ベータ確率変数の対数変換の 分布の形状について

1 ^ং࿦

༗ݶ۠ؒ(0,1)্ͷҰ༷෼෍͸ɼ࿈ଓܕͷ֬཰෼෍ͱͯ͠࠷΋جຊతͳ

෼෍ͷҰͭͰ͋ΔɽͦͷҰ༷෼෍ʹै͏֬཰ม਺ʢҰ༷֬཰ม਺ʣʹର͠

ͯෛͷର਺ม׵Λࢪͤ͹ɼ৽ͨʹແݶ۠ؒ(0,∞)ʹ஋ΛऔΔ֬཰ม਺͕ಋ

͔Εɼ͔ͭɼͦͷ෼෍͸ࢦ਺෼෍ʹҰக͢Δɽແݶ۠ؒ(0,∞)্ͷࢦ਺෼

෍΋·ͨҰ༷෼෍ͱಉ༷ʹ࿈ଓܕͷ֬཰෼෍ͱͯ͠جຊతͳ෼෍Ͱ͋Δɽ

্ड़ͷࣄ࣮ʹج͚ͮ͹ɼҰ༷෼෍Λಛघͳ৔߹ͱͯ͠แؚ͢Δϕʔλ෼

෍ʹै͏֬཰ม਺ʢϕʔλ֬཰ม਺ʣʹରͯ͠ಉ༷ͷม׵Λࢪͤ͹ɼࢦ਺

෼෍Λಛघͳ৔߹ͱͯ͠แؚ͢ΔΑΓҰൠతͳ֬཰෼෍଒ʢࢦ਺෼෍ͷҰ ൠԽͷҰྫʣ͕ಘΒΕΔ͜ͱ͕෼͔Δɽ͜ͷ֬཰෼෍଒ʹ͍ͭͯ͸ɼྫ͑

͹ɼJohnson, Kotz and Balakrishnan (1995, p.218)΍McDonald and Xu (1995a, pp.140–143), Nadarajah and Gupta (2004, p.127), ຸ୩ʢ2010ɼ pp.793–795ʣʹ͓͍ͯݴٴ͞Εɼࢦ਺ϕʔλ෼෍ͱ΋ݺ͹ΕΔɽ͔͠͠ɼ͍

ͣΕͷจݙʹ͓͍ͯ΋ɼ෼෍ͷ฼਺ͱີ౓ͷܗঢ়ͷؔ܎ʹ͍ͭͯମܥతʹ ݴٴ͞ΕΔ͜ͱ͸ͳ͍ɽ

ຊߘ͸ɼϕʔλ֬཰ม਺ʹରͯ͠ෛͷର਺ม׵Λࢪ͢͜ͱʹΑͬͯಘΒ ΕΔ֬཰෼෍ʢࢦ਺ϕʔλ෼෍ʣʹ͍ͭͯɼͦͷ฼਺ͱີ౓ͷܗঢ়ͷؔ܎Λ

∗෱Ԭେֶܦࡁֶ෦ɼe-mail: [email protected]

ベータ確率変数の対数変換の 分布の形状について

伴 原 理 人^＊

( １ )

ମܥతʹݕ౼͠ɼͦͷ݁Ռͱͯ͠ಘΒΕΔ஌ݟΛ੔ཧ͢Δʢ໋୊1ࢀরʣɽ

ͦΕͱ߹ͤͯɼ͜ͷࢦ਺ϕʔλ෼෍ʹै͏֬཰ม਺Λۊม׵͢Δ͜ͱʹΑͬ

ͯWeibull෼෍ͷҰൠԽͷҰྫʢҰൠԽࢦ਺ϕʔλ෼෍ʣΛߏ੒͢Δɽ͜ͷ

എܠͱͯ͠ɼWeibull෼෍͕ࢦ਺෼෍ʹै͏֬཰ม਺ͷۊม׵ͷ෼෍ͱͯ͠

ಘΒΕΔͱ͍͏ࣄ࣮ɼٴͼɼࢦ਺ϕʔλ෼෍͕ࢦ਺෼෍ͷҰൠԽͱͳͬͯ

͍Δࣄ࣮ʹ஫ҙ͢Δɽͳ͓ɼࢦ਺෼෍Λಛघͳ৔߹ͱͯ͠แؚ͢Δ֬཰෼

෍଒ͱͯ͠͸ɼຊߘͰͷٞ࿦ͷର৅ͱͳΔࢦ਺ϕʔλ෼෍ͷଞʹ΋ɼஶ໊

ͳྫͱͯ͠ΨϯϚ෼෍΍ઌड़ͷWeibull෼෍͕ڍ͛ΒΕΔɽΨϯϚ෼෍΍

Weibull෼෍ͷ฼਺ͱີ౓ͷܗঢ়ͷؔ܎ʹ͍ͭͯ͸ɼྫ͑͹ɼݤݪʢ2018ʣ

͸ɼͦΕΒͷ෼෍Λಛघͳ৔߹ͱͯ͠แؚ͢ΔΑΓҰൠతͳ֬཰෼෍଒Ͱ

͋ΔҰൠԽΨϯϚ෼෍ʹج͍ͮͯମܥతʹٞ࿦͢Δɽ

ҎԼɼຊߘ͸࣍ͷΑ͏ʹߏ੒͞ΕΔɽઌͣɼୈ2અʹ͓͍ͯɼϕʔλؔ

਺Λਖ਼نԽఆ਺ͱͯ͠ߏ੒͞ΕΔඪ४ϕʔλ෼෍ʹै͏֬཰ม਺ʹରͯ͠ɼ

ෛͷର਺ม׵Λࢪ͢͜ͱʹΑͬͯࢦ਺ϕʔλ෼෍Λಋೖ͢Δɽͦͷ্Ͱɼࢦ

਺ϕʔλ෼෍ʹै͏֬཰ม਺Λۊม׵͢Δ͜ͱʹΑͬͯҰൠԽࢦ਺ϕʔλ

෼෍Λߏ੒͠ɼߋʹɼҰൠԽࢦ਺ϕʔλ෼෍ʹै͏֬཰ม਺Λ1࣍ม׵͢

Δ͜ͱʹΑͬͯͦͷҐஔई౓෼෍଒Λߏ੒͢Δɽ࣍ʹɼୈ3અʹ͓͍ͯɼࢦ

਺ϕʔλ෼෍ͷີ౓ؔ਺ͷάϥϑͷܗঢ়͕ͦͷ฼਺ʹԠͯ͡ͲͷΑ͏ʹม Խ͢Δ͔Λݕ౼͠ɼͦͷ݁Ռͱͯ͠ಘΒΕͨ஌ݟΛ໋୊1ͱͯ͠੔ཧ͢Δɽ

ͦͷࡍɼີ౓ؔ਺ͷάϥϑ͸શͯิ࿦Aͱิ࿦Bʹਤࣔ͢Δɽਤࣔ͞Εͨ

άϥϑ͸શͯMaple 6ʹΑΔ࡞ਤͰ͋Δɽ࠷ޙʹɼୈ4અͰ݁࿦Λड़΂Δɽ

2 ࢦ਺ϕʔλ෼෍ͱͦͷҰൠԽ

೚ҙͷਖ਼ͷ࣮਺α, β >0ʹରͯ͠ੵ෼₁

0 x^α−1(1−x)^β−1dx͸ଘࡏ͢Δ ʢྫ͑͹ɼݘҪ1962ɼp.12΍ਿӜ1980ɼpp.295–296ࢀরʣɽ͜ͷࣄ࣮ʹ஫

ҙ͢Δͱɼ͜ͷੵ෼Λα, βͷؔ਺ͱݟ၏͢͜ͱ͕Ͱ͖ɼ͜ΕΛϕʔλؔ਺

ͱ͍͏ɿ

B(α, β) :=

₁

0

x^α−1(1−x)^β−1dx, α, β >0. (1)

