ベータ分布の形状について 利用統計を見る 福岡大学機関リポジトリ E6212 0051

1

ং࿦

ඪ४తͳϕʔλ෼෍͸ɼ2ͭͷ฼਺Λ࣋ͭ۠ؒ(0,1)্ͷ࿈ଓܕͷ֬཰෼

෍Ͱ͋Δɽϕʔλ෼෍ͷີ౓ؔ਺ͷάϥϑ͸ɼ2ͭͷ฼਺ͷ஋ʹԠͯ͡ॊ

ೈʹͦͷܗঢ়ΛมԽͤ͞Δɽ͜ͷ2ͭͷ฼਺͸͍ͣΕ΋ܗঢ়฼਺ͱݺ͹Ε

Δɽຊߘ͸ɼϕʔλ෼෍ͷີ౓ؔ਺ͷάϥϑͷܗঢ়͕2ͭͷܗঢ়฼਺ʹԠ

ͯ͡ͲͷΑ͏ʹมԽ͢Δ͔Λݕ౼͠ɼͦͷ݁Ռͱͯ͠ɼ2ͭͷܗঢ়฼਺ʹ

Ԡͯ͡ϕʔλ෼෍Λ5ྨܕʢҰ༷෼෍ɼ୯ๆܕɼUࣈܕɼ୯ௐ૿Ճܕɼ୯

ௐݮগܕʣʹ෼ྨ͢Δɽ

ຊߘͷ໋୊3ͱͯ͠ఏࣔ͞Εͨ஌ݟ͸ɼϕʔλ෼෍ͷ฼਺ͱܗঢ়ͷؔ܎ʹؔ

͢ΔݶΓɼ֬཰෼෍ʹ͍ͭͯͷઐ໳తͳจݙͰ͋Δຸ୩ʢ2010ʣ΍Stuart and

Ord (1994), Johnson, Kotz and Balakrishnan (1995), Balakrishnan and Nevzorov (2003), Gupta and Nadarajah (2004), Forbes, Evans, Hastings and Peacock (2011), Krishnamoorthy (2016) ΑΓ΋ৄࡉ΋͘͠͸ମܥత

Ͱ͋Δɽྫ͑͹ɼຊߘ(2)ࣜͰఆٛ͞ΕΔඪ४ϕʔλ෼෍ͷີ౓ؔ਺ͷά

ϥϑͷܗঢ়ʹ͍ͭͯɼBalakrishnan and Nevzorov (2003, pp.140–141) ͸ (α−1)(β−1)<0ͷ৔߹ʹݴٴͤͣɼಉ͡৔߹ʹ͓͍ͯɼStuart and Ord (1994, pp.220–221)ͱForbes, Evans, Hastings and Peacock (2011, p.55), Krishnamoorthy (2016, p.247) ͸୯ʹJࣈܕͱݴٴ͢ΔͷΈͰ͋Δɽ· ͨɼα = 1͔ͭβ < 1ͷ৔߹ͷϕʔλ෼෍ͷܗঢ়ʹ͍ͭͯɼStuart and

∗_{෱Ԭେֶܦࡁֶ෦ɼE-mail: kagihara@fukuoka-u.ac.jp}

ベータ分布の形状について

原 理 人

Ord (1994, pp.220–221)͸Uࣈܕʹ෼ྨ͠ɼForbes, Evans, Hastings and Peacock (2011, p.55)ͱKrishnamoorthy (2016, p.247)͸୯ๆܕʹ෼ྨ͢

ΔɽͦͷҰํͰɼα >1͔ͭβ <1ͷ৔߹ʹ͍ͭͯ͸ɼઌड़ͷ௨Γɼ૒ํ

ͷจݙڞʹJࣈܕʹ෼ྨ͢Δɽ͜ͷ఺ɼα= 1͔ͭβ <1ͷ৔߹ͱα >1

͔ͭβ <1ͷ৔߹ͷϕʔλ෼෍ͷܗঢ়͸͍ͣΕ΋୯ௐ૿Ճܕͱͯ͠ݴٴ͢

Δํ͕෼ྨͷ࢓ํͱͯ͠Ұ؏ੑ͕͋Ζ͏ɽຸ୩ʢ2010, p.630ʣͱJohnson,

Kotz and Balakrishnan (1995, p.219), Gupta and Nadarajah (2004, p.41)

͸ɼ(α−1)(β−1)≤0ͷ৔߹ͷϕʔλ෼෍ͷܗঢ়ʹ͍ͭͯɼ۠ؒ(0,1)Ͱ

࠷େ஋΋࠷খ஋΋औΒͳ͍Jࣈܕ/ٯJࣈܕͷ෼෍Ͱ͋Δͱݴٴ͢ΔͷΈͰ

͋Δɽ͜ͷ఺ɼ୯ௐ૿Ճܕͱ୯ௐݮগܕʹ۠෼ͯ͠ݴٴ͢Δํ͕෼෍ͷܗ ঢ়ͷ෼ྨͱͯ͠͸໌շͰ͋Ζ͏ɽ

ͦͷଞɼ্هͷઐ໳తจݙҎ֎ʹ΋ϕʔλ෼෍ͷ฼਺ͱܗঢ়ͷؔ܎ʹݴٴ ͢Δจݙ͸ଘࡏ͢Δ1_{ɽ͔͠͠ɼྫ͑͹ɼFeller (1971, p.50)΍Casella and} Berger (2002, p.107)͸(α−1)(β−1)<0ͷ৔߹ͷܗঢ়ʹ͍ͭͯݴٴͤͣɼ ླ໦ʢ1978ɼp.67ʣ΍஛಺ଞฤʢ1989ɼp.38ʣɼத࠺ʢ2007ɼp.58ʣɼAzzalini (1996, p.273)͸͍ͣΕ΋α= 1͔ͭβ <1ͷ৔߹ͱα <1͔ͭβ= 1ͷ৔ ߹ʹݴٴ͠ͳ͍ɽ·ͨɼླ໦ʢ1987, pp.67–68ʣ΍౉෦ʢ1999, pp.76–77ʣ

͸ମܥతʹݴٴ͢Δ΋ͷͷਤࣔͷΈͰ͋Δɽͳ͓ɼӳޠ൛Wikipedia͸ϕʔ

λ෼෍ͷ฼਺ͱܗঢ়ͷؔ܎ʹ͍ͭͯৄࡉ͔ͭମܥతʹݴٴ͢Δ͕ɼͦͷ෼

ྨͷ࢓ํ͸ຊߘͱҟͳΔ2_ɽ

ҎԼͰ͸ɼୈ2અʹ͓͍ͯɼඪ४తͳϕʔλ෼෍ͱͦͷҐஔई౓෼෍଒

Λಋೖ͠ɼୈ3અʹ͓͍ͯɼϕʔλ෼෍ͷີ౓ؔ਺ͷάϥϑͷܗঢ়͕฼਺

ʹԠͯ͡ͲͷΑ͏ʹมԽ͢Δ͔Λݕ౼͢Δɽͦͷ݁Ռͱͯ͠ɼ্ड़ͷΑ͏

ʹɼ2ͭͷܗঢ়฼਺ʹԠͯ͡ϕʔλ෼෍Λ5ྨܕʹ෼ྨ͢Δʢ໋୊3ʣɽҎ

্ͷաఔͰಘΒΕΔີ౓ؔ਺ͷ૿ݮද͸શͯิ࿦Aʹఏࣔ͠ɼີ౓ؔ਺ͷ

άϥϑ͸શͯิ࿦Bʹਤࣔ͢Δɽ࠷ޙʹɼୈ4અͰ݁࿦Λड़΂Δɽ

1_{౷ܭֶͷཧ࿦ॻͰϕʔλ෼෍ʹݴٴ͠ͳ͍จݙΛ୳͢ͷ͸ࠔ೉Ͱ͋Ζ͏ɽ͔͠͠ɼͦͷҰ} ํͰɼϕʔλ෼෍ͷ฼਺ͱܗঢ়ͷؔ܎ʹମܥతʹݴٴ͢ΔจݙΛ୳͢ͷ͸ͦΕఔʹ༰қͰ͸ͳ ͍ɽ

2_{Wikipedia “Beta disitribution” (last edited on 31 January 2018 at 17:58)ࢀরɽ} URL: https://en.wikipedia.org/wiki/Beta distribution#Shapes

