 (ε) = 128 [8ε(1 − ε) − 3] Q′′1 + 64Q′′′
1 + 9Q1 ,(4)B3 (δ) = 128 [8δ(1 − δ) − 3] Q′′3 − 64Q′′′
3 + 9Q31
ͷ߲͸ Q′′ ͷΈʹґଘ͢Δn• ਖ਼‫ن‬෼෍ͷͱ͖ Q′′1 < 0, Q′′3 > 0 ΑΓόΠΞε΋ͦΕͧΕෛɼਖ਼• Ұ༷෼෍ͷͱ͖ɼQ′′1 = Q′′3 = 0 ΑΓ઴ۙόΠΞε͸ n−2 ͷΦʔμʔ•• ඪ४‫ࠩޡ‬΍ RMSE ͸Ͳͷํ๏Ͱ΋͋·ΓมΘΓ͕ͳ͍• ਖ਼‫ن‬෼෍ͱҰ༷෼෍Ͱશ͘ҟͳΔ܏޲
35 / 4336 / 43઴ۙతͳධՁ઴ۙతͳධՁ

ञંɾૣ࡚ʢ౤ߘ४උதʣ: exclusive ͳਪఆྔͷ઴ۙόΠΞε઴ۙόΠΞεʢQ̂1 ʣBias(Q̂1 )⎧(4)′′′−512Q′1 +64Q′′
8Q′1 +3Q′′
1 +112Q1 +9Q1
1⎪++ O(n−3 )⎪32n2048n2
⎪
⎪
(4)
⎪
⎨ 3Q′′1 + −128Q′′1 +64Q′′′−31 +9Q1+O(n)232n2048n=(4)′′′−512Q′1 −192Q′′
8Q′1 +3Q′′⎪
1 +112Q1 +9Q1
1⎪++ O(n−3 )⎪32n2048n2⎪⎪(4)′′′⎩ 3Q′′1−384Q′′+64Q+9Q−3111++O(n)32n2048n2

઴ۙόΠΞεʢQ̂3 ʣn = 4jBias(Q̂3 )⎧(4)′′′
512Q′3 +64Q′′−8Q′3 +3Q′′
3 −112Q3 +9Q33⎪++ O(n−3 )⎪32n2048n2⎪⎪(4)′′′′′′′⎪⎨ 3Q3 + −128Q3 −64Q3 +9Q3 + O(n−3 )32n2048n2=(4)′′′′′′
512Q3−192Q3−112Q3+9Q3−8Q′3 +3Q′′⎪
3⎪++ O(n−3 )⎪32n2048n2⎪⎪(4)′′′′⎩ 3Q3−384Q3 −64Q′′′−33 +9Q3++ O(n )32n2048n2

n = 4j + 1
n = 4j + 2n = 4j + 31
ͷ߲͸ Q′ ͱ Q′′ ʹґଘ͢Δn• ਖ਼‫ن‬෼෍ͷͱ͖ɼQ′1 > 0, Q′′1 < 0 Ͱ͋Γɼn ͕‫਺ح‬ͷͱ͖όΠΞε•n = 4jn = 4j + 1
n = 4j + 2
n = 4j + 3