͜͜Ͱɼ౳߸:=͸ͦͷࠨลΛͦͷӈลʹΑͬͯఆٛ͢Δ͜ͱΛҙຯ͢Δɽ

೚ҙͷਖ਼ͷ࣮਺α, β >0ͱ(0,1)۠ؒ಺ͷ೚ҙͷ࣮਺x∈(0,1)ʹରͯ͠ɼ x^α−1(1−x)^β−1>0Ͱ͋ΔͷͰɼੵ෼ͷੑ࣭ʹΑΓɼϕʔλؔ਺͸ৗʹਖ਼஋

ΛऔΔ͜ͱ͕෼͔Δɽଈͪɼ೚ҙͷα, β >0ʹରͯ͠B(α, β)>0Ͱ͋Δɽ ϕʔλ෼෍ͱ͸ɼϕʔλؔ਺Λਖ਼نԽఆ਺ͱͯ͠ີ౓ؔ਺Λߏ੒ͨ֬͠

཰෼෍Ͱ͋Δɽͭ·Γɼ࣍ͷ(2)ࣜͱͯ۠ؒ͠(0,1)্ʹఆٛ͞ΕΔؔ਺f_∗ Λີ౓ؔ਺ʹ࣋ͭ֬཰෼෍͸ϕʔλ෼෍ʢಛʹɼඪ४ϕʔλ෼෍ʣͱݺ͹

Εɼαͱβ͸ܗঢ়฼਺ͱݺ͹ΕΔʢα, β >0ʣɿ f_∗(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

ͯ͠ɼz^α−1(1−z)^β−1>0Ͱ͋ΔͷͰf_∗(z|α, β)>0ΛಘΔɽ·ͨɼϕʔ λؔ਺ͷఆٛʹ஫ҙ͢Δͱɼ₁

0 f_∗(z|α, β)dz= 1ΛಘΔɽΑͬͯɼؔ਺f_∗

͸ɼඇෛ஋ੑͱਖ਼نԽ৚݅Λຬͨ͢ͷͰɼ͔֬ʹີ౓ؔ਺Ͱ͋Δɽ

ͯ͞ɼ༗ݶ۠ؒ(0,1)ʹ஋ΛऔΔҰ༷֬཰ม਺ʹରͯ͠ෛͷର਺ม׵Λࢪ

͢͜ͱʹΑͬͯಘΒΕΔ֬཰ม਺ͷ෼෍͸ɼແݶ۠ؒ(0,∞)্ʹఆٛ͞Ε Δࢦ਺෼෍Ͱ͋Γɼͦͷࢦ਺෼෍ʹै͏֬཰ม਺Λۊม׵͢Δ͜ͱʹΑͬ

ͯಘΒΕΔ֬཰ม਺ͷ෼෍͸ɼ(0,∞)্ͷWeibull෼෍Ͱ͋Δɽ

ຊઅͰ͸ɼ͜ͷࣄ࣮Λجʹͯ͠ɼઌͣɼୈ2.1અʹ͓͍ͯɼҰ༷෼෍Λಛ घͳ৔߹ͱͯ͠แؚ͢Δϕʔλ෼෍ʹै͏֬཰ม਺ʹରͯ͠ෛͷର਺ม׵

Λࢪ͢͜ͱʹΑͬͯɼࢦ਺෼෍Λಛघͳ৔߹ͱͯ͠แؚ͢ΔΑΓҰൠతͳ

֬཰෼෍଒ʢࢦ਺ϕʔλ෼෍ɼࢦ਺෼෍ͷҰൠԽͷҰྫʣΛߏ੒͢Δɽͦ

ͷ্Ͱɼୈ2.2અʹ͓͍ͯɼࢦ਺ϕʔλ෼෍ʹै͏֬཰ม਺ʹରͯ͠ۊม

׵Λࢪ͢͜ͱʹΑͬͯɼWeibull෼෍Λಛघͳ৔߹ͱͯ͠แؚ͢ΔΑΓҰ ൠతͳ֬཰෼෍଒ʢҰൠԽࢦ਺ϕʔλ෼෍ɼWeibull෼෍ͷҰൠԽͷҰྫʣ Λߏ੒͢Δɽͦͯ͠ɼୈ2.3અʹ͓͍ͯɼҰൠԽࢦ਺ϕʔλ෼෍ʹै͏֬

཰ม਺ʹରͯ͠1࣍ม׵Λࢪ͢͜ͱʹΑͬͯɼͦͷҐஔई౓෼෍଒Λߏ੒

͢Δɽ࠷ޙʹɼୈ2.4અʹ͓͍ͯɼ্ड़ͷΑ͏ʹͯ͠ߏ੒͞Εͨࢦ਺ϕʔ λ෼෍ʹؔ࿈͢Δز͔ͭͷ෼෍ʹ͍ͭͯݴٴ͢Δɽ

( ３ )

2.1 ࢦ਺ϕʔλ෼෍

ຊઅͰ͸ɼ(2)ࣜͰఆٛ͞ΕΔඪ४ϕʔλ෼෍ʹै͏֬཰ม਺ʹରͯ͠ෛ

ͷର਺ม׵Λࢪ͢͜ͱʹΑͬͯɼࢦ਺෼෍Λಛघͳ৔߹ͱͯ͠แؚ͢ΔΑ ΓҰൠతͳ֬཰෼෍଒ʢࢦ਺ϕʔλ෼෍ʣΛߏ੒͢Δɽଈͪɼ(2)ࣜͰఆٛ

͞ΕΔඪ४ϕʔλ෼෍ͷີ౓ؔ਺f_∗ʹରͯ͠Z ∼f_∗ͱͯ͠ɼͦͷෛͷର

਺ม׵X:=−logZͷ֬཰෼෍Λಋग़͢ΔɽXͷ෼෍ؔ਺ΛF₀ͱදه͢

ΔͱɼF₀͸ɼͦͷఆٛʹΑΓɼ೚ҙͷਖ਼ͷ࣮਺xʹରͯ࣍ࣜ͠ͱಘΒΕΔɿ F₀(x|α, β) :=P(X ≤x) =P

Z≥e^−x

= 1−F_∗

e^−x|α, β

, x >0.