2

ඪ४ϕʔλ෼෍ͱͦͷҐஔई౓෼෍଒

࣮਺α, β, xʹରͯ͠ɼxͷؔ਺g(x) :=xα−1_(1−x)β−1_{Λߟ͑Δɽ͜͜}

Ͱɼ౳߸:=͸ͦͷࠨลΛͦͷӈลʹΑͬͯఆٛ͢Δ͜ͱΛҙຯ͢Δɽ͞

ͯɼα ≥ 1͔ͭβ ≥ 1ͷ৔߹ɼؔ਺g(x)͸༗քด۠ؒ[0,1]Ͱ࿈ଓͰ͋

ΔͷͰɼఆੵ෼1

0 x

α−1_(1−x)β−1_{dx͸ଘࡏ͢ΔɽͦͷҰํͰɼα <1·} ͨ͸β < 1ͷ৔߹ɼؔ਺g(x)͸ͦΕͧΕx → 0·ͨ͸x → 1ͷ࣌ʹ

g(x) → ∞ͱൃࢄ͢ΔͷͰɼg(x)͸۠ؒ(0,1)Ͱ༗քͰ͸ͳ͍ɽ͔͠͠ɼ

͜ͷ৔߹Ͱ΋ɼे෼ʹখͳΔ೚ҙͷਖ਼ͷ࣮਺ǫ, δ >0ʹରͯ͠༗քด۠ؒ

[0 +ǫ,1−δ]Λߟ͑Ε͹ɼؔ਺g(x)͸۠ؒ[0 +ǫ,1−δ]Ͱ࿈ଓͳͷͰఆੵ෼ 1−δ

0+ǫ x

α−1_(1−x)β−1_{dx͸ଘࡏ͢Δɽ͜͜Ͱɼα >0͔ͭβ >0Ͱ͋Ε͹ɼ}

޿ٛੵ෼1

0 x α−1

(1−x)β−1

dx := limǫ→0, δ→0 1−δ

0+ǫ x α−1

(1−x)β−1 dx ͸

ଘࡏ͢Δʢྫ͑͹ɼݘҪ1962ɼp.12΍ਿӜ1980ɼpp.295–296ࢀরʣɽैͬ

ͯɼ೚ҙͷਖ਼ͷ࣮਺α, β >0ʹରͯ͠ɼੵ෼1

0 x α−1

(1−x)β−1

dx͸ଘࡏ

͢Δɽ͜ͷੵ෼Λα, βͷؔ਺ͱݟ၏ͨ͠΋ͷΛϕʔλؔ਺B(α, β)ͱ͍͏ɿ

B(α, β) := 1

0 xα−1

(1−x)β−1

dx, α, β >0. (1)

೚ҙͷਖ਼਺α, β > 0ͱ(0,1)۠ؒ಺ͷ೚ҙͷ࣮਺ x ∈ (0,1) ʹରͯ͠ɼ

xα−1_(1−x)β−1_{>0Ͱ͋ΔͷͰɼϕʔλؔ਺͸ৗʹਖ਼஋ΛऔΔɽଈͪɼ೚}

ҙͷα, β >0ʹରͯ͠B(α, β)>0Ͱ͋Δɽ

ϕʔλ෼෍ͱ͸ɼϕʔλؔ਺Λਖ਼نԽఆ਺ͱͯ͠ີ౓ؔ਺Λߏ੒ͨ֬͠

཰෼෍Ͱ͋Δɽͭ·ΓɼҎԼͷ(2)ࣜͰఆٛ͞ΕΔؔ਺f0 : (0,1)→R͸

α, β >0Λ฼਺ͱ͢Δີ౓ؔ਺Ͱ͋Γɼf0Λີ౓ؔ਺ʹ࣋ͭ֬཰෼෍͸ϕʔ

λ෼෍ʢಛʹɼඪ४ϕʔλ෼෍ʣͱݺ͹ΕΔɽ·ͨɼαͱβ͸ܗঢ়฼਺ͱݺ

͹ΕΔʢBalakrishnan and Nevzorov 2003, Leemis and McQueston 2008, Forbes, Evans, Hastings and Peacock 2011, Krishnamoorthy 2016ࢀরʣɽ

f0(z|α, β) = 1 B(α, β)z

α−1

(1−z)β−1

, 0< z <1. (2)

࣮ࡍɼα, β >0ͱB(α, β)>0ʹ஫ҙ͢Δͱɼ೚ҙͷ࣮਺z ∈(0,1)ʹର ͯ͠ɼzα−1_(1−z)β−1_{>0Ͱ͋ΔͷͰf}

λؔ਺ͷఆٛʹ஫ҙ͢Δͱɼ1

0 f0(z|α, β)dz= 1ΛಘΔɽΑͬͯɼؔ਺f0 ͸ɼඇෛ஋ੑͱਖ਼نԽ৚݅Λຬͨ͢ͷͰɼ͔֬ʹີ౓ؔ਺Ͱ͋Δɽ

ͯ͞ɼඪ४ϕʔλ෼෍ʹҐஔ฼਺ͱई౓฼਺Λಋೖ͢Δ͜ͱʹΑͬͯɼඪ

४ϕʔλ෼෍ͷҐஔई౓෼෍଒͕ಘΒΕΔɽZΛඪ४ϕʔλ෼෍ʹै͏֬

཰ม਺Z ∼f0ͱͯ͠ɼ࣮਺μͱਖ਼ͷ࣮਺σʹରͯ͠ZΛ1࣍ม׵͢Δɽ

ଈͪɼX :=μ+σZ, −∞< μ <∞, σ >0ͱ͢Δɽ͜ͷ࣌ɼX ͷ֬཰෼

෍ΛμΛҐஔ฼਺ɼσΛई౓฼਺ͱ͢Δඪ४ϕʔλ෼෍ͷҐஔई౓෼෍଒

ͱ͍͏ɽ͜͜Ͱɼ0< Z <1ʹΑΓμ < X < μ+σʹ஫ҙ͢ΔͱɼXͷ෼ ෍ؔ਺F͸μ < x < μ+σͳΔxʹରͯ࣍͠ͷΑ͏ʹಘΒΕΔɿ

F(x) :=P(X≤x) =P

Z ≤x−μ

σ

= 1

B(α, β) x−_σµ

0

zα−1

(1−z)β−1 dz.

Αͬͯɼμ < x < μ+σͳΔxʹରͯ͠ɼXͷີ౓ؔ਺f͸෼෍ؔ਺Fͷ

ಋؔ਺ͱͯ͠ಘΒΕΔ3_ʢ0

< α, β, σ <∞, −∞< μ <∞ʣɿ

f(x|α, β, μ, σ) = 1 σB(α, β)

_x−μ σ

α−1

1−x−μ

σ β−1

. (3)

ҎԼͰ͸ɼfΛີ౓ؔ਺ʹ࣋ͭ֬཰෼෍ʢඪ४ϕʔλ෼෍ͷҐஔई౓෼෍

଒ʣΛҰൠܕϕʔλ෼෍ͱݺ΅͏ɽҼΈʹɼҰൠܕϕʔλ෼෍ͷظ଴஋͸ E(X) =μ+σα/(α+β)ɼ෼ࢄ͸V(X) =σ2_{αβ/[(α+β)}2_{(α+β+ 1)]ͱ}

ͳΔɽ࣮ࡍɼඪ४ϕʔλ෼෍ʹै͏֬཰ม਺Z ∼f0ʹ͍ͭͯɼͦͷظ଴஋

E(Z) =α/(α+β)ͱ෼ࢄV(Z) =αβ/[(α+β)2_{(α+β+ 1)]ʹ஫ҙ͢Δ}

ͱɼX =μ+σZͷظ଴஋ͱ෼ࢄ͸ͦΕͧΕE(X) =μ+σE(Z), V(Z) =

σ2_{V(Z)ͱٻ·Δɽ}

ͯ͞ɼα=β= 1Ͱ͋ΔͳΒ͹ɼҰൠܕϕʔλ෼෍͸۠ؒ(μ, μ+σ)্ͷҰ

༷෼෍ʹؼண͢Δɽ࣮ࡍɼB(1,1) =1

0 dx= 1ʹ஫ҙ͢Δͱɼf(x|1,1, μ, σ) = 3_{M:=µ+σ ⇐⇒ σ=M−µͳΔ฼਺ม׵Λࢪ͢ͱɼX:=µ+σZ=µ+ (M−µ)Z} ͷີ౓ؔ਺f͸ɼµ < x < MͳΔxʹରͯ࣍ࣜ͠Ͱ༩͑ΒΕΔʢ0< α, β <∞, −∞< µ < M <∞ʣɿ

f(x|α, β, µ, M) = 1

B(α, β)

(x−µ)α−1

(M−x)β−1 (M−µ)α+β−1 .