ୈ 1 ࢛෼Ґ਺ͱಉ༷ͷ܏޲͸ফ͠߹͏ํ޲• Ұ༷෼෍ͷͱ͖ɼQ′1 > 0, Q′′1 = 0 ΑΓɼQ′ ʹΑΓόΠΞε͕૿͑Δ37 / 43

38 / 43͔͜͜ΒΘ͔Δ͜ͱ

QUARTILE.EXC ͷਪఆྔʹ͍ͭͯ• ਖ਼‫ن‬෼෍ͷͱ͖ Q1 ΛաখਪఆɼQ3 Λաେਪఆ⇒ ࢛෼ҐൣғΛ޿͘ਪఆ• Ұ༷෼෍ͷͱ͖όΠΞε͸ඇৗʹখ͍͞·ͱΊexclusive ͳਪఆྔʹ͍ͭͯ• ਖ਼‫ن‬෼෍ͷͱ͖ɼn ‫ͳ਺ۮ‬Β͹ɼQ1 ΛաେਪఆɼQ3 Λաখਪఆ⇒ ࢛෼ҐൣғΛ‫͘ڱ‬ਪఆ
QUARTILE.EXC ΑΓόΠΞεͷઈର஋͸খ͍͞• ਖ਼‫ن‬෼෍ͷͱ͖ɼn ‫ͳ਺ح‬Β͹ɼQ1 ΛաখਪఆɼQ3 Λաେਪఆ⇒ ࢛෼ҐൣғΛ޿͘ਪఆ
QUARTILE.EXC ͱશ͘ಉ͡ਪఆ݁Ռ• Ұ༷෼෍ͷͱ͖όΠΞε͸ QUARTILE.EXC ΑΓେ͖͍
39 / 43ຊ‫ڀݚ‬ͷ·ͱΊͱ͸͍͑‫ݱ‬৔ͷࠞཚ͕‫͍ͳ͖ى‬Α͏ɼ಺༰Λ੔ཧ͓͖͍ͯͨ͠ʂαʔϕΠ• ͳ࢛ͥ෼Ґ਺ͷఆ͕ٛෳ਺͋Δͷ͔⇒ ฼࢛෼Ґ਺ͷਪఆྔ͕͍ͭ͘΋ఏҊ͞Ε͍ͯΔ͔Βɽn ͰׂΔ෼ࢄͱ
n − 1 ͰׂΔ෼ࢄ͕͋Δͷͱಉ͡ɽ• தֶߍɾߴ౳ֶߍͰѻ͍ͬͯΔ΋ͷҎ֎ʹͲͷΑ͏ͳ΋ͷ͕͋Δ͔ɼͲͷΑ͏ͳҧ͍͕͋Δ͔• େࣄͳ͜ͱ͸࢛෼Ґ਺ͷ‫Ͱࢉܭ‬͸ͳ͍⇒ ώϯδ΍ Excel ͷํ๏ͳͲ঺հͨ͠ɽ҆৺࢛ͯ͠෼Ґ਺Λ࢖ͬͯ΄͍͠ʂ• େ·͔ʹ෼෍ͷ‫ܗ‬ঢ়΍ϘϦϡʔϜκʔϯͳͲΛಡΈऔΔͨΊʹ࢖͏Θ‫ڀݚ‬͚Ͱ͋Γɼࡉ͔ͳ਺ࣈͷҧ͍ΛͲ͏͜͏͍͏΂͖΋ͷͰ͸ͳ͍• ଞͷ࢛෼Ґ਺ͱൺ΂ͯ‫͠ࢉܭ‬΍͍͢ϝϦοτ͸͋Δ͕ɼଞʹϝϦοτɼσϝϦοτ͸ͳ͍ͷ͔⇒ όΠΞεͷઈର஋ࣗମ͸ Excel ͷ΋ͷΑΓখ͍͕͞ɼਖ਼ෛ͕ަ‫ʹޓ‬དྷΔ͜ͱʹ͸஫ҙɽ• ͦͷଞ‫ؾ‬Λ͚ͭΔ΂͖఺͸ͳ͍͔⇒ ࢛෼ҐൣғΛ n ͕ҟͳΔ‫Ͱ܈‬ൺֱ͢Δͱ͖ʹଟগ஫ҙ͕ඞཁɽ
41 / 43

42 / 43ࠓ‫͍ͯͭʹޙ‬• ࢛෼Ґ਺ʹ͍ͭͯ͸Ұఆͷ݁࿦Λಘͨͱ‫͍ͯ͡ײ‬Δʢinclusive ͳํ๏ͳͲʹ͍ͭͯ΋઴ۙόΠΞε͸ಘ͓͖͍͕ͯͨʣ• ଞʹ΋ɼཧ࿦తʹ੔ཧ͓ͯ͘͠΂͖಺༰͕͋ΔͷͰ͸ͳ͍͔ʁ ౷‫ܭ‬Ո͸΋ͬͱ͔ͬ͠ΓͱؔΘΔ΂͖Ͱ͸ʁ• ͢Ͱʹ͍͔ͭ͘͸ଞͷӃੜɼֶ෦ੜͨͪͱνϟϨϯδத• Կ͔ٙ໰ʹ‫͡ײ‬Δ͜ͱɼ਺ֶతʹ੔ཧ͓͖͍ͯͨ͜͠ͱͳͲ͋Ε͹͝ҙ‫͍ͩ͘͞ݟ‬ʂ43 / 43