͜͜ͰɼP(X≤x)͸֬཰ม਺Xͷ࣮ݱ஋͕࣮਺xҎԼʹͳΔͱ͍͏ࣄ৅

ͷ֬཰Λද͠ɼF_∗͸Zͷ෼෍ؔ਺Λද͢ɽΑͬͯɼ

F_∗

e^−x|α, β

= _e^−x

0

f_∗(z|α, β)dz

ʹ஫ҙ͢Ε͹ɼX ͷີ౓ؔ਺f₀͸ͦͷ෼෍ؔ਺F₀ͷಋؔ਺ͱͯ͠ಘΒ ΕΔɿ

f₀(x|α, β) = d

dxF₀(x|α, β) =e^−xf_∗

e^−xα, β).

ैͬͯɼ0 < x < ∞ͳΔxʹରͯ͠ɼX ͷີ౓ؔ਺f₀͸࣍ࣜͱͳΔ ʢα, β >0ʣɿ

f₀(x|α, β) = 1

B(α, β)e^−αx(1−e^−x)^β−1. (3)

͜ͷ֬཰෼෍଒͸ɼࢦ਺ϕʔλ෼෍ʢಛʹɼࢦ਺ϕʔλ෼෍ͷඪ४ܕʣͱ

΋ݺ͹ΕΔʢྫ͑͹ɼMcDonald and Xu 1995a΍Nadarajah and Gupta 2004, p.127,ຸ୩2010ɼpp.793–795ࢀরʣɽ͜ͷ໋໊ͷഎܠͱͯ͠ɼର਺

ਖ਼ن෼෍ʹै͏֬཰ม਺ͷର਺ม׵ͷ෼෍͕ਖ਼ن෼෍Ͱ͋Δࣄ࣮ɼٴͼɼີ

౓ؔ਺(3)ࣜʹै͏֬཰ม਺X ͷࢦ਺ม׵Z =e^−Xͷ෼෍͕ϕʔλ෼෍

Ͱ͋Δࣄ࣮ʹ஫ҙ͢ΔɽͦͷҰํͰɼWeibull෼෍ʹै͏֬཰ม਺ͷର਺

ม׵ͷ෼෍Λର਺Weibull෼෍ͱݺͿ͜ͱ΋͋Γʢྫ͑͹ɼJohnson, Kotz and Balakrishnan 1995, p.3΍Rinne 2009, pp.131–133౳ࢀরʣɼ͜Εʹ

ै͑͹ɼϕʔλ֬཰ม਺ͷର਺ม׵ͱͯ͠ಋग़͞ΕΔ(3)ࣜΛີ౓ؔ਺ʹ

࣋ͭ෼෍͸ɼࢦ਺ϕʔλ෼෍Ͱ͸ͳ͘ɼର਺ϕʔλ෼෍ͱݺͿ͜ͱʹͳΖ

͏ɽྫ͑͹ɼMcDonald and Xu (1995a) p.141΋·ͨ͜ͷ໋໊ͷՄೳੑΛ ഉআ͠ͳ͍ɽ͔͠͠ɼຊߘʹ͓͍ͯ͸ɼҎԼɼີ౓ؔ਺(3)ࣜʹΑͬͯఆٛ

͞ΕΔ෼෍Λࢦ਺ϕʔλ෼෍ͱݺͿ͜ͱʹ͢Δɽ

ͯ͞ɼࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺(3)ࣜ͸ɼα=β = 1ͱͨ͠৔߹ɼ f₀(x|1,1) =e^−x

ͱͳΔͷͰɼඪ४ࢦ਺෼෍ͷີ౓ؔ਺ʹؼண͢Δ͜ͱ͕෼͔Δɽ͜ΕʹΑ Γɼඪ४ϕʔλ෼෍ͷີ౓ؔ਺(2)͕ࣜα=β = 1ͷ࣌ʹ۠ؒ(0,1)্ͷ Ұ༷෼෍ʹؼண͢Δ͜ͱʹ஫ҙ͢ΔͱɼҰ༷֬཰ม਺ͷର਺ม׵ʹΑͬͯ

ඪ४ࢦ਺෼෍͕༠ಋ͞ΕΔ͜ͱ͕͔֬ʹཧղ͞ΕΔɽͳ͓ɼࢦ਺ϕʔλ෼

෍ͷີ౓ؔ਺(3)ࣜ͸ɼβ= 1ͱͨ͠৔߹ɼB(α,1) = 1/αʹ஫ҙ͢Δͱɼ f₀(x|α,1) =αe^−αx

ͱͳΔͷͰɼσ:= 1/α >0 ⇐⇒ α= 1/σͱ฼਺ม׵͢ΔͱɼσΛई౓฼

਺ͱ͢Δࢦ਺෼෍ʢඪ४ࢦ਺෼෍ͷई౓෼෍଒ʣʹؼண͢Δ͜ͱ͕෼͔Δɽ

͜ͷ࣌ɼϕʔλ෼෍ͷܗঢ়฼਺α͸ई౓฼਺ͷٯ਺ʹରԠ͢Δɽ

2.2 ҰൠԽࢦ਺ϕʔλ෼෍

ຊઅͰ͸ɼࢦ਺෼෍Λಛघͳ৔߹ͱͯ͠แؚ͢Δࢦ਺ϕʔλ෼෍ʹै͏֬

཰ม਺ʹରͯ͠ۊม׵Λࢪ͢͜ͱʹΑͬͯɼWeibull෼෍Λಛघͳ৔߹ͱ͠

ͯแؚ͢ΔΑΓҰൠతͳ֬཰෼෍଒ʢҰൠԽࢦ਺ϕʔλ෼෍ʣΛߏ੒͢Δɽ ଈͪɼ(3)ࣜͰఆٛ͞ΕΔࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺f₀ʹରͯ͠X ∼f₀ ͱͯ͠ɼਖ਼ͷ࣮਺γʹରͯ͠Xͷۊม׵Y :=X^γͷ֬཰෼෍Λಋग़͢Δɽ Y ͷ෼෍ؔ਺ΛF₁ͱදه͢Ε͹ɼͦͷఆٛʹΑΓɼ೚ҙͷਖ਼ͷ࣮਺yʹର

ͯ͠ɼ

F₁(y|α, β, γ) :=P(Y ≤y) =P

X≤y^1γ

= _y¹γ

0

f₀(x|α, β)dx

( ５ )

ͱͯ͠ද͞Εɼͦͷີ౓ؔ਺f₁͸F₁ͷಋؔ਺ͱͯ͠ಘΒΕΔɿ

f₁(y|α, β, γ) = d

dyF₁(y|α, β, γ) = 1

γf₀(y^γ1|α, β)y^γ1−1.

