ҰൠԽΨϯϚ෼෍͸ɼຸ୩ʢ2010ʣ΍ Stacy (1962), Johnson and Kotz (1972), Johnson, Kotz and Balakrishnan (1994), Khodabin and Ahmad- abadi (2010), Forbes, Evans, Hastings and Peacock (2011) ౳Ͱߟ࡯͞Ε

(1)

1 ^ং࿦

௨ৗͷΨϯϚ෼෍͸ɼແݶ۠ؒ (0, ∞ ) ্ʹఆٛ͞ΕΔ࿈ଓܕͷ֬཰෼෍

Ͱ͋Γɼܗঢ়฼਺ͱई౓฼਺ͷ 2 ͭͷ฼਺Λ࣋ͭɽΨϯϚ෼෍͸ɼͦͷಛ घͳ৔߹ͱͯ͠ɼࢦ਺෼෍΍ΧΠ 2 ৐෼෍Λแؚ͢Δɽ·ͨɼΨϯϚ෼෍

ʹै͏֬཰ม਺Λۊม׵͢Δ͜ͱʹΑͬͯಘΒΕΔ֬཰෼෍ΛҰൠԽΨϯ Ϛ෼෍ͱ͍͍ɼͦͷಛघͳ৔߹ͱͯ͠ɼΨϯϚ෼෍ͷଞʹ Weibull ෼෍Λ แؚ͢Δɽ

ҰൠԽΨϯϚ෼෍͸ɼຸ୩ʢ2010ʣ΍ Stacy (1962), Johnson and Kotz (1972), Johnson, Kotz and Balakrishnan (1994), Khodabin and Ahmad- abadi (2010), Forbes, Evans, Hastings and Peacock (2011) ౳Ͱߟ࡯͞Ε

͍ͯΔɽ͔͠͠ɼҰൠԽΨϯϚ෼෍ͷ฼਺ͱܗঢ়ͷؔ܎ʹؔ͢Δମܥతͳ ݴٴ͸ɼJohnson, Kotz and Balakrishnan (1994) ʹ͓͍ͯۇ͔ʹݟΒΕΔ ఔ౓Ͱ͋Δɽͦͯ͠ɼJohnson, Kotz and Balakrishnan (1994, p.389) ʹ͓

͍ͯ΋ɼຊߘͷ (11) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ʹ͍ͭͯɼͦͷ෼෍

ͷܗঢ়͸ γα > 1 ͳΒ͹௼৊ܕͰ͋Γɼͦ͏Ͱͳ͚Ε͹ٯ J ࣈܕͰ͋Δͱ

ݴٴ͞ΕΔͷΈͰ͋Δɽຊߘ͸ɼΑΓৄࡉͳܗͰɼҰൠԽΨϯϚ෼෍ͷີ

౓ؔ਺ͷάϥϑͷܗঢ়͕ͦͷ฼਺ʹԠͯ͡ͲͷΑ͏ʹมԽ͢Δ͔Λݕ౼͠ɼ

ͦͷ݁Ռͱͯ͠ಘΒΕͨ஌ݟΛ੔ཧ͢Δʢ໋୊ 1ɼ໋୊ 2 ࢀরʣɽ

∗෱Ԭେֶܦࡁֶ෦ɼE-mail: [email protected]

一般化ガンマ分布の形状について

伴原理人^＊

(2)

ຊߘ͸ҎԼͷΑ͏ʹߏ੒͞ΕΔɽઌͣɼୈ 2 અʹ͓͍ͯɼΨϯϚؔ਺Λਖ਼ نԽఆ਺ͱͯ͠ඪ४ΨϯϚ෼෍Λߏ੒্ͨ͠Ͱɼඪ४ΨϯϚ෼෍ʹै͏֬

཰ม਺Λۊม׵͢Δ͜ͱʹΑͬͯҰൠԽΨϯϚ෼෍Λಋೖ͢ΔɽߋʹɼҰ ൠԽΨϯϚ෼෍ʹै͏֬཰ม਺Λ 1 ࣍ม׵͢Δ͜ͱʹΑͬͯͦͷҐஔई౓

෼෍଒Λߏ੒͢Δɽ࣍ʹɼୈ 3 અʹ͓͍ͯɼҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺

ͷάϥϑͷܗঢ়͕ͦͷ฼਺ʹԠͯ͡ͲͷΑ͏ʹมԽ͢Δ͔Λݕ౼͠ɼͦͷ

݁Ռͱͯ͠ಘΒΕͨ஌ݟΛ໋୊ 1ɼ໋୊ 2 ͱͯ͠੔ཧ͢Δɽ·ͨɼҰൠԽΨ ϯϚ෼෍ͷಛघͳ৔߹ͱͯ͠ɼΨϯϚ෼෍ͱ Weibull ෼෍ͷີ౓ؔ਺ʹͭ

͍ͯ΋ಉ༷ͷߟ࡯Λߦ͏ʢܥ 1ɼܥ 2 ࢀরʣɽҎ্ͷݕ౼ͷաఔͰಘΒΕΔ

ີ౓ؔ਺ͷ૿ݮද͸શͯิ࿦ A ʹఏࣔ͠ɼີ౓ؔ਺ͷάϥϑ͸શͯิ࿦ B ʹఏࣔ͢Δɽ࠷ޙʹɼୈ 4 અͰ݁࿦Λड़΂Δɽ

2 ҰൠԽΨϯϚ෼෍ͱͦͷҐஔई౓෼෍଒

࣮਺ α, x ʹରͯ͠ɼx ͷؔ਺ g(x) := x

^α−1

e

^−x

Λߟ͑Δɽ͜͜Ͱɼ౳߸

:= ͸ͦͷࠨลΛͦͷӈลʹΑͬͯఆٛ͢Δ͜ͱΛҙຯ͢Δɽͯ͞ɼα > 1 ʹ ରͯ͠ g(0) = 0ɼ α = 1 ʹରͯ͠ g(0) = 1ɼα < 1 ʹରͯ͠ g(x) −−−−−→ ∞

^x→0+0

Ͱ͋Γɼ·ͨɼ೚ҙͷ α ∈ ( −∞ , ∞ ) ʹରͯ͠ g(x) −−−−→

^x→∞

0 Ͱ͋Δ͜ͱʹ

஫ҙ͢Δɽ·ͣɼα ≥ 1 ͷ৔߹ɼ೚ҙͷਖ਼ͷ࣮਺ M ʹରͯ͠ɼؔ਺ g(x) ͸ ด۠ؒ [0, M ] Ͱ࿈ଓͳͷͰఆੵ෼

_M

0

x

^α−1

e

^−x

dx ͸ଘࡏ͠ɼ͔ͭɼ޿ٛੵ

෼

_∞

0

x

^α−1

e

^−x

dx := lim

_M_→∞

_M

0

x

^α−1

e

^−x

dx ΋ଘࡏ͢Δʢྫ͑͹ɼݘҪ 1962ɼ p.1 ΍ਿӜ 1980ɼ pp.295–296 ࢀরʣɽ࣍ʹɼ α < 1 ͷ৔߹ɼ 0 < < M ͳΔ೚ҙͷ࣮਺ ʹରͯ͠ɼؔ਺ g(x) ͸ด۠ؒ [, M ] Ͱ࿈ଓͳͷͰఆੵ෼