1/σΛಘΔɽ·ͨɼμ= 0͔ͭσ= 1ͷ࣌ɼҰൠܕϕʔλ෼෍͸ඪ४ϕʔ λ෼෍ʹؼண͢Δɿf(x|α, β,0,1) =f0(x|α, β).

3

ϕʔλ෼෍ͷܗঢ়

ຊઅͰ͸ɼϕʔλ෼෍ͷີ౓ؔ਺ͷάϥϑͷܗঢ়ʹ͍ͭͯߟ࡯͢Δɽͦ ͷࡍɼ(3)ࣜͰఆٛ͞ΕΔҰൠܕϕʔλ෼෍ͷີ౓ؔ਺f(x|α, β, μ, σ), μ < x < μ+σͱ(2)ࣜͰఆٛ͞ΕΔඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0(z|α, β), 0< z <1ͱͷؒʹҎԼͷؔ܎͕ࣜ੒ཱ͢Δ͜ͱʹ஫ҙ͢Δɽୠ͠ɼҎԼͰ͸ɼ f(x) :=f(x|α, β, μ, σ), f0(z) :=f0(z|α, β)ͱུه͢Δɽ

f(x) = 1 σf0

x−μ

σ

.

͜ΕʹΑΓɼͦΕΒͷ1֊ಋؔ਺ͱ2֊ಋؔ਺ʹ͍ͭͯ͸

f′

(x) = 1 σ2f

′

0

_x−μ σ

, f′′

(x) = 1 σ3f

′′

0

_x−μ σ

ͳΔؔ܎͕੒ཱ͠ɼͦͯ͠ɼσ >0ʹ஫ҙ͢ΔͱҎԼΛಘΔɿ

f′

(x)≶0 ⇐⇒ f′

0

x−μ σ

≶0, f′′

(x)≶0 ⇐⇒ f′′

0

x−μ σ

≶0.

Αͬͯɼඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0(z)ͷάϥϑ{(z, f0(z))|z∈(0,1)}

ͷܗঢ়͕൑໌͢Ε͹ɼͦͷಠཱม਺zͷ஋Λx=σz+μͱஔ͖׵্͑ͨ

Ͱؔ਺஋f0(z)Λ1/σഒ͢Δ͜ͱʹΑͬͯɼҰൠܕϕʔλ෼෍ͷີ౓ؔ਺

f(x)ͷάϥϑ{(x, f(x))|x∈(μ, μ+σ)}ͷܗঢ়΋൑໌͢Δͱ෼͔Δɽैͬ

ͯɼҎԼͰ͸ɼ(2)ࣜͰఆٛ͞ΕΔඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0(z)ͷάϥ

ϑͷܗঢ়ʹ͍ͭͯߟ࡯͢Δʢ0< z < 1ʣɽͦͷࡍɼܗঢ়฼਺α, β >0ʹ

͍ͭͯɼα= 1·ͨ͸β = 1ͷ৔߹ʢୈ3.1અʣͱα= 1͔ͭβ= 1ͷ৔

߹ʢୈ3.2અʣͷ2ͭͷ৔߹ʹ෼͚ͯߟ࡯্ͨ͠ͰɼಘΒΕͨ஌ݟΛୈ3.3

અͰ૯߹͢Δɽ

3.1

_α

_{= 1}

·ͨ͸

_β

_{= 1}

ͷ৔߹

ຊઅͰ͸ɼα= 1·ͨ͸β= 1ͷ৔߹ʹ͓͍ͯʢα, β >0ʣɼඪ४ϕʔλ

෼෍ͷີ౓ؔ਺f0(z)ͷάϥϑͷܗঢ়ʹ͍ͭͯߟ࡯͢Δʢ0< z <1ʣɽ·

ͣɼα= 1͔ͭβ= 1ͷ৔߹ɼطʹୈ2અͷ࠷ޙͰ࿦ͨ͡Α͏ʹɼඪ४ϕʔ

λ෼෍͸۠ؒ(0,1)্ͷҰ༷෼෍ʹؼண͢Δɽ࣍ʹɼα·ͨ͸βͷ͍ͣΕ

͔Ұํ͕1Ͱͳ͍ͳΒ͹ɼඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0(z)ͱͦͷ1֊ಋ

ؔ਺f′

0(z)ͱ2֊ಋؔ਺f

′′

0(z)͸ҎԼͷΑ͏ʹٻ·Δɽଈͪɼα= 1͔ͭ

β= 1ͷ৔߹ɼ0< z <1ʹରͯ͠ɼ

f0(z) =β(1−z)β

−1 , f′

0(z) =−β(β−1)(1−z) β−2

, f′′

0(z) =β(β−1)(β−2)(1−z) β−3

Λಘͯɼα= 1͔ͭβ = 1ͷ৔߹ɼ0< z <1ʹରͯ͠ҎԼΛಘΔɿ

f0(z) =αzα

−1 , f′

0(z) =α(α−1)z α−2

, f′

0(z) =α(α−1)(α−2)z α−3

.

Αͬͯɼඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0(z)ͷ૿ݮද͸ɼα= 1͔ͭβ= 1ͷ

৔߹ʹද2ɼα= 1͔ͭβ = 1ͷ৔߹ʹද3ͱͯ͠ಘΒΕΔʢิ࿦Aʣɽ͜Ε

ʹΑΓɼඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0(z)ͷάϥϑͷܗঢ়ʹؔͯ͠ɼα= 1

·ͨ͸β= 1ͷ৔߹ʹ͓͚Δ஌ݟ͕ಘΒΕΔʢ໋୊1ʣɽ

໋୊ 1(ඪ४ϕʔλ෼෍ͷܗঢ়ɿα_{= 1}·ͨ͸_{β = 1}ͷ৔߹). _{α= 1}·ͨ

͸β= 1ͷ৔߹ʢα, β >0ʣɼ(2)ࣜͰఆٛ͞ΕΔඪ४ϕʔλ෼෍ͷີ౓ؔ

਺f0ͷάϥϑ{(z, f0(z))| z∈(0,1)}ͷܗঢ়ʹ͍ͭͯҎԼ͕੒ཱ͢Δɽਤ

ʹ͍ͭͯ͸ิ࿦BΛࢀর͞Ε͍ͨɽ

α= 1͔ͭβ= 1ͷ৔߹ʢਤ1ʣີ౓ؔ਺_f0͸Ұఆ஋ΛऔΔʢҰ༷෼෍ʣɿ

f0(z) = 1, ∀z∈(0,1). ैͬͯɼf0ͷάϥϑͷܗঢ়͸ਫฏͱͳΔɽ

α= 1͔ͭβ= 1ͷ৔߹ ີ౓ؔ਺f0͸୯ௐؔ਺Ͱ͋Δɿ

1. α= 1͔ͭβ <1ͷ৔߹ʢਤ8ʣɿf0͸୯ௐ૿Ճͷತؔ਺Ͱ͋Γɼ

ͦͷάϥϑͷܗঢ়͸Լʹತͷӈ্ΓͰ͋Δɽ·ͨɼz →0ͷ࣌

ʹf0(z)→βɼz→1ͷ࣌ʹf0(z)→ ∞Ͱ͋Δɽ

2. α= 1͔ͭβ >1ͷ৔߹ʢਤ11ʣɿf0͸୯ௐݮগؔ਺Ͱ͋Γɼͦ

ͷάϥϑͷܗঢ়͸ӈԼΓͰ͋Δɽ·ͨɼz→0ͷ࣌ʹf0(z)→βɼ

x→1ͷ࣌ʹf0(z)→0Ͱ͋Δɽ

(a) β <2ͷ৔߹ɼf0͸୯ௐݮগͷԜؔ਺Ͱ͋Γɼͦͷάϥϑ ͷܗঢ়͸্ʹತͷӈԼΓͰ͋Δɽ

(b) β = 2ͷ৔߹ɼf0ͷάϥϑͷܗঢ়͸ӈԼΓͷ௚ઢͰ͋Δɽ

(c) β >2ͷ৔߹ɼf0͸୯ௐݮগͷತؔ਺Ͱ͋Γɼͦͷάϥϑ ͸ԼʹತͷӈԼΓͰ͋Δɽ

α= 1͔ͭβ = 1ͷ৔߹ ີ౓ؔ਺f0͸୯ௐؔ਺Ͱ͋ΔʢϕΩؔ਺෼෍/ϕ

Ω෼෍ʣɿ

1. α <1͔ͭβ = 1ͷ৔߹ʢਤ12ʣɿf0͸୯ௐݮগͷತؔ਺Ͱ͋

Γɼͦͷάϥϑͷܗঢ়͸ԼʹತͷӈԼΓͰ͋Δɽ·ͨɼz→0ͷ

࣌ʹf0(z)→ ∞ɼz→1ͷ࣌ʹf0(z)→αͰ͋Δɽ

2. α >1͔ͭβ= 1ͷ৔߹ʢਤ9ʣɿf0͸୯ௐ૿Ճؔ਺Ͱ͋Γɼͦͷ

άϥϑͷܗঢ়͸ӈ্ΓͰ͋Δɽ·ͨɼz→0ͷ࣌ʹf0(z)→0ɼ

z→1ͷ࣌ʹf0(z)→αͰ͋Δɽ

(a) α <2ͷ৔߹ɼf0͸୯ௐ૿ՃͷԜؔ਺Ͱ͋Γɼͦͷάϥϑ ͷܗঢ়͸্ʹತͷӈ্ΓͰ͋Δɽ

(b) α= 2ͷ৔߹ɼf0ͷάϥϑͷܗঢ়͸ӈ্Γͷ௚ઢͰ͋Δɽ

(c) α >2ͷ৔߹ɼf0͸୯ௐ૿Ճͷತؔ਺Ͱ͋Γɼͦͷάϥϑ ͷܗঢ়͸Լʹತͷӈ্ΓͰ͋Δɽ

3.2

_α

_{= 1}

͔ͭ

_β

_{= 1}

ͷ৔߹

ຊઅͰ͸ɼα= 1͔ͭβ = 1ͷ৔߹ʹ͓͍ͯʢα, β >0ʣɼඪ४ϕʔλ෼

෍ͷີ౓ؔ਺f0(z)ͷάϥϑͷܗঢ়ʹ͍ͭͯߟ࡯͢Δʢ0< z <1ʣɽα= 1

͔ͭβ= 1ͷ࣌ɼඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0ͷ1֊ඍ෼͸

f′

0(z) = 1 B(α, β)z

α−2

(1−z)β−2

[(2−α−β)z+α−1]

ͱͳΓɼͦͷූ߸͸ҎԼͷΑ͏ʹDͷූ߸Ͱܾఆ͞ΕΔɿ

f′

0(z)≶0 ⇐⇒ D:= [(2−α−β)z+α−1]≶0 ⇐⇒ (2−α−β)z≶1−α.

α= 1͔ͭβ= 1ͷ࣌ɼDͷූ߸͸฼਺αͱβʹԠͯ͡ҎԼͷΑ͏ʹܾఆ

͞ΕΔʢα, β >0ʣɿ

1. α+β = 2ͷ৔߹ɼD=α−1Ͱ͋ΔɽैͬͯɼҎԼΛಘΔɿ

D≶0 ⇐⇒ α≶1.

2. α+β <2ͷ৔߹ɼ2−α−β >0Ͱ͋ΓҎԼΛಘΔɿ

D≶0 ⇐⇒ z≶ 1−α

2−α−β =:z

∗

.

͜͜Ͱɼz∗

≷0 ⇐⇒ 1−α≷0 ⇐⇒ α≶1Ͱ͋Γɼz∗

≶1 ⇐⇒

β≶1Ͱ͋Δ͜ͱʹ஫ҙ͢ΔͱɼҎԼΛಘΔɿ

(a) α <1͔ͭβ <1ͳΒ͹0< z∗

<1ͱͳΓɼz < z∗_Ͱ

D <0ɼ z > z∗_Ͱ

D >0Ͱ͋Δɽ

(b) α > 1ͳΒ͹ʢ͜ͷ࣌β < 1Ͱ͋Δʣz∗

< 0ͱͳΓɼ೚ҙͷ

z∈(0,1)ͰD >0Ͱ͋Δɽ

(c) β > 1ͳΒ͹ʢ͜ͷ࣌α < 1Ͱ͋Δʣz∗

> 1ͱͳΓɼ೚ҙͷ

z∈(0,1)ͰD <0Ͱ͋Δɽ

3. α+β >2ͷ৔߹ɼ2−α−β <0Ͱ͋ΓҎԼΛಘΔɿ

D≶0 ⇐⇒ z≷ α−1

α+β−2 =

1−α 2−α−β =z

∗

.

͜͜Ͱɼz∗

≷0 ⇐⇒ α−1≷0 ⇐⇒ α≷1Ͱ͋Γɼz∗

≶1 ⇐⇒

β≷1Ͱ͋Δ͜ͱʹ஫ҙ͢ΔͱɼҎԼΛಘΔɿ

(a) α >1͔ͭβ >1ͳΒ͹0< z∗

<1ͱͳΓɼz < z∗

ͰD >0ɼ z > z∗_Ͱ

(b) α < 1ͳΒ͹ʢ͜ͷ࣌β > 1Ͱ͋Δʣz∗

< 0ͱͳΓɼ೚ҙͷ

z∈(0,1)ͰD <0Ͱ͋Δɽ

(c) β < 1 ͳΒ͹ʢ͜ͷ࣌α > 1Ͱ͋Δʣz∗

> 1 ͱͳΓ೚ҙͷ

z∈(0,1)ͰD >0Ͱ͋Δɽ

Ҏ্Λ·ͱΊΔͱɼα= 1͔ͭβ= 1ͷ࣌ɼҎԼΛಘΔʢα, β >0ʣɿ

1. α <1͔ͭβ <1ͷ৔߹ɿD≶0 ⇐⇒ z≶z∗

.

2. α <1͔ͭβ >1ͷ৔߹ɿ∀z∈(0,1), D <0. 3. α >1͔ͭβ <1ͷ৔߹ɿ∀z∈(0,1), D >0. 4. α >1͔ͭβ >1ͷ৔߹ɿD≶0 ⇐⇒ z≷z∗

.

Ҏ্ͷٞ࿦ʹΑΓɼඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0(z)ͷ૿ݮද͸ɼα <1

͔ͭβ < 1ͷ৔߹ʹද4ɼα < 1͔ͭβ > 1ͷ৔߹ʹද5ɼα > 1͔ͭ β <1ͷ৔߹ʹද6ɼα >1͔ͭβ >1ͷ৔߹ʹද7ͱͯ͠ಘΒΕΔʢิ࿦

Aʣɽ͜ΕʹΑΓɼඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0(z)ͷάϥϑͷܗঢ়ʹؔ͠

ͯɼα= 1͔ͭβ= 1ͷ৔߹ʹ͓͚Δ஌ݟ͕ಘΒΕΔʢ໋୊2ʣɽ

໋୊ 2 (ඪ४ϕʔλ෼෍ͷܗঢ়ɿα_{= 1}͔ͭ_{β = 1}ͷ৔߹). _{α= 1}͔ͭ

β = 1ͷ৔߹ʢα, β >0ʣɼ(2)ࣜͰఆٛ͞ΕΔඪ४ϕʔλ෼෍ͷີ౓ؔ਺

f0ͷάϥϑ{(z, f0(z))| z ∈(0,1)}ͷܗঢ়ʹ͍ͭͯҎԼ͕੒ཱ͢Δɽਤʹ

͍ͭͯ͸ิ࿦BΛࢀর͞Ε͍ͨɽ

α <1͔ͭβ <1ͷ৔߹ʢਤ5ɼਤ6ɼਤ7ʣ_{ີ౓ؔ਺f0ͷάϥϑ͸Uࣈ}

ܕͷܗঢ়Λࣔ͢ɽͭ·Γɼf0(z)͸z=z∗Ͱ࠷খ஋ΛऔΓɼͦͷάϥ

ϑͷܗঢ়͸z < z∗_{ͰӈԼΓɼz > z}∗_{Ͱӈ্ΓͱͳΔɽ·ͨɼz}

→0

·ͨ͸z→1ͷ࣌ɼf0(z)→ ∞Ͱ͋Δɽ

z∗

= 1−α 2−α−β =

1−α (1−α) + (1−β).