ैͬͯɼ0 < y < ∞ͳΔy ʹରͯ͠ɼY ͷີ౓ؔ਺f₁ ͸࣍ࣜͱͳΔ ʢ0< α, β, γ <∞ʣɿ

f₁(y|α, β, γ) = 1

γB(α, β)y^γ1−1e^−αy

γ1

1−e^−y

γ1

_β−1

. (4)

δ:= 1/γͳΔ฼਺ม׵Λࢪ͢ͱɼY :=X^γ =X^1/δͷີ౓ؔ਺f₁ͷผදݱ

͕ಘΒΕΔʢ0< α, β, δ <∞ʣɿ

f₁(y|α, β, δ) = δ

B(α, β)y^δ−1e^−αyδ

1−e^−yδ _β−1

, 0< y <∞. (5) Ҏ্ͷΑ͏ʹɼ(4)ࣜ΋͘͠͸(5)ࣜͰఆٛ͞Εͨີ౓ؔ਺f₁Λ࣋ͭ֬཰

෼෍ΛҰൠԽࢦ਺ϕʔλ෼෍ʢಛʹɼҰൠԽࢦ਺ϕʔλ෼෍ͷඪ४ܕʣͱ

͍͏ɽҎԼʹ͓͍ͯ͸ɼಛʹஅΒͳ͍ݶΓɼҰൠԽࢦ਺ϕʔλ෼෍ͷີ౓

ؔ਺f₁ͱͯ͠͸(5)ࣜͷදݱΛ༻͍Δɽ

ҰൠԽࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺(5)ࣜ͸ɼδ= 1ͱͨ͠৔߹ɼࢦ਺ϕʔ λ෼෍ͷີ౓ؔ਺(3)ࣜʹؼண͢Δɿ

f₁(y|α, β,1) = 1

B(α, β)e^−αy(1−e^−y)^β−1=f₀(y|α, β).

·ͨɼβ = 1ͱͨ͠৔߹ɼB(α,1) = 1/αʹ஫ҙ͢Δͱɼ

f₁(y|α,1, δ) =αδy^δ−1e^−αyδ (6) ͱͳΔͷͰɼσ := 1/α¹δ >0 ⇐⇒ α= 1/σ^δ ͱ฼਺ม׵͢Δ͜ͱʹΑͬ

ͯɼσΛई౓฼਺ͱ͢ΔWeibull෼෍ʹؼண͢Δ͜ͱ͕෼͔Δɽ͜ΕʹΑ Γɼࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺(3)͕ࣜβ = 1ͷ࣌ʹࢦ਺෼෍ʹؼண͢Δ

͜ͱʹ஫ҙ͢Δͱɼࢦ਺෼෍ͷۊม׵ʹΑͬͯWeibull෼෍͕༠ಋ͞ΕΔ

͜ͱ͕͔֬ʹཧղ͞ΕΔɽ·ͨɼ͜ͷ࣌ɼ(6)ࣜʹΑΓɼϕʔλ෼෍ͷܗঢ়

฼਺α͸ई౓฼਺ͷۊ৐ͷٯ਺ʹରԠ͢Δͱ෼͔Δɽ

2.3 ҰൠԽࢦ਺ϕʔλ෼෍ͷҐஔई౓෼෍଒

ຊઅ͸ɼ(5)ࣜͰఆٛ͞ΕΔҰൠԽࢦ਺ϕʔλ෼෍ͷඪ४ܕʹҐஔ฼਺ͱ ई౓฼਺Λಋೖ͢ΔɽଈͪɼY ΛҰൠԽࢦ਺ϕʔλ෼෍ͷඪ४ܕʹै͏֬

཰ม਺Y ∼f₁ͱͯ͠ɼ࣮਺μͱਖ਼ͷ࣮਺σʹରͯ͠Y Λ1࣍ม׵͢Δɿ W :=μ+σY, −∞< μ <∞, σ >0. ͜ͷ࣌ɼW ͷ֬཰෼෍ΛμΛҐஔ

฼਺ɼσΛई౓฼਺ͱ͢ΔҰൠԽࢦ਺ϕʔλ෼෍ͷҐஔई౓෼෍଒ʢ͋Δ

͍͸ɼ୯ʹҰൠԽࢦ਺ϕʔλ෼෍ʣͱ͍͏ɽͦͷ෼෍ؔ਺F ͸ɼͦͷఆٛ

ʹΑΓɼw > μͳΔwʹରͯ͠ɼ

F(w|α, β, δ, μ, σ) :=P(W ≤w) =P Y ≤w−μ σ

= ^w−μ

σ

0

f₁(y|α, β, δ)dy Ͱ͋Γɼͦͷີ౓ؔ਺f͸F ͷಋؔ਺ͱͯ͠ಋग़͞ΕΔɿ

f(w|α, β, δ, μ, σ) = d

dwF(w|α, β, δ, μ, σ) = 1

σf₁ w−μ σ

α, β, δ

.

ैͬͯɼw > μͳΔwʹରͯ͠ɼWͷີ౓ؔ਺f͸࣍ࣜͱͯ͠ಘΒΕΔɿ f(w|α, β, δ, μ, σ) =

δ σB(α, β)

w−μ σ

_δ−1 exp

−α w−μ σ

_δ

1−exp

− w−μ σ

_δβ−1

(7)

͜͜ͰɼҐஔ฼਺μ͸೚ҙͷ࣮਺ͰΑ͍ͷʹରͯ͠ɼई౓฼਺σΛ࢝Ίͱ͢Δ

ͦͷଞͷ฼਺α, β, δ͸ਖ਼஋Ͱ͋Δ͜ͱʹ஫ҙ͢Δɿμ∈(−∞,∞), σ, α, β, δ∈ (0,∞). μ= 0, σ= 1ͷ৔߹ɼҰൠԽࢦ਺ϕʔλ෼෍(7)ࣜ͸ҰൠԽࢦ਺ϕʔ λ෼෍ͷඪ४ܕ(5)ࣜʹؼண͢Δɿf(w|α, β, δ,0,1) = f₁(w|α, β, δ), w >

μ= 0.

ͯ͞ɼҰൠԽࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺(7)ࣜ͸ɼδ = 1ͷ৔߹ɼࢦ਺

ϕʔλ෼෍ͷີ౓ؔ਺

f(w|α, β,1, μ, σ) = 1

σB(α, β)e^{−ασ(w−μ)}

1−e^−w−μσ _β−1

(8)

( ７ )

ͱͳΓɼߋʹɼμ= 0, σ= 1ͷ৔߹ɼࢦ਺ϕʔλ෼෍ͷඪ४ܕͷີ౓ؔ਺

(3)ࣜʹؼண͢Δɿf(w|α, β,1,0,1) =f₀(w|α, β).