_M

x

^α−1

e

^−x

dx ͸ଘࡏ͢Δɽ͜͜Ͱɼߋʹɼα > 0 Ͱ͋ΔͳΒ͹ɼ޿ٛੵ

෼

_∞

0

x

^α−1

e

^−x

dx := lim

_→0,M→∞

_M

x

^α−1

e

^−x

dx ͸ଘࡏ͢ΔʢݘҪ 1962ɼ p.1ɼਿӜ 1980ɼpp.295–296 ࢀরʣɽैͬͯɼ೚ҙͷ α > 0 ʹରͯ͠ɼੵ

෼

_∞

0

x

^α−1

e

^−x

dx ͕ఆٛ͞ΕΔɽ͜ͷੵ෼Λਖ਼ͷ࣮਺ α ͷؔ਺ͱݟ၏ͨ͠

΋ͷΛΨϯϚؔ਺ͱ͍͍ɼΓ(α) ͱදه͢Δɿ

Γ(α) :=

_∞

0

x

^α−1

e

^−x

dx, α > 0. (1)

(3)

೚ҙͷਖ਼ͷ࣮਺ x ∈ (0, ∞ ) ʹରͯ͠ x

^α−1

e

^−x

> 0 Ͱ͋ΔͷͰɼ೚ҙͷ α > 0 ʹରͯ͠ɼΨϯϚؔ਺͸ৗʹਖ਼஋ΛऔΔɽଈͪɼ೚ҙͷਖ਼ͷ࣮਺ α > 0 ʹ ରͯ͠ Γ(α) > 0 Ͱ͋Δɽ

ҎԼɼୈ 2.1 અʹ͓͍ͯɼΨϯϚؔ਺Λਖ਼نԽఆ਺ͱͯ͠ඪ४ΨϯϚ෼෍

Λߏ੒্ͨ͠Ͱɼඪ४ΨϯϚ෼෍ʹै͏֬཰ม਺Λۊม׵͢Δ͜ͱʹΑͬ

ͯҰൠԽΨϯϚ෼෍Λಋೖ͢Δɽଓ͍ͯɼୈ 2.2 અʹ͓͍ͯɼҰൠԽΨϯ Ϛ෼෍ʹै͏֬཰ม਺Λ 1 ࣍ม׵͢Δ͜ͱʹΑͬͯͦͷҐஔई౓෼෍଒Λ ߏ੒͢Δɽ

2.1 ඪ४ΨϯϚ෼෍ͱҰൠԽΨϯϚ෼෍

ΨϯϚ෼෍ͱ͸ɼΨϯϚؔ਺Λਖ਼نԽఆ਺ͱͯ͠ߏ੒͞ΕΔ֬཰෼෍Ͱ

͋ΔɽଈͪɼҎԼͰఆٛ͞ΕΔີ౓ؔ਺ f

_∗

: (0, ∞ ) → ( −∞ , ∞ ) Λ࣋ͭ֬

཰෼෍ΛΨϯϚ෼෍ʢಛʹɼඪ४ΨϯϚ෼෍ʣͱ͍͍ɼα Λܗঢ়฼਺ͱ͍

͏ʢα > 0ʣ ɿ

f

_∗

(z | α) := 1

Γ(α) z

^α−1

e

^−z

, z > 0. (2)

࣮ࡍɼ೚ҙͷਖ਼ͷ࣮਺ α, z > 0 ʹରͯ͠ɼz

^α−1

e

^−z

> 0 ʹͯ͠ Γ(α) > 0 Ͱ

͋Δ͜ͱʹ஫ҙ͢Δͱɼf

_∗

(z | α) > 0 ΛಘΔɽ·ͨɼΨϯϚؔ਺ͷఆٛʹ஫

ҙ͢Δͱɼ

_∞

0

f

_∗

(z | α)dz = 1 ΛಘΔɽΑͬͯɼؔ਺ f

_∗

͸ɼඇෛ஋ੑͱਖ਼ن Խ৚݅Λຬͨ͢ͷͰɼ͔֬ʹີ౓ؔ਺Ͱ͋Δɽ

ͯ͞ɼZ Λඪ४ΨϯϚ෼෍ʹै͏֬཰ม਺ Z ∼ f

_∗

ͱͯ͠ɼਖ਼ͷ࣮਺ β ʹରͯ͠ Z Λۊม׵͢ΔɽଈͪɼX := Z

^β

, β > 0 ͱ͢Δɽ͜ͷ࣌ɼX ͷ

֬཰෼෍ΛҰൠԽΨϯϚ෼෍ͱ͍͍ɼβ ͸ α ͱಉ͘͡ܗঢ়฼਺ͱݺ͹ΕΔɽ

͜͜Ͱɼ0 < Z < ∞ , β > 0 ʹΑΓ 0 < X < ∞ ஫ҙ͢ΔͱɼX ͷ෼෍ؔ

਺ F

₀

͸ɼͦͷఆٛʹΑΓɼ0 < x < ∞ ͳΔ x ʹରͯ͠ɼ

F

₀

(x) := P(X ≤ x) = P

Z ≤ x

^β¹

=

_x¹β

0

f

_∗

(z | α)dz

(4)

ͱͯ͠ද͞Εɼͦͷີ౓ؔ਺ f

₀

͸ F

₀

ͷಋؔ਺ͱͯ͠ಘΒΕΔɿ f

₀

(x | α, β) = d

dx F

₀

(x) = 1

β f

_∗

(x

^β¹

| α)x

^β¹⁻¹

.

ैͬͯɼ0 < x < ∞ ͳΔ x ʹରͯ͠ɼX ͷີ౓ؔ਺ f

₀

͸࣍ࣜͱͳΔ ʢ0 < α, β < ∞ ʣ ɿ

f

₀

(x | α, β) = 1

β Γ(α) x

^α^β⁻¹

e

^−x

1β

. (3)

γ := 1/β ͳΔ฼਺ม׵Λࢪ͢ͱɼX := Z

^β

= Z

^1/γ

ͷີ౓ؔ਺ f

₀

ͷผදݱ

͕ಘΒΕΔʢ0 < α, γ < ∞ ʣ ɿ f

₀

(x | α, γ) = γ

Γ(α) x

^γα−1

e

^−x^γ

, 0 < x < ∞ . (4) Ҏ্ͷΑ͏ʹɼ(3) ࣜ΋͘͠͸ (4) ࣜͰఆٛ͞Εͨີ౓ؔ਺ f

₀

Λ࣋ͭ֬཰

෼෍ΛҰൠԽΨϯϚ෼෍ʢಛʹɼҰൠԽΨϯϚ෼෍ͷඪ४ܕʣͱ͍͍ɼα ͱ

βʢ΋͘͠͸ γʣΛܗঢ়฼਺ͱ͍͏ɽҎԼʹ͓͍ͯ͸ɼಛʹஅΒͳ͍ݶΓɼ

ҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ f

₀

ͱͯ͠͸ (4) ࣜͷදݱΛ༻͍Δɽ

ͯ͞ɼγ = 1ʢ ⇐⇒ β = 1ʣͷ৔߹ɼҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺͸ඪ

४ΨϯϚ෼෍ͷີ౓ؔ਺ʹؼண͢Δɿ f

₀

(x | α, 1) = f

_∗

(x | α) = 1

Γ(α) x

^α−1

e

^−x

, 0 < x < ∞ . (5)

·ͨɼα = 1 ͷ৔߹ɼΓ(1) = 1 ʹ஫ҙ͢ΔͱɼҰൠԽΨϯϚ෼෍ͷີ౓ؔ

਺ f

₀

͸ඪ४ Weibull ෼෍ͷີ౓ؔ਺ʹؼண͢Δɿ

f

₀

(x | 1, γ) = γx

^γ−1

e

^−x^γ

, 0 < x < ∞ . (6) ߋʹɼα = γ = 1 ͷ৔߹ɼҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ f