෼෍ͷ࠷খ఺z∗

͸ɼα=βͷ৔߹ɼz∗

= 1/2ʹΑΓ۠ؒ(0,1)ͷਅ தʹҐஔ͠ʢਤ5ʣɼα < βͷ৔߹ɼ1−α > 1−β ⇒z∗

ΑΓ۠ؒ(0,1)ͷӈدΓʹҐஔ͠ʢਤ6ʣɼα > βͷ৔߹ɼ1−α < 1−β ⇒z∗

<1/2ʹΑΓ۠ؒ(0,1)ͷࠨدΓʹҐஔ͢Δʢਤ7ʣɽ

α <1͔ͭβ >1ͷ৔߹ʢਤ13ʣີ౓ؔ਺_f0͸୯ௐݮগؔ਺Ͱ͋Γɼͦ

ͷάϥϑͷܗঢ়͸ӈԼΓͰ͋Δɽ·ͨɼz → 0ͷ࣌ʹf0(z) → ∞ɼ

z →1ͷ࣌ʹf0(z)→0Ͱ͋Δɽୠ͠ɼz →1ʹ͓͚Δf0(z) →0

΁ͷऩଋ͸ɼβ <2Ͱ܏͖͕−∞ɼβ = 2Ͱ܏͖͕ఆ਺−α(α+ 1)ɼ

β >2Ͱ܏͖͕0ͱ͍͏ܗΛऔΔɽ

α >1͔ͭβ <1ͷ৔߹ʢਤ10ʣີ౓ؔ਺_f0͸୯ௐ૿Ճؔ਺Ͱ͋Γɼͦͷ

άϥϑͷܗঢ়͸ӈ্ΓͰ͋Δɽ·ͨɼz→0ͷ࣌ʹf0(z)→0ɼz→1

ͷ࣌ʹf0(z)→ ∞Ͱ͋Δɽୠ͠ɼz→0ʹ͓͚Δf0(z)→0΁ͷऩ

ଋ͸ɼα <2Ͱ܏͖͕+∞ɼα= 2Ͱ܏͖͕ఆ਺β(β+ 1)ɼα >2Ͱ

܏͖͕0ͱ͍͏ܗΛऔΔɽ

α >1͔ͭβ >1ͷ৔߹ʢਤ2ɼਤ3ɼਤ4ʣີ౓ؔ਺_f0ͷάϥϑ͸୯ๆ

ܕͷܗঢ়Λࣔ͢ɽͭ·Γɼf0(z)͸z=z

∗

Ͱ࠷େ஋ΛऔΓʢଈͪɼz∗

͸࠷ස஋ʣɼͦͷάϥϑͷܗঢ়͸z < z∗_{Ͱӈ্Γɼz > z}∗_ͰӈԼΓ

ͱͳΔɽ·ͨɼz→0·ͨ͸z→1ͷ࣌ɼf0(z)→0Ͱ͋Δ4ɽ

z∗

= α−1 α+β−2 =

α−1 (α−1) + (β−1).

෼෍ͷ࠷େ఺ʢ࠷ස஋ʣz∗_͸ɼα

= β ͷ৔߹ɼz∗

= 1/2ʹΑΓ۠ ؒ(0,1)ͷਅதʹҐஔ͠ʢਤ2ʣɼα < βͷ৔߹ɼα−1< β−1⇒

z∗

<1/2ʹΑΓ۠ؒ(0,1)ͷࠨدΓʹҐஔ͠ʢਤ3ʣɼα > βͷ৔߹ɼ α−1> β−1⇒z∗

>1/2ʹΑΓ۠ؒ(0,1)ͷӈدΓʹҐஔ͢Δʢਤ

4ʣɽ

4_z

→0ʹ͓͚Δf0(z)→0΁ͷऩଋ͸ɼα <2Ͱ܏͖͕+∞ɼα= 2Ͱ܏͖͕ఆ਺

β(β+ 1)ɼα >2Ͱ܏͖͕0ͱ͍͏ܗΛऔΓʢਤ2ࢀরʣɼz→1ʹ͓͚Δf0(z)→0΁ͷ ऩଋ͸ɼβ <2Ͱ܏͖͕−∞ɼβ= 2Ͱ܏͖͕ఆ਺−α(α+ 1)ɼβ >2Ͱ܏͖͕0ͱ͍͏ܗ ΛऔΔʢਤ2ɼਤ3ɼਤ4ࢀরʣɽ