2.4 ࢦ਺ϕʔλ෼෍ʹؔ࿈͢Δز͔ͭͷ෼෍

ຊઅ͸ɼҎ্ͷٞ࿦ʹΑͬͯߏ੒͞Εͨࢦ਺ϕʔλ෼෍ʹؔ࿈͢Δزͭ

͔ͷ෼෍ʹ͍ͭͯݴٴ͢Δɽ

2.4.1 ࢦ਺ҰൠԽϕʔλ෼෍

McDonald and Xu (1995a)͸ɼ(2)ࣜͰఆٛ͞ΕΔୈ1छͷϕʔλ෼෍

͚ͩͰͳ͘ɼୈ2छͷϕʔλ෼෍Λ΋แؚ͢Δ෼෍଒Λߟ࡯ର৅ͱ্ͨ͠

Ͱɼࢦ਺ҰൠԽϕʔλ෼෍ΛఏҊͨ͠ɽ͔͠͠ɼؔ৺Λୈ1छͷϕʔλ෼

෍ʹݶఆ͢ΔͳΒ͹ɼຊߘʹΑͬͯߏ੒͞ΕΔ෼෍଒ͷํ͕ΑΓҰൠతͰ

͋Δɽ

࣮ࡍɼMcDonald and Xu (1995a)͸ɼ༗ݶ۠ؒ(0, c), c >0ʹ஋ΛऔΔ ϕʔλ֬཰ม਺ͷۊม׵ʹΑͬͯୈ1छͷҰൠԽϕʔλ෼෍Λߏ੒্ͨ͠

Ͱɼୈ1छͷҰൠԽϕʔλ෼෍ʹै͏֬཰ม਺ͷର਺ม׵ʹΑͬͯୈ1छ ͷࢦ਺ҰൠԽϕʔλ෼෍Λߏ੒͢Δɽ͜ͷΑ͏ʹͯ͠ߏ੒͞ΕΔୈ1छͷ ࢦ਺ҰൠԽϕʔλ෼෍͸ɼຊߘʹ͓͚Δ(8)ࣜɼͭ·Γɼࢦ਺ϕʔλ෼෍ʹ ରԠ͢Δɽैͬͯɼຊߘ(7)ࣜͷҰൠԽࢦ਺ϕʔλ෼෍͸ɼMcDonald and

Xu (1995a)ͷୈ1छͷࢦ਺ҰൠԽϕʔλ෼෍Λಛघͳ৔߹ͱͯ͠แؚ͢Δ

ΑΓҰൠతͳ෼෍଒Ͱ͋Δͱ෼͔Δɽ

ͳ͓ɼ෼෍଒ͷ໊ʹף͞ΕΔҰൠԽͱ͍͏ޠ͸ɼ௨ৗͷ৔߹ɼ֬཰ม਺ͷ ۊม׵ʹΑͬͯߏ੒͞ΕΔ෼෍଒ʹରͯ͠༻͍ΒΕΔɽ͜ΕʹΑΓɼϕʔ λ֬཰ม਺ʹରͯ͠ɼMcDonald and Xu (1995a)ͷࢦ਺ҰൠԽϕʔλ෼෍

ʹ͓͍ͯ͸ɼۊม׵ʢҰൠԽϕʔλ෼෍ʣͷޙʹର਺ม׵ʢࢦ਺ҰൠԽϕʔ λ෼෍ʣΛࢪ͢ͷʹରͯ͠ɼຊߘͰٞ࿦͢ΔҰൠԽࢦ਺ϕʔλ෼෍ʹ͓͍

ͯ͸ɼର਺ม׵ʢࢦ਺ϕʔλ෼෍ʣͷޙʹۊม׵ʢҰൠԽࢦ਺ϕʔλ෼෍ʣ Λࢪ͢ͱ͍͏૬ҧ͕͋Δͱ෼͔Δɽ

2.4.2 ϕʔλࢦ਺෼෍ͱϕʔλWeibull෼෍

ඍ෼Մೳͳ෼෍ؔ਺F :R → [0,1]ʹରͯ͠ɼ(2)ࣜͰఆٛ͞ΕΔඪ४ ϕʔλ෼෍ͷີ౓ؔ਺f_∗Λ༻͍ͯ৽ͨͳ෼෍ؔ਺G:R→[0,1],

G(x) :=

_F(x)

0

f_∗(z)dz

Λੜ੒͢Δ͜ͱ͕Ͱ͖ɼ͜ͷΑ͏ʹͯ͠ੜ੒͞ΕΔ෼෍͸ϕʔλF෼෍ͱ

͍͏ܗͰݺ͹ΕΔʢEugene, Lee and Famoye 2002ɼNadarajah and Gupta 2004, p.146, Nadarajah and Kotz 2006, Alexander, Cordeiro, Ortega and Sarabia 2012౳ࢀরʣɽͳ͓ɼͦͷີ౓ؔ਺g(x) :=G(x) =f_∗(F(x))F(x)

͸࣍ࣜͱͯ͠ಘΒΕΔɿ g(x) = 1

B(α, β)[F(x)]^α−1[1−F(x)]^β−1F(x).

͜Ε͸ɼFʹै͏ಠཱಉ෼෍ͷ֬཰ม਺ྻͷॱং౷ܭྔͷີ౓ؔ਺ͷҰൠԽ ʹͳ͍ͬͯΔ͜ͱʹ஫ҙ͢Δɽ࣮ࡍɼX₁, . . . , X_n, i.i.d.∼Fʹରͯͦ͠ͷ ॱং౷ܭྔΛX₍₁₎, . . . , X_(n)ͱදه͢Δͱɼr൪໨ͷॱং౷ܭྔX_(r)ͷີ

౓ؔ਺͸࣍ࣜͱಘΒΕΔ͜ͱ͕෼͔Δʢྫ͑͹ɼJohnson, Kemp and Kotz 2005, p.61ࢀরʣɿ

f_(r)(x) = 1

B(r, n−r+ 1)[F(x)]^r−1[1−F(x)]^n−rF(x), 1≤r≤n.

ͦͷࡍɼϕʔλؔ਺BͱΨϯϚؔ਺Γͷؔ܎ࣜB(α, β) = Γ(α)Γ(β)/Γ(α+

β)ɼٴͼɼΨϯϚؔ਺ͷੑ࣭Γ(α) = (α−1)Γ(α−1)ʹ஫ҙ͢Δɿ

B(r, n−r+ 1) = (r−1)!(n−r)!

n! .

ͯ͞ɼFͱͯ͠ࢦ਺෼෍ͷ෼෍ؔ਺F(x) = 1−e^−xΛద༻͢Δ͜ͱͰੜ

੒͞ΕΔ෼෍͸ϕʔλࢦ਺෼෍ͱݺ͹Εɼͦͷີ౓ؔ਺͸࣍ࣜͱͳΔɿ g(x) = 1

B(α, β)

1−e^−x_α−1 e^−βx.