₀

͸ඪ४ࢦ਺෼෍

ͷີ౓ؔ਺ʹؼண͢Δɿ

f

₀

(x | 1, 1) = e

^−x

, 0 < x < ∞ . (7) Ҏ্ʹΑΓɼҰൠԽΨϯϚ෼෍ͷ 2 ͭͷܗঢ়฼਺ α ͱ γʢ΋͘͠͸ β ʣʹ

͍ͭͯɼα ͸ΨϯϚ෼෍ͷܗঢ়฼਺ɼγʢ΋͘͠͸ βʣ͸ Weibull ෼෍ͷܗ

ঢ়฼਺ʹରԠ͢Δͱ෼͔Δɽ

(5)

࠷ޙʹɼ(4) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ͷඪ४ܕʹै͏֬཰ม਺ X ͷجຊతͳಛੑ஋ͱͯ͠ɼͦͷظ଴஋ͱ෼ࢄ͸࣍ͷΑ͏ʹٻ·Δɿ

E(X) = Γ(α + 1/γ )

Γ(α) , V (X) = Γ(α + 2/γ) Γ(α) −

Γ(α + 1/γ) Γ(α)

₂

. (8)

࣮ࡍɼ೚ҙͷਖ਼ͷ࣮਺ k ʹରͯ͠ɼX ͷ k ࣍Ϟʔϝϯτ͸ɼ E

X

^k

=

_∞

0

x

^k

f

₀

(x | α, γ)dx

= Γ(α + k/γ) Γ(α)

_∞

0

γ

Γ(α + k/γ ) x

^{γ(α+k/γ)−1}

e

^−x^γ

dx = Γ(α + k/γ ) Γ(α) ͱٻΊΒΕΔɽ͜͜Ͱɼ࠷ޙͷ౳ࣜ͸ɼͦͷࠨลͷඃੵ෼ؔ਺͕ (4) ࣜͰ ఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ͳ͍ͬͯΔ͜ͱɼͭ·Γɼ

f

₀

(x | α + k/γ, γ) = γ

Γ(α + k/γ ) x

^{γ(α+k/γ)−1}

e

^−x^γ

Ͱ͋Δ͜ͱʹ஫ҙ͢Δͱɼີ౓ؔ਺ͷਖ਼نԽ৚݅ʹΑͬͯ੒ཱ͢Δ͜ͱ͕

෼͔Δɽͳ͓ɼ෼ࢄެࣜ V (X) = E(X

²

) − [E(X)]

²

ʹ஫ҙ͢Δɽ

ͯ͞ɼҰൠԽΨϯϚ෼෍ͷඪ४ܕͷظ଴஋ͱ෼ࢄ (8) ࣜ͸ɼγ = 1 ͷ৔

߹ɼͭ·Γɼඪ४ΨϯϚ෼෍ (5) ࣜͷ৔߹ɼΓ(α + 1) = αΓ(α) ʹ஫ҙ͢Δ ͱɼ࣍ͷΑ͏ʹ؆୯Խ͞ΕΔɿ

E(X) = α, V (X ) = α. (9)

·ͨɼα = 1 ͷ৔߹ɼͭ·Γɼඪ४ Weibull ෼෍ (6) ࣜͷ৔߹ɼΓ(1) = 1 ʹ

஫ҙ͢Δͱɼ࣍ͱͳΔɿ

E(X ) = Γ

1 + 1 γ

, V (X ) = Γ

1 + 2 γ

−

Γ

1 + 1 γ

₂

. (10)

2.2 ҰൠԽΨϯϚ෼෍ͷҐஔई౓෼෍଒

ຊઅ͸ɼ(4) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ͷඪ४ܕʹҐஔ฼਺ͱई౓

฼਺Λಋೖ͢ΔɽX ΛҰൠԽΨϯϚ෼෍ͷඪ४ܕʹै͏֬཰ม਺ X ∼ f

₀

(6)

ͱͯ͠ɼ࣮਺ μ ͱਖ਼ͷ࣮਺ σ ʹରͯ͠ Z Λ 1 ࣍ม׵͢ΔɽଈͪɼY :=

μ + σX, −∞ < μ < ∞ , σ > 0 ͱ͢Δɽ͜ͷ࣌ɼY ͷ֬཰෼෍Λ μ ΛҐஔ

฼਺ɼσ Λई౓฼਺ͱ͢ΔҰൠԽΨϯϚ෼෍ͷҐஔई౓෼෍଒ͱ͍͏ɽͦ

ͷ෼෍ؔ਺ F ͸ɼͦͷఆٛʹΑΓɼy > μ ͳΔ y ʹରͯ͠ɼ

F (y) := P (Y ≤ y) = P

X ≤ y − μ σ

=

^y−μ

σ

0

f

₀

(x | α, γ)dx Ͱ͋Γɼͦͷີ౓ؔ਺ f ͸ F ͷಋؔ਺ͱͯ͠ಋग़͞ΕΔɿ

f (y | α, γ, μ, σ) = d

dy F (y) = 1 σ f

₀

y − μ σ

α, γ

.

ैͬͯɼy > μ ͳΔ y ʹରͯ͠ɼY ͷີ౓ؔ਺ f ͸࣍ࣜͱͯ͠ಘΒΕΔɿ f (y | α, γ, μ, σ) = γ

σΓ(α)

y − μ σ

_γα−1

e

⁻

(

^y−μ_σ

)

^γ

. (11)

͜͜ͰɼҐஔ฼਺ μ ͸೚ҙͷ࣮਺ͰΑ͍ͷʹରͯ͠ɼई౓฼਺ σ ͱ 2 ͭͷܗ ঢ়฼਺ α ͱ γ ͸ਖ਼஋Ͱ͋Δ͜ͱʹ஫ҙ͢Δɿ μ ∈ ( −∞ , ∞ ), σ, α, γ ∈ (0, ∞ ).

μ = 0, σ = 1 ͷ৔߹ɼҰൠԽΨϯϚ෼෍ͷҐஔई౓෼෍଒͸ҰൠԽΨϯϚ

෼෍ͷඪ४ܕ (4) ࣜʹؼண͢Δɿf (y | α, γ, 0, 1) = f

₀

(y | α, γ), y > μ = 0.

ͯ͞ɼୈ 2.1 અͰࢦఠͨ͠௨Γɼ(11) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍

ͷҐஔई౓෼෍଒ʹ͓͍ͯɼ2 ͭͷܗঢ়฼਺ α ͱ γ ͸ͦΕͧΕΨϯϚ෼෍

ͱ Weibull ෼෍ͷܗঢ়฼਺ʹରԠ͢Δɽ

ઌͣɼγ = 1 ͷ৔߹ɼ(11) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ͷҐஔई౓

෼෍଒ͷີ౓ؔ਺ f ͸ඪ४ΨϯϚ෼෍ (5) ࣜͷҐஔई౓෼෍଒ͷີ౓ؔ਺

ʹؼண͢Δʢy > μʣ ɿ

f (y | α, 1, μ, σ) = 1 σΓ(α)

y − μ σ

_α−1

e

⁻

(

^y−μ_σ

) . (12) ߋʹɼ೚ҙͷਖ਼ͷ࣮਺ n ʹରͯ͠ʢ0 < n < ∞ ʣɼα = n/2, σ = 2, μ = 0 Ͱ͋ΔͳΒ͹ɼࣗ༝౓ n ͷΧΠ 2 ৐෼෍ͷີ౓ؔ਺ʹؼண͢Δʢy > 0ʣ ɿ

f (y | n/2, 1, 0, 2) = 1

2

^n/2

Γ(n/2) y

ⁿ²⁻¹

e

⁻^y²

.