3.3

ϕʔλ෼෍ͷܗঢ়

Ҏ্ͷٞ࿦Λ౿·͑Δͱɼϕʔλ෼෍ͷ฼਺ͱܗঢ়ͷؔ܎ʹ͍ͭͯɼҎԼ

ͷ໋୊3ͱ໋୊4ΛಘΔɽ·ͨɼද1͸໋୊3Λදʹ·ͱΊͨ΋ͷͰ͋Δɽ

໋୊3(ඪ४ϕʔλ෼෍ͷܗঢ়). ₍₂₎ࣜͰఆٛ͞ΕΔඪ४ϕʔλ෼෍ͷີ౓

ؔ਺f0 ͷάϥϑ{(z, f0(z))|z∈(0,1)}ͷܗঢ়͸ɼਖ਼஋ͷܗঢ়฼਺αͱβ

ʹԠͯ͡ʢα, β > 0ʣɼҎԼͷΑ͏ʹ5ͭͷܕʹ෼ྨͰ͖Δʢද1ࢀরʣɽ

ਤʹ͍ͭͯ͸ิ࿦BΛࢀর͞Ε͍ͨɽ

1. α=β= 1ͷ৔߹ʢৄ͘͠͸໋୊1ࢀরʣɿҰ༷෼෍ʢਤ1ʣɽ

2. α >1͔ͭβ >1ͷ৔߹ʢৄ͘͠͸໋୊2ࢀরʣɿ୯ๆܕͷܗঢ়ɽ෼

෍ͷ࠷େ఺ʢ࠷ස஋ʣz∗_͸

(α−1)/(α+β−2)Ͱɼα=β ͳΒ͹ z∗_{= 1/2ʢਤ}

2ʣɼα≶βʹԠͯ͡z∗

≶1/2ʢਤ3ɼਤ4ʣͰ͋Δɽ·

ͨɼz→0·ͨ͸z→1ͷ࣌ɼf0(z)→0Ͱ͋Δɽ

3. α <1͔ͭβ <1ͷ৔߹ʢৄ͘͠͸໋୊2ࢀরʣɿUࣈܕͷܗঢ়ɽ෼

෍ͷ࠷খ఺z∗_͸

(1−α)/(2−α−β)Ͱɼα=βͳΒ͹z∗_{= 1/2ʢਤ}

5ʣɼα≶βʹԠͯ͡z∗

≷1/2ʢਤ6ɼਤ7ʣͰ͋Δɽ·ͨɼz→0·

ͨ͸z→1ͷ࣌ɼf0(z)→ ∞Ͱ͋Δɽ

4. α≥1͔ͭβ ≤1ͷ৔߹ʢα=βʣɿ୯ௐ૿ՃܕͰӈ্Γͷܗঢ়ɽ

(a) α= 1͔ͭβ <1ͷ৔߹ʢৄ͘͠͸໋୊1ࢀরʣɿz→0ͷ࣌ʹ f0(z)→βɼz→1ͷ࣌ʹf0(z)→ ∞Ͱ͋Δʢਤ8ʣɽ

(b) α >1͔ͭβ = 1ͷ৔߹ʢৄ͘͠͸໋୊1ࢀরʣɿz→0ͷ࣌ʹ f0(z)→0ɼz→1ͷ࣌ʹf0(z)→αͰ͋Δ5ʢਤ9ʣɽ

(c) α >1͔ͭβ <1ͷ৔߹ʢৄ͘͠͸໋୊2ࢀরʣɿz→0ͷ࣌ʹ f0(z)→0ɼz→1ͷ࣌ʹf0(z)→ ∞Ͱ͋Δʢਤ10ʣɽ

5. α≤1͔ͭβ ≥1ͷ৔߹ʢα=βʣɿ୯ௐݮগܕͰӈԼΓͷܗঢ়ɽ

5_{͜ͷ৔߹ɼz∈[0,1]ͱ͢Ε͹z= 1Λ࠷ස஋ͱ͢Δ୯ๆܕʹ෼ྨ͢Δ͜ͱ΋Ͱ͖Δɽ}

(a) α= 1͔ͭβ >1ͷ৔߹ʢৄ͘͠͸໋୊1ࢀরʣɿz→0ͷ࣌ʹ f0(z)→βɼz→1ͷ࣌ʹf0(z)→0Ͱ͋Δ6ʢਤ11ʣɽ

(b) α <1͔ͭβ = 1ͷ৔߹ʢৄ͘͠͸໋୊1ࢀরʣɿz→0ͷ࣌ʹ f0(z)→ ∞ɼz→1ͷ࣌ʹf0(z)→αͰ͋Δʢਤ12ʣɽ

(c) α <1͔ͭβ >1ͷ৔߹ʢৄ͘͠͸໋୊2ࢀরʣɿz→0ͷ࣌ʹ f0(z)→ ∞ɼz→1ͷ࣌ʹf0(z)→0Ͱ͋Δʢਤ13ʣɽ

·ͨɼඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0͸ɼα≥1͔ͭβ ≥1ͷ৔߹ʹ༗քͰ

͋Γɼα <1·ͨ͸β <1ͷ৔߹ʹ༗քͰ͸ͳ͍ɿ

1. z→0ͷ࣌ɼα <1ͷ৔߹͸f0(z)→ ∞ɼα= 1ͷ৔߹͸f0(z)→βɼ α >1ͷ৔߹͸f0(z)→0ͱͳΔɽ

2. z→1ͷ࣌ɼβ <1ͷ৔߹͸f0(z)→ ∞ɼβ = 1ͷ৔߹͸f0(z)→αɼ β >1ͷ৔߹͸f0(z)→0ͱͳΔɽ

ҼΈʹɼα=βͷ৔߹ɼඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0(z)͸ɼ

f0(z) = 1

B(α, α)[z(1−z)] α−1

= 1

B(α, α)

1 4 −

z−1

2 2α

−1

ͱͳΔͷͰɼz= 1/2Λத৺ͱͯ͠ࠨӈରশͱͳΔɽ

ද1: ඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0(z)ͷάϥϑͷܗঢ়ͷ෼ྨ

β <1 : β= 1 : β >1 : f(z)−−−→ ∞z→1 f(z)−−−→z→1 α f(z)−−−→z→1 0

α <1 : f(z)−−−→ ∞z→0 U ࣈܕ ୯ௐݮগܕ ୯ௐݮগܕ α= 1 : f(z)−−−→z→0 β ୯ௐ૿Ճܕ Ұ༷෼෍ ୯ௐݮগܕ α >1 : f(z)−−−→z→0 0 ୯ௐ૿Ճܕ ୯ௐ૿Ճܕ ୯ๆܕ

6_{͜ͷ৔߹ɼz∈[0,1]ͱ͢Ε͹z= 0Λ࠷ස஋ͱ͢Δ୯ๆܕʹ෼ྨ͢Δ͜ͱ΋Ͱ͖Δɽ}

໋୊4(Ұൠܕϕʔλ෼෍ͷܗঢ়). ₍₃₎ࣜͰఆٛ͞ΕΔҰൠܕϕʔλ෼෍ͷ

ີ౓ؔ਺f ͷάϥϑ{(x, f(x))|x∈(μ, μ+σ), μ∈(−∞,∞), σ∈(0,∞)}

ͷܗঢ়͸ɼਖ਼஋ͷܗঢ়฼਺αͱβʹԠͯ͡ʢα, β >0ʣɼ໋୊3ͱಉ͡ܗͰ

5ͭͷܕʹ෼ྨͰ͖Δɽୠ͠ɼີ౓ؔ਺f ͷάϥϑͷܗঢ়͕୯ๆܕʢα >1

͔ͭβ >1ʣ·ͨ͸Uࣈܕʢα <1͔ͭβ <1ʣͳΔ৔߹ʹ͓͍ͯɼີ౓

ؔ਺f(x)ͷ࠷େ఺ʢଈͪɼ࠷ස஋ʣ΋͘͠͸࠷খ఺͸࣍ͱͳΔɿ

x∗

=μ+ σ(α−1) α+β−2.

·ͨɼҰൠܕϕʔλ෼෍ͷີ౓ؔ਺fͷ༗քੑʹ͍ͭͯ΋໋୊3ͱಉ༷ʹ

੒ཱ͢Δɽଈͪɼx→μͷ࣌ɼα <1ͷ৔߹͸f(x)→ ∞ɼα= 1ͷ৔߹͸

f(x)→β/σɼα >1ͷ৔߹͸f(x)→0Ͱ͋Γɼx→μ+σͷ࣌ɼβ <1ͷ ৔߹͸f(x)→ ∞ɼβ = 1ͷ৔߹͸f(x)→α/σɼβ >1ͷ৔߹͸f(x)→0 Ͱ͋Δɽ

࠷ޙʹɼ(2)ࣜͰఆٛ͞ΕΔඪ४ϕʔλ෼෍ͷ฼਺ͱಛੑ஋ͷؔ܎ʹͭ

͍ͯݴٴ͢Δɽಛʹɼطड़ͷظ଴஋ʢҎԼɼμͱදه͢Δʣͱ࠷େ΋͘͠

͸࠷খ఺z∗ _{ͷେখؔ܎ɼฒͼʹɼ෼෍ͷ࿪Έʹ͍ͭͯݴٴ͢Δɽطʹड़}

΂ΒΕͨΑ͏ʹɼμ=α/(α+β), z∗

= (α−1)/(α+β−2)Ͱ͋Γɼ0 < z∗

<1 ⇐⇒ (α−1)(β−1)>0 ⇐⇒[α > 1͔ͭβ >1]·ͨ͸[α < 1 ͔ͭβ <1]Ͱ͋ΔɽΑͬͯɼμͱz∗_{ͷେখؔ܎Λൺֱ͢Δͱɼα >}

1͔ ͭβ > 1ͷ৔߹ʢ෼෍ͷܗঢ়͸୯ๆܕʣɼz∗_{͸࠷େ఺ʢ࠷ස஋ʣʹͯ͠}

μ≶z∗

⇐⇒ α≷βͰ͋Γɼα <1͔ͭβ <1ͷ৔߹ʢ෼෍ͷܗঢ়͸Uࣈ

ܕʣɼz∗_{͸࠷খ఺ʹͯ͠}

μ≶z∗

⇐⇒ α≶βͰ͋Δɽ͜͜Ͱɼඪ४ϕʔλ

෼෍ͷ࿪౓Λskͱදه͢Δͱɼsk≶0 ⇐⇒ α≷βͰ͋Δ͜ͱʹ஫ҙ͢Δ

ʢྫ͑͹ɼBalakrishnan and Nevzorov 2003, p.147, Gupta and Nadarajah

2004, p.42౳ࢀরʣɽҎ্ʹΑΓɼඪ४ϕʔλ෼෍ͷ฼਺α, βͱಛੑ஋ʢظ

଴஋μɼ࠷େ΋͘͠͸࠷খ఺z∗_{ɼ࿪౓skʣͷؔ܎ʹ͍ͭͯҎԼΛಘΔɿ}

α=β= 1ͷ৔߹ʢҰ༷෼෍ʣμ= 1/2, sk= 0.

α >1͔ͭβ >1ͷ৔߹ʢ୯ๆܕʣz∗_{ɿ࠷େ఺ʢ࠷ස஋ʣ}

α < βͷ৔߹ z∗

< μ <1/2, sk >0.

α=βͷ৔߹ μ=z∗

= 1/2, sk= 0.

α > βͷ৔߹ 1/2< μ < z∗

, sk <0.

α <1͔ͭβ <1ͷ৔߹ʢUࣈܕʣ_z∗_ɿ࠷খ఺

α < βͷ৔߹ μ <1/2< z∗

, sk >0.

α=βͷ৔߹ μ=z∗

= 1/2, sk= 0.

α > βͷ৔߹ z∗

<1/2< μ, sk <0.

α≥1͔ͭβ ≤1, α=βͷ৔߹ʢ୯ௐ૿Ճܕʣ1/2< μ, sk <0.

α≤1͔ͭβ ≥1, α=βͷ৔߹ʢ୯ௐݮগܕʣμ <1/2, sk >0.