( ９ )

͜ͷີ౓ؔ਺͸ɼϕʔλؔ਺ͷରশੑB(α, β) =B(β, α)ʹ஫ҙ͢Ε͹ɼ(3)

ࣜͰఆٛ͞Εͨࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺f₀ʹଞͳΒͳ͍͜ͱ͕෼͔Δɽ

·ͨɼFͱͯ͠Weibull෼෍ͷ෼෍ؔ਺F(x) = 1−e^−xδΛద༻͢Δ͜

ͱͰੜ੒͞ΕΔ෼෍͸ϕʔλWeibull෼෍ͱݺ͹Εɼͦͷີ౓ؔ਺͸࣍ࣜ

ͱͳΔɿ

g(x) = δx^δ−1 B(α, β)

1−e^−xδ _α−1

e^−βxδ.

͜ͷີ౓ؔ਺͸ɼϕʔλؔ਺ͷରশੑB(α, β) =B(β, α)ʹ஫ҙ͢Ε͹ɼ(5)

ࣜͰఆٛ͞ΕͨҰൠԽࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺f₁ʹଞͳΒͳ͍͜ͱ͕෼

͔ΔɽNadarajah and Kotz (2006)͸ɼϕʔλWeibull෼෍ʹ͍ͭͯ͸ݴٴ

͢ΔͷΈͰ͋Δ͕ɼϕʔλࢦ਺෼෍ʹ͍ͭͯ͸ͦͷੑ࣭Λৄࡉʹ෼ੳ͢Δɽ

2.4.3 ࢦ਺෼෍ͱWeibull෼෍ͷॱং౷ܭྔͷ෼෍

ຊઅ͸ɼࢦ਺෼෍ͱWeibull෼෍ͷॱং౷ܭྔͷ෼෍ʹ͍ͭͯߟ࡯͢Δɽ ಛʹɼWeibull෼෍ͷॱং౷ܭྔͷ෼෍͕ҰൠԽࢦ਺ϕʔλ෼෍ʹؼண͢

Δ͜ͱΛ֬ೝ͢Δɽ͜ΕʹΑͬͯɼͦͷಛघͳ৔߹ͱͯ͠ɼࢦ਺෼෍ͷॱ

ং౷ܭྔͷ෼෍͕ࢦ਺ϕʔλ෼෍ʹؼண͢Δ͜ͱ΋൑໌͢Δɽ

ͯ͞ɼ(6)ࣜʹ͓͍ͯα= 1ͱͨ͠ඪ४Weibull෼෍ʹै͏ಠཱಉ෼෍ͷ

֬཰ม਺ྻY₁, . . . , Y_nʹରͯ͠ɼͦͷॱং౷ܭྔΛY₍₁₎, . . . , Y_(n)ͱදه͢Δ ͱɼલઅͷٞ࿦ʹΑΓɼr൪໨ͷॱং౷ܭྔY_(r)ͷີ౓ؔ਺f_(r), 1≤r≤n

͸࣍ࣜͱͳΔ͜ͱ͕෼͔Δɿ f_(r)(y) = 1

B(r, n−r+ 1)δy^δ−1e^{−(n−r+1)yδ}

1−e^−yδ _r−1

, 0< y <∞. ϕʔλؔ਺ͷରশੑʹ஫ҙ͢Δͱɼ͜Ε͸ҰൠԽࢦ਺ϕʔλ෼෍ͷີ౓ؔ

਺(5)ࣜʹ͓͍ͯα =n−r+ 1, β =rͱஔ͍ͨ΋ͷʹଞͳΒͳ͍ͱ෼

͔Δɽ

Ҏ্ʹΑΓɼWeibull෼෍͔ΒಘΒΕͨॱং౷ܭྔͷ෼෍͕ҰൠԽࢦ਺

ϕʔλ෼෍ʹؼண͢Δ͜ͱ͕֬ೝ͞Εͨɽ·ͨɼ্ड़ͷٞ࿦ͷಛघͳ৔߹

ͱͯ͠ɼࢦ਺෼෍͔ΒಘΒΕΔॱং౷ܭྔͷ෼෍͕ࢦ਺ϕʔλ෼෍ʹؼண

͢Δ͜ͱ͕֬ೝ͞ΕΔɽ

3 ^{ࢦ਺ϕʔλ෼෍ͷܗঢ়}

ຊઅͰ͸ɼ(5)ࣜͰఆٛ͞ΕΔҰൠԽࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺ͷάϥϑ ͷܗঢ়ʹ͍ͭͯٞ࿦͢Δɽಛʹɼͦͷδ= 1ͱͨ͠৔߹ɼଈͪɼ(3)ࣜͰද ݱ͞ΕΔࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺ʹ͍ͭͯ͸ɼͦͷάϥϑͷܗঢ়͕ͦͷ

฼਺αͱβʹԠͯ͡ͲͷΑ͏ʹมԽ͢Δ͔Λମܥతʹݕ౼͠ɼͦͷ݁Ռͱ

ͯ͠ಘΒΕͨ஌ݟΛ໋୊1ͱͯ͠੔ཧ͢Δɽ

ͯ͞ɼ(3)ࣜͰఆٛ͞ΕΔࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺f₀ͱ(5)ࣜͰఆٛ

͞ΕΔҰൠԽࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺f₁ͱͷؒʹҎԼͷؔ܎͕ࣜ੒ཱ͢

Δ͜ͱʹ஫ҙ͢Δɽୠ͠ɼҎԼͰ͸ɼ฼਺Λಛʹ໌ࣔ͢Δඞཁ͕ͳ͍৔߹ɼ f₀(x) :=f₀(x|α, β), f₁(y) :=f₁(y|α, β, δ)ͱུه͢Δɽ

f₁(y) =δy^δ−1f₀(y^δ), y∈(0,∞).

ΑͬͯɼͦΕͧΕͷಋؔ਺f₀ ͱf₁ ͷؒʹ͸࣍ͷؔ܎͕ࣜ੒ཱ͢Δɿ f₁(y) =δy^δ−2

(δ−1)f₀(y^δ) +δy^δf₀(y^δ) .