(7)

͜͜Ͱɼࣗ༝౓ n ͷΧΠ 2 ৐෼෍͸ɼඞͣ͠΋ͦͷࣗ༝౓Λࣗવ਺ʹݶΔ ඞཁ͸ͳ͘ɼ೚ҙͷਖ਼ͷ࣮਺ n ʹରͯ͠ఆٛ͞ΕಘΔ͜ͱʹ஫ҙ͢Δɽ

࣍ʹɼα = 1 ͷ৔߹ɼҰൠԽΨϯϚ෼෍ͷҐஔई౓෼෍଒ͷີ౓ؔ਺ f ͸ ඪ४ Weibull ෼෍ (6) ࣜͷҐஔई౓෼෍଒ͷີ౓ؔ਺ʹؼண͢Δʢy > μʣ ɿ

f(y | 1, γ, μ, σ) = γ σ

y − μ σ

_γ−1

e

⁻

(

^y−μ_σ

)

^γ

. (13)

·ͨɼα = γ = 1 ͷ৔߹ɼҰൠԽΨϯϚ෼෍ͷҐஔई౓෼෍଒ͷີ౓ؔ਺

f ͸ඪ४ࢦ਺෼෍ (7) ࣜͷҐஔई౓෼෍଒ͷີ౓ؔ਺ʹؼண͢Δʢy > μʣ ɿ f (y | 1, 1, μ, σ) = 1

σ e

⁻^y−μ^σ

. (14)

࠷ޙʹɼҰൠԽΨϯϚ෼෍ͷҐஔई౓෼෍଒ (11) ࣜʹै͏֬཰ม਺ Y ͷ ظ଴஋ͱ෼ࢄ͸ɼY := μ + σX, X ∼ f

₀

ʹ஫ҙ͢Δͱɼ(8) ࣜʹΑΓ

E(Y ) = μ + σ Γ(α + 1/γ)

Γ(α) , V (Y ) = σ

²

Γ(α + 2/γ) Γ(α) −

Γ(α + 1/γ) Γ(α)

₂

ͱٻ·Γɼ͜Ε͸ɼγ = 1 ͷ৔߹ɼͭ·Γɼඪ४ΨϯϚ෼෍ͷҐஔई౓෼෍

଒ (12) ࣜͷ৔߹ɼ

E(Y ) = μ + σα, V (Y ) = σ

²

α

ͱ؆୯Խ͞Εɼα = 1 ͷ৔߹ɼͭ·Γɼඪ४ Weibull ෼෍ͷҐஔई౓෼෍

଒ (13) ࣜͷ৔߹ɼ࣍ͱͳΔɿ

E(Y ) = μ + σΓ

1 + 1 γ

, V (Y ) = σ

²

Γ

1 + 2 γ

−

Γ

1 + 1 γ

₂

.

3 ҰൠԽΨϯϚ෼෍ͷܗঢ়

ຊઅͰ͸ɼҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ͷάϥϑͷܗঢ়ʹ͍ͭͯߟ࡯͢

ΔɽͦͷࡍɼҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ʹ͍ͭͯɼ(4) ࣜͰఆٛ͞ΕΔඪ

(8)

४ܕͷີ౓ؔ਺ f

₀

(x | α, γ), 0 < x < ∞ ͱ (11) ࣜͰఆٛ͞ΕΔͦͷҐஔई

౓෼෍଒ͷີ౓ؔ਺ f (y | α, γ, μ, σ), μ < y < ∞ ͱͷؒʹҎԼͷؔ܎͕ࣜ੒

ཱ͢Δ͜ͱʹ஫ҙ͢Δɽୠ͠ɼҎԼͰ͸ɼ฼਺Λಛʹ໌ࣔ͢Δඞཁ͕ͳ͍

৔߹ɼf

₀

(x) := f

₀

(x | α, γ), f (y) := f(y | α, γ, μ, σ) ͱུه͢Δɽ f (y) = 1

σ f

₀

y − μ σ

.

͜ΕʹΑΓɼͦΕΒͷ 1 ֊ಋؔ਺ʹ͍ͭͯ͸

f

(y) = 1 σ

²

f

₀

y − μ σ

ͳΔؔ܎͕੒ཱ͠ɼͦͯ͠ɼσ

²

> 0 ʹ஫ҙ͢Δͱɼ࣍ͷಉ஋ؔ܎ΛಘΔɿ f

(y) ≶ 0 ⇐⇒ f

₀

y − μ σ

≶ 0.

ΑͬͯɼҰൠԽΨϯϚ෼෍ͷඪ४ܕͷີ౓ؔ਺ f

₀

(x) ͷάϥϑ { (x, f

₀

(x)) | x ∈ (0, ∞ ) } ͷܗঢ়͕൑໌͢Ε͹ɼͦͷಠཱม਺ x ͷ஋Λ y = σx + μ ͱஔ

͖׵্͑ͨͰؔ਺஋ f

₀

(x) Λ 1/σ ഒ͢Δ͜ͱʹΑͬͯɼҰൠԽΨϯϚ෼෍

ͷҐஔई౓෼෍଒ͷີ౓ؔ਺ f (y) ͷάϥϑ { (y, f(y)) | y ∈ (μ, ∞ ) } ͷܗঢ়

΋൑໌͢Δͱ෼͔Δɽ

ैͬͯɼҎԼͰ͸ɼ(4) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ͷඪ४ܕͷີ౓

ؔ਺ f

₀

ͷάϥϑͷܗঢ়͕ͦͷܗঢ়฼਺ α ͱ γ ʹԠͯ͡ͲͷΑ͏ʹมԽ͢

Δ͔Λݕ౼্ͨ͠ͰɼͦͷҐஔई౓෼෍଒ͷີ౓ؔ਺ f ͷάϥϑͷܗঢ়͕

ͦͷҐஔ฼਺ μ ͱई౓฼਺ σ ʹԠͯ͡ͲͷΑ͏ʹมԽ͢Δ͔ʹ͍ͭͯݕ౼ ΛՃ͑Δɽͦͷࡍɼୈ 3.1 અͰҰൠͷ৔߹ʹ͍ͭͯٞ࿦ͨ͠ޙɼୈ 3.2 અͰ ز͔ͭͷಛघͳ৔߹ʹ͍ͭͯٞ࿦͢Δɽ

3.1 Ұൠͷ৔߹ɿҰൠԽΨϯϚ෼෍

(4) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ͷඪ४ܕͷີ౓ؔ਺ f

₀

ͷ 1 ֊ಋؔ

਺ͱͯ͠ҎԼ͕ಘΒΕΔɿ f

₀

(x) = γ

Γ(α) [(γα − 1) − γx

^γ

] x

^γα−2

e

^−x^γ

, x > 0.

(9)