4

݁࿦

໋୊3Ͱࣔͨ͠Α͏ʹɼ(2)ࣜͰఆٛ͞ΕΔඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0

ͷάϥϑͷܗঢ়͸େ͖͘5ྨܕʹ෼ྨ͞ΕΔɽଈͪɼϕʔλ෼෍ͷܗঢ়͸ɼ

α=β = 1ͷ৔߹ʹҰ༷෼෍ɼα >1͔ͭβ >1ͷ৔߹ʹ୯ๆܕɼα <1 ͔ͭβ <1ͷ৔߹ʹUࣈܕɼα=βʹͯ͠α≥1͔ͭβ≤1ͷ৔߹ʹ୯ௐ

૿Ճܕɼα=βʹͯ͠α≤1͔ͭβ≥1ͷ৔߹ʹ୯ௐݮগܕͱͳΔɽ·ͨɼ

ඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0͸ɼα≥1͔ͭβ ≥1ͷ৔߹ʹ༗քͰ͋Γɼ

α <1·ͨ͸β <1ͷ৔߹ʹ༗քͰ͸ͳ͍ɽಛʹɼα <1ͷ৔߹͸z→0 ͷ࣌ʹf0(z)→ ∞Ͱ͋Γɼβ <1ͷ৔߹͸z→1ͷ࣌ʹf0(z)→ ∞Ͱ͋

Δɽಉ༷ͷ݁Ռ͸ɼ(3)ࣜͰఆٛ͞ΕΔҰൠܕϕʔλ෼෍ʹ͍ͭͯ΋੒ཱ͢

Δʢ໋୊4ʣɽ

ँࣙ

ຊߘͷࣥචʹࡍͯ͠ɼถాਗ਼ڭतʢ෱Ԭେֶʣͱ܀ాߴޫڭतʢ෱Ԭେ ֶʣ͔ΒوॏͳίϝϯτΛ௖ଷͨ͠ɽ೔ࠒͷҙݟަ׵ͷػձͱ߹Θͤͯɼ͜ ͜ʹهͯ͠ਂ͘ײँΛਃ্͍͛ͨ͠ɽ

ࢀߟจݙ

ݘҪమ࿠ʢ1962ʣʰಛघവ਺ʱؠ೾ॻళɽ

ਿӜޫ෉ʢ1980ʣʰղੳೖ໳Iʱ౦ژେֶग़൛ձɽ

ླ໦ઇ෉ʢ1978ʣʰ౷ܭղੳʱஜຎॻ๪ɽ ླ໦ઇ෉ʢ1987ʣʰ౷ܭֶʱே૔ॻళɽ

஛಺ܒଞฤʢ1989ʣʰ౷ܭֶࣙయʱ౦༸ܦࡁ৽ใࣾɽ த࠺র༤ʢ2007ʣʰೖ໳ϕΠζ౷ܭֶʱே૔ॻళɽ

ຸ୩ઍ႗඙ʢ2010ʣʰ౷ܭ෼෍ϋϯυϒοΫʢ૿ิ൛ʣʱே૔ॻళɽ ౉෦༸ʢ1999ʣʰϕΠζ౷ܭֶೖ໳ʱ෱ଜग़൛ɽ

Azzalini, A. (1996)Statistical Inference: Based on the Likelihood, Chap-man & Hall.

Balakrishnan, N. and V. B. Nevzorov (2003) A Primer on Statistical Distributions, John Wiley & Sons.

Casella, G. and R. L. Berger (2002) Statistical Inference, 2nd edition, Duxbury.

Feller, W. (1971)An Introduction to Probability Theory and Its Applica-tions, Volume 2, 2nd edition, John Wiley & Sons.

Forbes, C., M. Evans, N. Hastings and B. Peacock (2011) Statistical Distributions, 4th edition, John Wiley & Sons.

Gupta, A. and S. Nadarajah (2004) “Mathematical properties of the beta distribution”, In Handbook of Beta Distribution and Its Applications, edited by A. K. Gupta and S. Nadarajah, Marcel Dekker, pp.33–53.

Johnson, N. L., S. Kotz and N. Balakrishnan (1995) Continuous Uni-variate Distributions, Volume 2, 2nd edition, John Wiley & Sons.

Krishnamoorthy, K. (2016) Handbook of Statistical Distributions with Applications, 2nd edition, Taylor & Francis Group.

Leemis, M. L. and J. T. McQueston (2008) “Univariate distribution re-lationships”,The American Statistician, 62, 45–53.

Stuart, A. and J. K. Ord (1994)Kendall’s Advanced Theory of Statistics, Volume 1: Distribution Theory, 6th edition, Arnold.

A

ඪ४ϕʔλ෼෍ͷີ౓ؔ਺ͷ૿ݮද

ຊิ࿦͸ɼ(2)ࣜͰఆٛ͞ΕΔඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0(z), 0< z <1

ͷ૿ݮදΛఏࣔ͢Δɽಛʹɼิ࿦A.1͸ୈ3.1અͰಘΒΕͨ૿ݮදΛఏࣔ

͠ɼิ࿦A.2͸ୈ3.2અͰಘΒΕͨ૿ݮදΛఏࣔ͢Δɽ͜ΕʹΑΓɼ2ͭ

ͷ฼਺α, β >0ͱີ౓ؔ਺f0(z)ͷάϥϑͷܗঢ়ͱͷؔ܎͕໌Β͔ʹͳΔ

ʢ໋୊1ɼ໋୊2ʣɽ

A.1

_α

_{= 1}

·ͨ͸

_β

_{= 1}

ͷ৔߹

ද2: ີ౓ؔ਺f0ͷ૿ݮදʢα= 1͔ͭβ= 1ͷ৔߹ʣɿ0< β <1ʢ্ஈ ࠨʣɼ1< β <2ʢ্ஈӈʣɼβ = 2ʢԼஈࠨʣɼβ >2ʢԼஈӈʣ

z 0 · · · 1

f′

0 β(1−β) + ∞

f′′

0 β(1−β)(2−β) + ∞

f0 β ր ∞

z 0 · · · 1

f′

0 β(1−β) − −∞

f′′

0 β(1−β)(2−β) − −∞

f0 β ց 0

z 0 · · · 1

f′

0 −2 −2 −2 f′′

0 0 0 0

f0 2 ց 0

z 0 · · · 1

f′

0 β(1−β) − 0

f′′

0 β(1−β)(2−β) + ∗

f0 β ց 0

∗:∞(β <3ͷ࣌), 6 (β= 3ͷ࣌), 0 (β >3ͷ࣌)

ද3: ີ౓ؔ਺f0ͷ૿ݮදʢα= 1͔ͭβ = 1ͷ৔߹ʣɿ0< α <1ʢ্ஈ ࠨʣɼ1< α <2ʢ্ஈӈʣɼα= 2ʢԼஈࠨʣɼα >2ʢԼஈӈʣ

z 0 · · · 1

f′

0 −∞ − α(α−1)

f′′

0 ∞ + α(α−1)(α−2)

f0 ∞ ց α

z 0 · · · 1

f′

0 ∞ + α(α−1)

f′′

0 −∞ − α(α−1)(α−2)

f0 0 ր α

z 0 · · · 1

f′

0 2 + 2

f′′

0 0 0 0

f0 0 ր 2

z 0 · · · 1

f′

0 0 + α(α−1)

f′′

0 ∗ + α(α−1)(α−2) f0 0 ր α

A.2

_α

_{= 1}

͔ͭ

_β

_{= 1}

ͷ৔߹

ද 4: ີ౓ؔ਺f0ͷ૿ݮදʢα <1͔ͭβ <1ͷ৔߹ʣ

z 0 · · · z∗

· · · 1

f′

0 −∞ − 0 + ∞

f0 ∞ ց ր ∞

ද 5: ີ౓ؔ਺f0ͷ૿ݮදʢα <1͔ͭβ > 1ͷ৔߹ʣɿ1 < β <2ʢࠨ දʣɼβ = 2ʢதදʣɼβ >2ʢӈදʣ

z 0 · · · 1

f′

0 −∞ − −∞

f0 ∞ ց 0

z 0 · · · 1

f′

0 −∞ − −α(α+ 1) f0 ∞ ց 0

z 0 · · · 1

f′

0 −∞ − 0

f0 ∞ ց 0

ද 6: ີ౓ؔ਺f0ͷ૿ݮදʢα >1͔ͭβ <1ͷ৔߹ʣɿ1 < α <2ʢࠨ දʣɼα= 2ʢதදʣɼα >2ʢӈදʣ

z 0 · · · 1

f′

0 ∞ + ∞

f0 0 ր ∞

z 0 · · · 1

f′

0 β(β+ 1) + ∞

f0 0 ր ∞

z 0 · · · 1

f′

0 0 + ∞

f0 0 ր ∞

ʢ্ஈࠨදʣ/β= 2ʢ্ஈதදʣ/β >2ʢ্ஈӈදʣɼα= 2͔ͭβ <2ʢத ஈࠨදʣ/β= 2ʢதஈதදʣ/β >2ʢதஈӈදʣɼα >2͔ͭβ <2ʢԼஈ ࠨදʣ/β= 2ʢԼஈதදʣ/β >2ʢԼஈӈදʣ