͜͜Ͱɼࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺f₀ͷఆٛࣜ(3)ʹ஫ҙ͢Δͱɼͦͷಋ

ؔ਺͸

f₀(x) = f₀(x) 1−e^−x

(α+β−1)e^−x−α

(9) ͱಘΒΕΔͷͰɼ࣍ࣜΛಘΔɿ

f₁(y) = δf₁(y)

y 1−1

δ

+ y^δ 1−e^−yδ

(α+β−1)e^−yδ−α

. (10) ҎԼͰ͸ɼઌͣɼୈ3.1અʹ͓͍ͯɼδ= 1ͷ৔߹ɼଈͪɼ(3)ࣜͰఆٛ͞

ΕΔࢦ਺ϕʔλ෼෍ͷඪ४ܕͷີ౓ؔ਺f₀ͷάϥϑͷܗঢ়͕ͦͷ฼਺αͱ βʹԠͯ͡ͲͷΑ͏ʹมԽ͢Δ͔ʹ͍ͭͯମܥతʹݕ౼͠ɼͦͷ݁Ռͱ͠

ͯಘΒΕͨ஌ݟΛ໋୊1ͱͯ͠੔ཧ͢Δɽ࣍ʹɼୈ3.2અʹ͓͍ͯɼδ= 1 ͷ৔߹ɼଈͪɼ(5)ࣜͰఆٛ͞ΕΔҰൠԽࢦ਺ϕʔλ෼෍ͷඪ४ܕͷີ౓ؔ

਺f₁ͷάϥϑͷܗঢ়ʹ͍ͭͯҰఆͷݕ౼ΛՃ͑Δɽ࠷ޙʹɼୈ3.3અʹ͓

͍ͯɼͦΕΒͷҐஔई౓෼෍଒ͷܗঢ়ʹ͍ͭͯߟ࡯͢Δɽ

( 11 )

3.1 δ = 1ͷ৔߹ɿࢦ਺ϕʔλ෼෍

(5)ࣜͰఆٛ͞ΕΔҰൠԽࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺f₁͸ɼδ= 1ͷ৔

߹ɼ(3)ࣜͰఆٛ͞ΕΔࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺f₀ʹؼண͢Δɽͭ·Γɼ ҰൠԽࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺f₁ͷಋؔ਺(10)ࣜ͸ɼδ= 1ͷ৔߹ɼࢦ

਺ϕʔλ෼෍ͷີ౓ؔ਺f₀ͷಋؔ਺(9)ࣜʹؼண͢Δɽ

ͯ͞ɼ೚ҙͷx∈(0,∞)ʹରͯ͠f₀(x)>0, 1−e^−x>0Ͱ͋Γɼ·ͨɼ

α >0Ͱ͋Δ͜ͱʹ஫ҙ͢ΔͱɼҎԼͷಉ஋ؔ܎ΛಘΔɿ

f₀(x)≶0 ⇐⇒ (α+β−1)e^−x−α≶0

⇐⇒ 1 + β−1 α

e^−x≶1.

͜ΕʹΑΓɼ೚ҙͷx∈(0,∞)ʹରͯ͠e^−x∈(0,1)Ͱ͋Δ͜ͱʹ஫ҙ

͢Δͱɼβ≤1ͷ৔߹ɼ࣍ͷෆ౳͕ࣜ੒ཱ͢Δ͜ͱ͕෼͔Δɿ 1 + β−1

α

e^−x≤e^−x<1.

ैͬͯɼβ≤1ͷ৔߹ɼ೚ҙͷx∈(0,∞)ʹରͯ͠

1 +β−1 α

e^−x<1 ⇐⇒ f₀(x)<0

ΛಘΔͷͰɼ(3)ࣜͰఆٛ͞ΕΔࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺f₀(x)͸xͷ୯ ௐݮগؔ਺Ͱ͋Δͱ෼͔Δɽ

࣍ʹɼβ > 1 ͷ৔߹Λߟ࡯͢Δɽ͜ͷ࣌ɼ1 + (β −1)/α > 1ͳͷͰ log[1 + (β−1)/α]>0Ͱ͋Δɽ·ͨɼx >0Ͱ͋Δ͜ͱʹ஫ҙ͢ΔɽΑͬ

ͯɼಉ஋ؔ܎

f₀(x)≶0 ⇐⇒ log 1 +β−1 α

≶x

ʹΑΓɼβ >1ͷ৔߹ɼ(3)ࣜͰఆٛ͞ΕΔࢦ਺ϕʔλ෼෍ͷඪ४ܕͷີ౓

ؔ਺f₀(x)͸ɼ

x^∗:= log 1 + β−1 α

>0

ʹରͯ͠ɼx < x^∗ͷ࣌ʹ୯ௐ૿Ճɼx > x^∗ͷ࣌ʹ୯ௐݮগͰ͋Γɼx=x^∗ Ͱ࠷େ஋ΛऔΔ͜ͱ͕෼͔Δɽଈͪɼີ౓ؔ਺f₀(x)ͷܗঢ়͸x=x^∗Λ࠷

େ఺ʢ࠷ස஋ʣͱ͢Δ୯ๆܕͰ͋Δɽ

Ҏ্ͷٞ࿦ʹΑͬͯɼࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺f₀ͷܗঢ়͸ɼβ≤1ͷ

৔߹ʹ୯ௐݮগܕɼβ > 1ͷ৔߹ʹ୯ๆܕͰ͋Δ͜ͱ͕൑໌ͨ͠ʢ໋୊1

ࢀরʣɽ

໋୊1(ࢦ਺ϕʔλ෼෍ͷܗঢ়). (3)ࣜͰఆٛ͞ΕΔࢦ਺ϕʔλ෼෍ͷີ౓ؔ

਺f₀ͷάϥϑ{(x, f₀(x|α, β))|x∈(0,∞)}ͷܗঢ়͸ɼਖ਼஋ͷ฼਺βʹԠ͡

ͯҎԼͷΑ͏ʹఆ·Δʢਤ1–ਤ6ࢀরʣɽୠ͠ɼҎԼͰ͸f₀(x) :=f₀(x|α, β) ͱུه͢Δɽͳ͓ɼຊ໋୊Ͱݴٴ͞ΕΔਤ͸શͯิ࿦Aʹఏࣔ͞ΕΔɽ

1. β≤1ͷ৔߹ʢ୯ௐݮগܕʣ

ີ౓ؔ਺f₀(x)͸xͷ୯ௐݮগؔ਺Ͱ͋Γɼͦͷάϥϑ͸ӈԼΓͷ ܗঢ়Λࣔ͢ɽಛʹɼx→ ∞ͷ৔߹ʹ͓͍ͯf₀(x)→0Ͱ͋Δɽ

(a) β <1ͷ৔߹ʢඇ༗քɼਤ7ʣ

x→0ʹ͓͍ͯf₀(x)→ ∞Ͱ͋Δɽ (b) β = 1ͷ৔߹ʢ༗քɿࢦ਺෼෍ɼਤ8ʣ

x→0ʹ͓͍ͯf₀(x)→αͰ͋Δɽ͜ͷ৔߹ɼࢦ਺ϕʔλ෼

෍͸σ:= 1/αΛई౓฼਺ͱ͢Δࢦ਺෼෍ʹؼண͢Δɽ

2. β >1ͷ৔߹ʢ୯ๆܕɼਤ9–ਤ12ʣ

ີ౓ؔ਺f₀(x)͸x=x^∗Λ࠷େ఺ʢ࠷ස஋ʣͱͯ͠ɼͦͷάϥϑ

͸୯ๆܕͷܗঢ়Λࣔ͢ɿ

x^∗= log 1 + β−1 α

.