೚ҙͷ α, γ > 0 ͱ x > 0 ʹରͯ͠ɼγx

^γα−2

e

^−x^γ

/Γ(α) > 0 Ͱ͋ΔͷͰɼ f

₀

(x) ≶ 0 ⇐⇒ (γα − 1) − γx

^γ

≶ 0

Ͱ͋Δͱ෼͔ΔɽҎԼʹ͓͍ͯɼγα − 1 ≤ 0 ͷ৔߹ͱ γα − 1 > 0 ͷ৔߹

ͱʹେผͯٞ͠࿦͢Δɽ

ୈҰʹɼγα − 1 ≤ 0 ⇐⇒ γα ≤ 1 ͷ৔߹ɼ೚ҙͷ x ∈ (0, ∞ ) ʹରͯ͠

γx

^γ

> 0 Ͱ͋Δ͜ͱʹ஫ҙ͢Δͱɼ(γα − 1) − γx

^γ

< 0 ΛಘΔɽΑͬͯɼ (γα − 1) − γx

^γ

< 0 ⇐⇒ f

₀

(x) < 0

ʹΑΓɼγα ≤ 1 ͷ৔߹ɼҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ f

₀

(x) ͸ x ͷ୯ௐ ݮগؔ਺Ͱ͋Δͱ෼͔Δɽ͜͜Ͱɼߋʹɼγα < 1 ͷ৔߹ͱ γα = 1 ͷ৔߹

ͷ 2 ͭʹ෼͚ͯߟ͑Δɽઌͣɼγα < 1 ͷ৔߹ʹ͍ͭͯߟ͑Δͱɼx → 0 ͷ

࣌ɼx

^γα−1

→ ∞ , x

^γα−2

→ ∞ , x

^γ

→ 0 Ͱ͋Γɼx → ∞ ͷ࣌ɼx

^γα−1

→ 0, x

^γα−2

→ 0, x

^γ

→ ∞ , x

^γ

= o(e

^x^γ

) Ͱ͋Δ͜ͱʹ஫ҙ͢ΔͱɼҎԼΛ ಘΔɿ

f

₀

(x) −−−→ ∞

^x→0

, f

₀

(x) −−−−→

^x→∞

0, f

₀

(x) −−−→ −∞

^x→0

, f

₀

(x) −−−−→

^x→∞

0. ࣍ʹɼγα = 1 ͷ৔߹Λߟ͑ΔͱɼҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ f

₀

(x) ͱͦ

ͷ 1 ֊ಋؔ਺ f

₀

(x) ͸࣍ͷΑ͏ʹ୯७Խ͞ΕΔɿ f

₀

(x) = γ

Γ(α) e

^−x^γ

, f

₀

(x) = − γ

²

Γ(α) x

^γ−1

e

^−x^γ

. Αͬͯɼγ = 1/α, αΓ(α) = Γ(α + 1) ʹ஫ҙ͢Δͱɼ

f

₀

(x) −−−→

^x→0

1 Γ(α + 1) , f

₀

(x) −−−−→

^x→∞

0 ΛಘΔɽߋʹɼγ < 1 ͳΒ͹ɼ

f

₀

(x) −−−→ −∞

^x→0

, f

₀

(x) −−−−→

^x→∞

0

(10)

Ͱ͋Γɼγ = 1 ͳΒ͹ɼΓ(1) = 1 ʹ஫ҙ͢Δͱɼ f

₀

(x) −−−→ −

^x→0

1, f

₀

(x) −−−−→

^x→∞

0 Ͱ͋Γɼγ > 1 ͳΒ͹ɼ࣍ΛಘΔɿ

f

₀

(x) −−−→

^x→0

0, f

₀

(x) −−−−→

^x→∞

0. Ҏ্ʹΑΓɼγα − 1 ≤ 0 ͷ৔߹ɼҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ f

₀

(x) ͸ x ͷ୯ௐݮগؔ਺Ͱ͋Γɼͦͷ૿ݮදͱͯ͠ γα − 1 < 0 ͷ৔߹ʹ͸ද 1ɼ γα − 1 = 0 ͷ৔߹ʹ͸ද 2 ͕ಘΒΕΔʢิ࿦ A ࢀরʣɽ

ୈೋʹɼγα − 1 > 0 ⇐⇒ γα > 1 ͷ৔߹ɼ

(γα − 1) − γx

^γ

≶ 0 ⇐⇒ x ≷

γα − 1 γ

¹_γ

=: x

_m

ʹΑΓɼҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ f

₀

(x) ͸ɼ 0 < x < x

_m

ͳΔ x ʹରͯ͠

͸୯ௐ૿Ճɼx

_m

< x < ∞ ͳΔ x ʹରͯ͠͸୯ௐݮগͱͳΓɼ x = x

_m

Λ࠷

େ఺ʢ࠷ස஋ʣͱ͢Δ୯ๆܕͷܗঢ়Ͱ͋Δɽ·ͨɼx → 0 ͷ࣌ʹ x

^γα−1

→ 0 ͱͳΓɼx → ∞ ͷ࣌ʹ x

^γα−1

→ ∞ ʹͯ͠ x

^γα−1

= o(e

^x^γ

) ͱͳΔ͜ͱʹ

஫ҙ͢Δͱɼ࣍ΛಘΔɿ

f

₀

(x) −−−→

^x→0

0, f

₀

(x) −−−−→

^x→∞

0. 1 ֊ಋؔ਺ f

₀

(x) ʹ͍ͭͯಉ༷ͷߟ࡯Λ܁Γฦ͢ͱɼ1 < γα < 2 ͷ৔߹ʹ f

₀

(x) −−−→ ∞

^x→0

, f

₀

(x) −−−−→

^x→∞

0,

γα = 2 ͷ৔߹ʹ

f

₀

(x) −−−→

^x→0

2 Γ(α + 1) , f

₀

(x) −−−−→

^x→∞

0, γα > 2 ͷ৔߹ʹҎԼΛಘΔɿ

f

₀

(x) −−−→

^x→0

0, f

₀

(x) −−−−→

^x→∞

0.

(11)

Ҏ্ʹΑΓɼγα − 1 > 0 ͷ৔߹ɼҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ f

₀

(x) ͷ૿

ݮදͱͯ͠ද 3 ͕ಘΒΕΔʢิ࿦ A ࢀরʣɽ

Ҏ্ͷٞ࿦ʹΑͬͯɼҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ f

₀

ͷάϥϑͷܗঢ়ʹ

͍ͭͯҎԼͷ໋୊ΛಘΔɽ

໋୊ 1 (ҰൠԽΨϯϚ෼෍ͷܗঢ়ɿඪ४ܕͷ৔߹) . (4) ࣜͰఆٛ͞ΕΔҰൠ

ԽΨϯϚ෼෍ͷີ౓ؔ਺ f

₀

ͷάϥϑ { (x, f

₀

(x | α, γ)) | x ∈ (0, ∞ ) } ͷܗঢ়

͸ɼਖ਼஋ͷܗঢ়฼਺ α ͱ γ ʹԠͯ͡ҎԼͷΑ͏ʹఆ·Δʢα, γ > 0ʣɽୠ

͠ɼҎԼͰ͸ f

₀

(x) := f (x | α, γ) ͱུه͢Δɽͳ͓ɼਤʹ͍ͭͯ͸ิ࿦ B Λࢀর͞Ε͍ͨɽ

1. γα ≤ 1 ͷ৔߹ʢ୯ௐݮগܕʣ

ີ౓ؔ਺ f

₀

(x) ͸ x ͷ୯ௐݮগؔ਺Ͱ͋Γɼͦͷάϥϑ͸ӈԼΓͷ ܗঢ়Λࣔ͢ɽಛʹɼx → ∞ ͷ৔߹ʹ͓͍ͯ f

₀

(x) → 0 Ͱ͋Δɽ

(a) γα < 1 ͷ৔߹ʢඇ༗քɿਤ 1 ʣ x → 0 ʹ͓͍ͯ f

₀

(x) → ∞ Ͱ͋Δɽ (b) γα = 1 ͷ৔߹ʢ༗քɿਤ 2 ʣ

x → 0 ʹ͓͍ͯ f

₀

(x) → 1/Γ(α + 1) Ͱ͋Δ

¹

ɽ 2. γα > 1 ͷ৔߹ʢ୯ๆܕɿਤ 3 ɼਤ 4 ʣ

ີ౓ؔ਺ f

₀

(x | α, γ) ͸ x = x

_m

Λ࠷େ఺ʢ࠷ස஋ʣͱͯ͠ɼͦͷά ϥϑ͸୯ๆܕͷܗঢ়Λࣔ͢ɿ

x

_m

=

γα − 1 γ

¹_γ

.