β <2 : β= 2 : β >2 :

α <2 :

z 0 · · · z∗

· · · 1

f′

0 ∞ + 0 − −∞

f0 0 ր ց 0

z 0 · · · z∗

· · · 1

f′

0 ∞ + 0 − −α(α+ 1)

f0 0 ր ց 0

z 0 · · · z∗

· · · 1

f′

0 ∞ + 0 − 0

f0 0 ր ց 0

α= 2 :

z 0 · · · z∗

· · · 1

f′

0 β(β+ 1) + 0 − −∞

f0 0 ր ց 0

z 0 · · · 1/2 · · · 1

f′

0 6 + 0 − −6

f0 0 ր ց 0

z 0 · · · z∗

· · · 1

f′

0 β(β+ 1) + 0 − 0

f0 0 ր ց 0

α >2 :

z 0 · · · z∗

· · · 1

f′

0 0 + 0 − −∞

f0 0 ր ց 0

z 0 · · · z∗

· · · 1

f′

0 0 + 0 − −α(α+ 1)

f0 0 ր ց 0

z 0 · · · z∗

· · · 1

f′

0 0 + 0 − 0

f0 0 ր ց 0

ベータ分布の形状について（

原

）

−6

9−

(1

B

ඪ४ϕʔλ෼෍ͷີ౓ؔ਺ͷάϥϑ

ຊิ࿦͸ɼ(2)ࣜͰఆٛ͞ΕΔඪ४ϕʔλ෼෍ͷີ౓ؔ਺f0(z), 0< z <1

ͷάϥϑΛ໋୊3Ͱఏࣔ͞Εͨ෼ྨʹैͬͯਤࣔ͢Δʢԣ࣠ɿzɼॎ࣠ɿf0(z)ʣɽ

ຊิ࿦Ͱఏࣔ͢Δਤ͸શͯMaple 6ʹΑΔ΋ͷͰ͋Δɽ

B.1

_α

₌

_β

_{= 1}

ͷ৔߹ɿҰ༷෼෍

0 0.5 1 1.5 2

0.2 0.4 0.6 0.8 1

ਤ 1: ඪ४ϕʔλ෼෍ͷີ౓ؔ਺ʢα=β = 1ʣ

B.2

_{α >}

₁

͔ͭ

_{β >}

₁

ͷ৔߹ɿ୯ๆܕ

0 0.5 1 1.5 2 2.5 3

0.2 0.4 0.6 0.8 1

ਤ 2: ඪ४ϕʔλ෼෍ͷີ౓ؔ਺ʢα=β >1ʣɿα= 1.5ʢ࣮ઢʣɼα= 2

ʢ఺ઢʣɼα= 3ʢࡉ࣮ઢʣɼα= 5ʢࡉ఺ઢʣ

0 0.5 1 1.5 2 2.5 3

0.2 0.4 0.6 0.8 1

ਤ3: ඪ४ϕʔλ෼෍ͷີ౓ؔ਺ʢβ ≥α >1ɼα= 1.5ʣɿβ = 1.5ʢ࣮ઢʣɼ

β = 2ʢ఺ઢʣɼβ = 3ʢࡉ࣮ઢʣɼβ= 5ʢࡉ఺ઢʣ

0 0.5 1 1.5 2 2.5 3

0.2 0.4 0.6 0.8 1

ਤ 4: ඪ४ϕʔλ෼෍ͷີ౓ؔ਺ʢα≥β >1ɼα= 5ʣɿβ = 1.5ʢ࣮ઢʣɼ

β = 2ʢ఺ઢʣɼβ = 3ʢࡉ࣮ઢʣɼβ= 5ʢࡉ఺ઢʣ

B.3

_{α <}

₁

͔ͭ

_{β <}

₁

ͷ৔߹ɿ

U

ࣈܕ

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.2 0.4 0.6 0.8 1

ਤ 5: ඪ४ϕʔλ෼෍ͷີ౓ؔ਺ʢα= β < 1ʣɿα = β = 0.2ʢ࣮ઢʣɼ

α=β= 0.5ʢ఺ઢʣɼα=β= 0.8ʢࡉ࣮ઢʣ

0 0.2 0.4 0.6 0.8 1

0.2 0.4 0.6 0.8 1

ਤ6: ඪ४ϕʔλ෼෍ͷີ౓ؔ਺ʢα≤β <1, α= 0.2ʣɿβ= 0.2ʢ࣮ઢʣɼ

β = 0.5ʢ఺ઢʣɼβ= 0.8ʢࡉ࣮ઢʣ

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0.2 0.4 0.6 0.8 1

ਤ7: ඪ४ϕʔλ෼෍ͷີ౓ؔ਺ʢβ ≤α <1, α= 0.8ʣɿβ= 0.2ʢ࣮ઢʣɼ

β = 0.5ʢ఺ઢʣɼβ= 0.8ʢࡉ࣮ઢʣ

B.4

_α

_≥

₁

͔ͭ

_β

_≤

₁

ͷ৔߹ʢ

_α

₌

_β

ʣɿ୯ௐ૿Ճܕ

0 0.5 1 1.5 2 2.5 3 3.5

0.2 0.4 0.6 0.8 1

ਤ 8: ඪ४ϕʔλ෼෍ͷີ౓ؔ਺ʢα= 1͔ͭβ <1ʣɿβ = 0.2ʢ࣮ઢʣɼ

β = 0.5ʢ఺ઢʣɼβ= 0.8ʢࡉ࣮ઢʣ

0 1 2 3 4

0.2 0.4 0.6 0.8 1

ਤ 9: ඪ४ϕʔλ෼෍ͷີ౓ؔ਺ʢα > 1͔ͭβ = 1ʣɿα= 1.5ʢ࣮ઢʣɼ

α= 2ʢ఺ઢʣɼα= 3ʢࡉ࣮ઢʣɼα= 5ʢࡉ఺ઢʣ

0 0.5 1 1.5 2 2.5 3

0.2 0.4 0.6 0.8 1

ਤ10: ඪ४ϕʔλ෼෍ͷີ౓ؔ਺ʢα >1͔ͭβ= 0.5<1ʣɿα= 1.5ʢ࣮

ઢʣɼα= 2ʢ఺ઢʣɼα= 3ʢࡉ࣮ઢʣɼα= 5ʢࡉ఺ઢʣ

B.5

_α

_≤

₁

͔ͭ

_β

_≥

₁

ͷ৔߹ʢ

_α

₌

_β

ʣɿ୯ௐݮগܕ

0 1 2 3 4

0.2 0.4 0.6 0.8 1

ਤ 11: ඪ४ϕʔλ෼෍ͷີ౓ؔ਺ʢα= 1͔ͭβ >1ʣɿβ = 1.5ʢ࣮ઢʣɼ

β = 2ʢ఺ઢʣɼβ = 3ʢࡉ࣮ઢʣɼβ= 5ʢࡉ఺ઢʣ

0 1 2 3 4 5

0.2 0.4 0.6 0.8 1

ਤ 12: ඪ४ϕʔλ෼෍ͷີ౓ؔ਺ʢα <1͔ͭβ = 1ʣɿα= 0.2ʢ࣮ઢʣɼ

α= 0.5ʢ఺ઢʣɼα= 0.8ʢࡉ࣮ઢʣ

0 0.5 1 1.5 2 2.5 3

0.2 0.4 0.6 0.8 1

ਤ13: ඪ४ϕʔλ෼෍ͷີ౓ؔ਺ʢα= 0.5<1͔ͭβ >1ʣɿβ= 1.5ʢ࣮

ઢʣɼβ= 2ʢ఺ઢʣɼβ= 3ʢࡉ࣮ઢʣɼβ = 5ʢࡉ఺ઢʣ