·ͨɼx→0΋͘͠͸x→ ∞ͷ͍ͣΕͷ৔߹ʹ͓͍ͯ΋f₀(x)→0 Ͱ͋Δ¹ɽ

1x→0ʹ͓͍ͯີ౓ؔ਺f0(x)͕0ʹऩଋ͢Δࡍɼͦͷ܏͖͸ɼβ <2ͷ৔߹ʹແݶେ

ʹൃࢄ͠ʢf₀(x)−−−→ ∞^x→0 ʣɼβ= 2ͷ৔߹ʹఆ਺ʹऩଋ͠ʢf₀(x)−−−→^x→0 α(α+ 1)ʣɼβ >2 ͷ৔߹ʹ0ʹऩଋ͢Δʢf₀(x)−−−→^x→0 0ʣͱ͍͏ܗΛऔΔʢਤ1ɼਤ3ɼਤ5ɼਤ9–ਤ12ࢀ রʣɽ

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ূ໌. (3)ࣜͰఆٛ͞ΕΔີ౓ؔ਺f₀ͷܗঢ়͕β ≤1ͷ৔߹ʹ୯ௐݮগܕɼ

β >1ͷ৔߹ʹ୯ๆܕͱͳΔ͜ͱʹ͍ͭͯ͸ɼ্ड़ͷٞ࿦ʹ͓͍ͯطʹ໌

Β͔ʹ͞ΕͨɽΑͬͯɼҎԼͰ͸ɼີ౓ؔ਺f₀(x)ͱͦͷಋؔ਺f₀(x)ʹͭ

͍ͯɼx→0ͱx→ ∞ʹ͓͚ΔऩଋઌΛߟ࡯͢Δɽ

ୈҰʹɼx→ ∞ʹ͓͚Δऩଋઌʹ͍ͭͯߟ͑Δɽ೚ҙͷα >0ʹରͯ͠

e^{−αx x}−−−−→^→∞ 0Ͱ͋Γɼ·ͨɼ1−e^{−x x}−−−−→^→∞ 1ʹΑΓ೚ҙͷβ >0ʹର͠

ͯ(1−e^−x)^β−1−−−−→^x→∞ 1Ͱ͋Δ͜ͱʹ஫ҙ͢Δͱɼ f₀(x) = 1

B(α, β)e^−αx(1−e^−x)^β−1−−−−→^x→∞ 0

Λಘͯɼߋʹɼ(9)ࣜʹ஫ҙ͢Δͱ(α+β−1)e^−x−α−−−−→ −^x→∞ αʹΑΓɼ f₀(x) = f₀(x)

1−e^−x

(α+β−1)e^−x−α _x→∞

−−−−→0 ΛಘΔɽ

ୈೋʹɼx→0ʹ͓͚Δऩଋઌʹ͍ͭͯߟ͑Δɽ೚ҙͷα >0ʹରͯ͠

e^{−αx x}−−−→^→0 1Ͱ͋Δɽ·ͨɼ1−e^{−x x}−−−→^→0 0ʹΑΓɼ(1−e^−x)^β−1ͷऩଋ ઌ͸ɼβ <1, β = 1, β >1ͷ֤৔߹ʹԠͯͦ͡ΕͧΕ∞, 1, 0ͱͳΔɽ ΑͬͯɼB(α,1) = 1/αʹ஫ҙ͢Δͱɼ

f₀(x) = 1

B(α, β)e^−αx(1−e^−x)^β−1 −−−→^x→0

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∞ if β <1 α if β = 1 0 if β >1 ΛಘΔɽߋʹɼ(9)ࣜʹ஫ҙ͢Δͱɼ(α+β−1)e^−x−α−−−→^x→0 β−1ʹΑ Γɼβ <1ͷ৔߹ɼ

f₀(x) = f₀(x) 1−e^−x

(α+β−1)e^−x−α _x→0

−−−→ −∞

ΛಘΔ͕ɼβ ≥1ͷ৔߹͸ෆఆܗͱͳΔҝɼۃݶͷධՁʹ͸ผͳΔ޻෉Λ ཁ͢Δɽઌͣɼβ = 1ͷ৔߹ɼB(α,1) = 1/αʹ஫ҙ͢Δͱɼಋؔ਺f₀(x)

͸ҎԼͷΑ͏ʹ؆୯Խ͞ΕΔɿ

f₀(x) =−αf₀(x) =−α²e^{−αx x}−−−→ −^→0 α².

࣍ʹɼβ > 1ͷ৔߹Λߟ͑Δɽ(9)ࣜʹ(3)ࣜΛ୅ೖ͢Δͱɼ࣍ͷදݱΛ ಘΔɿ

f₀(x) = e^−αx(1−e^−x)^β−2 B(α, β)

(α+β−1)e^−x−α .

͜͜Ͱɼx→0ʹ͓͚Δ(1−e^−x)^β−2ͷऩଋઌ͸ɼβ <2, β= 2, β >2 ͷ֤৔߹ʹԠͯͦ͡ΕͧΕ∞, 1, 0ͱͳΔ͜ͱʹ஫ҙ͠ɼ·ͨɼB(α,2) = 1/[α(α+ 1)]ʹ஫ҙ͢ΔͱɼҎԼΛಘΔɿ

f₀(x) −−−→^x→0

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

∞ if β <2, α(α+ 1) if β= 2, 0 if β >2.

Ҏ্ͷٞ࿦ʹΑͬͯɼີ౓ؔ਺f₀(x)ͷ૿ݮද͸ද1ͱͯ͠ಘΒΕΔɽ͜

ΕʹΑΓɼ໋୊ͷ੒ཱ͸͔֬ΊΒΕͨɽ

ද 1: ࢦ਺ϕʔλ෼෍ͷີ౓ؔ਺f₀ʢ(3)ࣜʣͷ૿ݮදɿ্ஈࠨදʢβ <1 ͷ৔߹ʣɼ্ஈӈදʢβ= 1ͷ৔߹ɼࢦ਺෼෍ʣɼԼஈදʢβ >1ͷ৔߹ʣ

x 0 · · · ∞

f₀ −∞ − 0

f₀ ∞ 0

x 0 · · · ∞ f₀ −α² − 0

f₀ α 0

x 0 · · · log[1 + (β−1)/α] · · · ∞

f₀ ∗ + 0 − 0

f₀ 0 0

∗ · · · ∞ʢβ <2ͷ࣌ʣ, α(α+ 1)ʢβ= 2ͷ࣌ʣ, 0ʢβ >2ͷ࣌ʣ

໋୊1ʹΑΓɼβ >1ͷ৔߹ɼଈͪɼ(3)ࣜͰఆٛ͞ΕΔࢦ਺ϕʔλ෼

෍ͷີ౓ؔ਺f₀ ͷάϥϑͷܗঢ়͕୯ๆܕͷ৔߹ɼͦͷ࠷େ఺ʢ࠷ස஋ʣ

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