·ͨɼ͜ͷ৔߹ɼx → 0 ΋͘͠͸ x → ∞ ͷ͍ͣΕͷ৔߹ʹ͓͍ͯ΋

f

₀

(x | α, γ) → 0 Ͱ͋Δ

²

ɽ

1x→0ʹ͓͍ͯີ౓ؔ਺f0(x)͕1/Γ(α+ 1)ʹऩଋ͢Δࡍɼͦͷ܏͖͸ɼγ <1 ⇐⇒

α >1ͷ৔߹ʹෛͷແݶେʹൃࢄ͠ʢf₀(x)−−−→ −∞^x→0 ʣɼγ= 1 ⇐⇒ α= 1ͷ৔߹ʹఆ਺

ʹऩଋ͠ʢf₀(x)−−−→ −^x→0 1ʣɼγ >1 ⇐⇒ α <1ͷ৔߹ʹ0ʹऩଋ͢Δʢf₀(x)−−−→^x→0 0ʣ ͱ͍͏ܗΛऔΔʢਤ2ࢀরʣɽ

2x→0ʹ͓͍ͯີ౓ؔ਺f0(x)͕0ʹऩଋ͢Δࡍɼͦͷ܏͖͸ɼγα <2ͷ৔߹ʹແݶ

େʹൃࢄ͠ʢf₀(x)−−−→ ∞^x→0 ʣɼγα= 2ͷ৔߹ʹఆ਺ʹऩଋ͠ʢf₀(x)−−−→^x→0 2/Γ(α+ 1)ʣɼ γα >2ͷ৔߹ʹ0ʹऩଋ͢Δʢf₀(x)−−−→^x→0 0ʣͱ͍͏ܗΛऔΔʢਤ3ɼਤ4ࢀরʣɽ

(12)

Ҏ্ʹ͓͍ͯɼҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ͷܗঢ়͕ 2 ͭͷܗঢ়฼਺ʹ Ԡͯ͡ͲͷΑ͏ʹมԽ͢Δ͔͕໌Β͔ʹͳͬͨɽຊઅͷ࠷ޙʹɼҰൠԽΨ ϯϚ෼෍ͷີ౓ؔ਺ͷܗঢ়ͱҐஔ฼਺ɼई౓฼਺ͷؔ܎Λ໌Β͔ʹ͠Α͏ɽ

(11) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ͷҐஔई౓෼෍଒ͷີ౓ؔ਺ f ʹ

͓͍ͯɼҐஔ฼਺ μ ͕ͦͷάϥϑΛ μ ͚ͩࠨӈʹฏߦҠಈͤ͞Δ͜ͱ͸໌

Β͔Ͱ͋Ζ͏ɽैͬͯɼҎԼʹ͓͍ͯ͸ɼई౓฼਺ͱີ౓ؔ਺ f ͷܗঢ়ͷ

ؔ܎ʹ͍ͭͯݕ౼͢Δɽ

0 < σ < τ ͱͯ͠ɼ(11) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ͷҐஔई౓෼

෍଒ͷີ౓ؔ਺ f ͕೚ҙͷ y ∈ (μ, ∞ ) ʹରͯ͠ਖ਼஋Ͱ͋Δ͜ͱʹ஫ҙ͢Δ ͱɼ࣍ͷಉ஋ؔ܎ΛಘΔɿ

f (y | α, γ, μ, σ) ≶ f (y | α, γ, μ, τ) ⇐⇒ f (y | α, γ, μ, σ) f (y | α, γ, μ, τ) ≶ 1.

·ͨɼର਺ม׵͕୯ௐม׵Ͱ͋Δ͜ͱʹ஫ҙ͢Δͱɼ

f (y | α, γ, μ, σ) ≶ f (y | α, γ, μ, τ) ⇐⇒ log f (y | α, γ, μ, σ) f (y | α, γ, μ, τ) ≶ 0 ΛಘΔɽͯ͞ɼ

f (y | α, γ, μ, σ) f(y | α, γ, μ, τ ) =

τ σ

_γα

e

⁻^τγ−σγ^{σγ τγ} ^(y−μ)^γ

ʹΑΓɼҎԼΛಘΔɿ

log f (y | α, γ, μ, σ)

f (y | α, γ, μ, τ) = γα(log τ − log σ) − τ

^γ

− σ

^γ

σ

^γ

τ

^γ

(y − μ)

^γ

≶ 0.

⇐⇒ (y − μ)

^γ

≷ γασ

^γ

τ

^γ

τ

^γ

− σ

^γ

(log τ − log σ).

͜͜Ͱɼγ > 0, y > μ, γασ

^γ

τ

^γ

(log τ − log σ)/(τ

^γ

− σ

^γ

) > 0 Ͱ͋Δ͜ͱʹ

஫ҙ͠ɼਖ਼ͷ࣮਺ʹର͢Δਖ਼ͷۊม׵͕୯ௐม׵Ͱ͋Δ͜ͱʹ஫ҙ͢Δͱɼ

log f (y | α, γ, μ, σ)

f (y | α, γ, μ, τ ) ≶ 0 ⇐⇒ y − μ ≷

γασ

^γ

τ

^γ

τ

^γ

− σ

^γ

(log τ − log σ)

_γ¹

(13)

ΛಘΔɽҎ্ʹΑΓɼ࣍ͷಉ஋ؔ܎ΛಘΔɿ f(y | α, γ, μ, σ) ≶ f(y | α, γ, μ, τ) ⇐⇒ y ≷ μ+

γασ

^γ

τ

^γ

τ

^γ

− σ

^γ

(log τ − log σ)

¹_γ

.

͜ΕʹΑΓɼ(11) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ͷҐஔई౓෼෍଒ʹ͓

͍ͯɼई౓฼਺͕େ͖͘ͳΔఔɼͦͷີ౓ؔ਺͸ӈ੄͕ް͘ͳΓɼࠨ੄͕

ബ͘ͳΔ͜ͱ͕൑໌͢Δɽ

Ҏ্ͷٞ࿦ʹΑͬͯɼҰൠԽΨϯϚ෼෍ͷҐஔई౓෼෍଒ͷີ౓ؔ਺ f ͷάϥϑͷܗঢ়ʹ͍ͭͯ࣍ͷ໋୊ΛಘΔɽ

໋୊ 2 (ҰൠԽΨϯϚ෼෍ͷܗঢ়ɿҐஔई౓෼෍଒ͷ৔߹) . (11) ࣜͰఆٛ͞Ε

ΔҰൠԽΨϯϚ෼෍ͷҐஔई౓෼෍଒ͷີ౓ؔ਺f ͷάϥϑ { (y, f(y | α, γ, μ, σ)) | y ∈ (μ, ∞ ) } ͷܗঢ়͸ɼ 4 ͭͷ฼਺ α, γ, μ, σ ʹԠͯ͡ҎԼͷΑ͏ʹఆ·

Δɽୠ͠ɼα, γ, σ ∈ (0, ∞ ), μ ∈ ( −∞ , ∞ ) Ͱ͋Δɽਤʹ͍ͭͯ͸ิ࿦ B Λ

ࢀর͞Ε͍ͨɽ

1. 2 ͭͷܗঢ়฼਺ α ͱ γ ʹԠͯ͡ɼີ౓ؔ਺ f (y | α, γ, μ, σ) ͷάϥϑͷ ܗঢ়͸ɼඪ४ܕͷີ౓ؔ਺ f

₀

(x | α, γ) ͱಉ༷ͷܗͰܾఆ͞ΕΔʢ໋୊

1 ࢀরʣɽୠ͠ɼಠཱม਺ͷ࠲ඪ஋͸ y = μ + σx ͱม׵͞Εɼؔ਺஋

ʢີ౓ؔ਺ͷߴ͞ʣ͸ 1/σ ഒ͞Εͳ͚Ε͹ͳΒͳ͍ɽ

2. Ґஔ฼਺ μ ͕େ͖͘ͳΔͱɼີ౓ؔ਺ f (y | α, γ, μ, σ) ͷάϥϑ͸ӈʹ ฏߦҠಈ͠ɼখ͘͞ͳΔͱࠨʹฏߦҠಈ͢Δʢਤ 7 ɼਤ 8 ɼਤ 15 ɼਤ 16 ɼਤ 20 ࢀরʣɽ

3. ई౓฼਺ σ ͕େ͖͘ͳΔͱɼີ౓ؔ਺ f (y | α, γ, μ, σ) ͷάϥϑ͸ͦͷ ӈ੄͕ް͘ͳΓɼࠨ੄͕ബ͘ͳΔʢਤ 9 ɼਤ 10 ɼਤ 11 ɼਤ 17 ɼਤ 18 ɼ ਤ 19 ɼਤ 21 ࢀরʣɽΑΓਖ਼֬ʹදݱ͢Δͱɼσ < τ Ͱ͋Δ 2 ͭͷई

౓฼਺ σ ͱ τ ʹରͯ͠ɼy ͕Ұఆͷ஋ μ + y

_στ

ΑΓখ͍͞৔߹ʹ͸

f (y | α, γ, μ, σ) > f(y | α, γ, μ, τ)ɼେ͖͍৔߹ʹ͸ f (y | α, γ, μ, σ) <

f (y | α, γ, μ, τ) Ͱ͋Γɼy = μ + y

_στ

ͷ৔߹ʹ͸ f (y | α, γ, μ, σ) = f (y | α, γ, μ, τ) Ͱ͋Δɿ

y

_στ

:=

γασ

^γ

τ

^γ

τ

^γ

− σ

^γ

log τ

σ

_γ¹

.

(14)

3.2 ز͔ͭͷಛघͳ৔߹

(4) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ͷඪ४ܕͷີ౓ؔ਺ f

₀

͸ɼγ = 1 ͱ

ͨ͠৔߹ɼ(5) ࣜͰݟͨΑ͏ʹඪ४ΨϯϚ෼෍ͷີ౓ؔ਺ʹؼண͠ɼα = 1 ͱͨ͠৔߹ɼ(6) ࣜͰݟͨΑ͏ʹඪ४ Weibull ෼෍ͷີ౓ؔ਺ʹؼண͢Δɽ ຊઅͰ͸ɼ͜ͷಛघͳ৔߹ʹ͓͍ͯɼҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ͷܗঢ় ΛվΊͯߟ࡯͢Δɽͳ͓ɼͦͷଞͷಛघͳ৔߹ͱͯ͠ɼα = 1/2 ͷ৔߹ͱ α = 2 ͷ৔߹ɼγ = 1/2 ͷ৔߹ɼγ = 2 ͷ৔߹ʹ͍ͭͯɼҰൠԽΨϯϚ෼෍

ͷີ౓ؔ਺ͷάϥϑΛͦΕͧΕਤ 22ɼਤ 23ɼਤ 24ɼਤ 25 ʹఏࣔ͢Δʢิ

࿦ B ࢀরʣɽ

3.2.1 γ = 1 ͷ৔߹ɿΨϯϚ෼෍

(4) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ͷඪ४ܕͷີ౓ؔ਺ f

₀

͸ɼγ = 1 ͱ

ͨ͠৔߹ɼ(5) ࣜͰݟͨΑ͏ʹඪ४ΨϯϚ෼෍ͷີ౓ؔ਺ʹؼண͢ΔɽΑͬ

ͯɼ໋୊ 1 ͷܥͱͯ͠ҎԼͷ݁Ռʢܥ 1ʣ͕ಘΒΕΔɽ࣮ࡍɼγ = 1 ͷ৔߹ɼ ҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ f

₀

͸ඪ४ΨϯϚ෼෍ͷີ౓ؔ਺ʹؼண͢Δͷ Ͱɼͦͷ 1 ֊ಋؔ਺͸

f

₀

(x) = 1

Γ(α) [(α − 1) − x] x

^α−2

e

^−x

ͱ؆୯Խ͞Εɼͦͷ૿ݮද͸ද 4 ʹ༩͑ΒΕΔʢิ࿦ A ࢀরʣɽ

ܥ 1 (ඪ४ΨϯϚ෼෍ͷܗঢ়) . (5) ࣜͰఆٛ͞ΕΔඪ४ΨϯϚ෼෍ͷີ

౓ؔ਺ f

₀

ͷάϥϑ { (x, f

₀

(x | α, 1)) | x ∈ (0, ∞ ) } ͷܗঢ়͸ɼਖ਼஋ͷܗঢ়

฼਺ α > 0 ʹԠͯ͡ҎԼͷΑ͏ʹఆ·Δʢਤ 5 ࢀরʣɽୠ͠ɼҎԼͰ͸

f

₀

(x) := f

₀

(x | α, 1) ͱུه͢Δɽͳ͓ɼਤʹ͍ͭͯ͸ิ࿦ B Λࢀর͞Ε

͍ͨɽ

1. α ≤ 1 ͷ৔߹ʢ୯ௐݮগܕʣ

ີ౓ؔ਺ f

₀

(x) ͸୯ௐݮগؔ਺Ͱ͋Γɼͦͷάϥϑ͸ӈԼΓͷܗঢ়

Λࣔ͢ɽಛʹɼx → ∞ ͷ৔߹ʹ͓͍ͯ f

₀

(x) → 0 Ͱ͋Δɽ

(15)

(a) α < 1 ͷ৔߹ʢඇ༗քʣ

x → 0 ʹ͓͍ͯ f

₀

(x) → ∞ Ͱ͋Δɽ (b) α = 1 ͷ৔߹ʢ༗քɼඪ४ࢦ਺෼෍ʣ

x → 0 ʹ͓͍ͯ f

₀

(x) → 1 Ͱ͋Δɽ͜ͷ৔߹ɼඪ४ΨϯϚ෼

෍͸ඪ४ࢦ਺෼෍ʹؼண͢Δɽ 2. α > 1 ͷ৔߹ʢ୯ๆܕʣ

ີ౓ؔ਺ f

₀

(x) ͸ x = x

_m

Λ࠷େ఺ʢ࠷ස஋ʣͱͯ͠ɼͦͷάϥϑ

͸୯ๆܕͷܗঢ়Λࣔ͢ɿ

x

_m

= α − 1.

·ͨɼ͜ͷ৔߹ɼx → 0 ΋͘͠͸ x → ∞ ͷ͍ͣΕͷ৔߹ʹ͓͍ͯ΋

f

₀

(x) → 0 Ͱ͋Δ

³

ɽ

͜ΕʹΑΓɼܗঢ়฼਺ α ͕େ͖͘ͳΔʹͭΕͯɼີ౓ؔ਺ͷ࠷େ఺ʢ࠷

ස஋ʣ΋େ͖͘ͳΓແݶେʹൃࢄ͢Δ͜ͱ͕෼͔Δɿ x

_m

= α − 1 −−−−→ ∞

^α→∞

.

ͳ͓ɼີ౓ؔ਺ͷ࠷େ஋͸ f

₀

(x

_m

) = (α − 1)

^α−1

e

^−(α−1)

/Γ(α) ͱͳΔɽ·

ͨɼඪ४ΨϯϚ෼෍ (5) ࣜͷظ଴஋ͱ෼ࢄʹ͍ͭͯ΋ (9) ࣜʹΑΓ α → ∞ ͷ࣌ʹແݶେʹൃࢄ͢Δͱ෼͔Δɿ

E(X) = V (X) = α −−−−→ ∞

^α→∞

.

͜ΕΒͷࣄ࣮ʹ͍ͭͯ͸ɼਤ 5 ΍ਤ 6 ͔Β΋ཧղ͞ΕΔʢิ࿦ B ࢀরʣɽ

ͯ͞ɼඪ४ΨϯϚ෼෍ͷҐஔई౓෼෍଒ͷີ౓ؔ਺ (12) ࣜͷάϥϑͷܗ ঢ়ʹ͍ͭͯ͸ɼ໋୊ 2 ʹ͓͍ͯ γ = 1 ͱஔ͚͹ɼͦͷ݁࿦͸ͦͷ··ʹ੒

ཱ͢Δɽͭ·ΓɼҐஔ฼਺ μ ͱई౓฼਺ σ ͕ີ౓ؔ਺ (12) ࣜͷάϥϑͷܗ ঢ়ʹٴ΅͢Өڹ͸໋୊ 2 ͷ݁࿦ͱશ͘ಉ͡Ͱ͋ΔɽΑͬͯɼҐஔ฼਺͸ີ

3x→0ʹ͓͍ͯີ౓ؔ਺f0(x)͕0ʹऩଋ͢Δࡍɼͦͷ܏͖͸ɼα <2ͷ৔߹ʹແݶେ

ʹൃࢄ͠ʢf₀(x)−−−→ ∞^x→0 ʣɼα= 2ͷ৔߹ʹఆ਺ʹऩଋ͠ʢf₀(x)−−−→^x→0 1ʣɼα >2ͷ৔

߹ʹ0ʹऩଋ͢Δʢf₀(x)−−−→^x→0 0ʣͱ͍͏ܗΛऔΔʢਤ5ɼਤ10ɼਤ11ࢀরʣɽ

(16)

౓ؔ਺ͷάϥϑΛࠨӈʹฏߦҠಈͤ͞Δʢਤ 7ɼਤ 8ɼਤ 20 ࢀরʣɽ·ͨɼ ई౓฼਺ʹ͍ͭͯ͸ɼσ < τ Ͱ͋Δ 2 ͭͷई౓฼਺ σ ͱ τ ʹରͯ͠ɼy ͕ Ұఆͷ஋ μ + y

_στ

ΑΓখ͍͞৔߹ʹ͸ f (y | α, 1, μ, σ) > f (y | α, 1, μ, τ )ɼେ

͖͍৔߹ʹ͸ f (y | α, 1, μ, σ) < f (y | α, 1, μ, τ) Ͱ͋Γɼy = μ + y

_στ

ͷ৔

߹ʹ͸ f (y | α, 1, μ, σ) = f(y | α, 1, μ, τ) Ͱ͋Δʢਤ 9ɼਤ 10ɼਤ 11ɼਤ 21

ࢀরʣ ɿ

y

_στ

:= αστ τ − σ log τ

σ .

ຊઅͷ࠷ޙʹɼඪ४ΨϯϚ෼෍ͷҐஔई౓෼෍଒ͷಛघͳ৔߹ͱͯ͠ɼࣗ༝

౓ n ͷΧΠ 2 ৐෼෍ͷܗঢ়ʹ͍ͭͯݴٴ͢Δɽࣗ༝౓ n ͷΧΠ 2 ৐෼෍ͱ͸ɼ ඪ४ΨϯϚ෼෍ͷҐஔई౓෼෍଒ (12) ࣜʹ͓͍ͯ α = n/2, σ = 2, μ = 0 ͱͨ͠৔߹ʹଞͳΒͳ͍ͷͰɼͦͷີ౓ؔ਺ͷάϥϑͷܗঢ়͸ɼܥ 1 ౳ʹ ΑΓɼࣗ༝౓͕ 2 ҎԼͰ͋Ε͹୯ௐݮগͰ͋Γɼࣗ༝౓͕ 2 ΑΓେͰ͋Ε

͹୯ๆܕͰ͋Δͱ෼͔Δʢਤ 12 ࢀরʣɽ

3.2.2 α = 1 ͷ৔߹ɿ Weibull ෼෍

(4) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ f

₀

͸ɼα = 1 ͱͨ͠৔

߹ɼ(6) ࣜͰݟͨΑ͏ʹඪ४ Weibull ෼෍ͷີ౓ؔ਺ʹؼண͢ΔɽΑͬͯɼ

໋୊ 1 ͷܥͱͯ͠ҎԼͷ݁Ռʢܥ 2ʣ͕ಘΒΕΔɽ࣮ࡍɼα = 1 ͷ৔߹ɼҰ ൠԽΨϯϚ෼෍ͷີ౓ؔ਺ f

₀

͸ඪ४ Weibull ෼෍ͷີ౓ؔ਺ʹؼண͢Δͷ Ͱɼͦͷ 1 ֊ಋؔ਺͸

f

₀

(x) = γ[(γ − 1) − γx

^γ

]x

^γ−2

e

^−x^γ

ͱ؆୯Խ͞Εɼͦͷ૿ݮද͸ද 5 ʹ༩͑ΒΕΔʢิ࿦ A ࢀরʣɽ

ܥ 2 (ඪ४ Weibull ෼෍ͷܗঢ়) . (6) ࣜͰఆٛ͞ΕΔඪ४ Weibull ෼෍ͷ

ີ౓ؔ਺ f

₀

ͷάϥϑ { (x, f

₀

(x | 1, γ)) | x ∈ (0, ∞ ) } ͷܗঢ়͸ɼਖ਼஋ͷܗঢ়

฼਺ γ > 0 ʹԠͯ͡ҎԼͷΑ͏ʹఆ·Δʢਤ 13 ࢀরʣɽୠ͠ɼҎԼͰ͸

f

₀

(x) := f

₀

(x | 1, γ) ͱུه͢Δɽͳ͓ɼਤʹ͍ͭͯ͸ิ࿦ B Λࢀর͞Ε

͍ͨɽ

(17)

1. γ ≤ 1 ͷ৔߹ʢ୯ௐݮগܕʣ

ີ౓ؔ਺ f

₀

(x) ͸୯ௐݮগؔ਺Ͱ͋Γɼͦͷάϥϑ͸ӈԼΓͷܗঢ় Λࣔ͢ɽಛʹɼx → ∞ ͷ৔߹ʹ͓͍ͯ f

₀

(x) → 0 Ͱ͋Δɽ

(a) γ < 1 ͷ৔߹ʢඇ༗քʣ

x → 0 ʹ͓͍ͯ f

₀

(x) → ∞ Ͱ͋Δɽ (b) γ = 1 ͷ৔߹ʢ༗քɼඪ४ࢦ਺෼෍ʣ

x → 0 ʹ͓͍ͯ f

₀

(x) → 1 Ͱ͋Δɽ͜ͷ৔߹ɼඪ४ Weibull ෼

෍͸ඪ४ࢦ਺෼෍ʹؼண͢Δɽ 2. γ > 1 ͷ৔߹ʢ୯ๆܕʣ

ີ౓ؔ਺ f

₀

(x) ͸ x = x

_m

Λ࠷େ఺ʢ࠷ස஋ʣͱͯ͠ɼͦͷάϥϑ

͸୯ๆܕͷܗঢ়Λࣔ͢ɿ

x

_m

=

γ − 1 γ

¹_γ

.

·ͨɼ͜ͷ৔߹ɼx → 0 ΋͘͠͸ x → ∞ ͷ͍ͣΕͷ৔߹ʹ͓͍ͯ΋

f

₀

(x) → 0 Ͱ͋Δ

⁴

ɽ

ܗঢ়฼਺ γ ͕େ͖͘ͳΔʹͭΕͯɼີ౓ؔ਺ͷ࠷େ఺ʢ࠷ස஋ʣ͸ 1 ʹ ऩଋ͢Δʢྫ͑͹ɼJohnson, Kotz and Balakrishnan 1994, p.630 ࢀরʣ ɿ

x

_m

=

1 − 1 γ

_γ¹

−−−−→

γ→∞

1. ͳ͓ɼີ౓ؔ਺ͷ࠷େ஋͸ f

₀

(x

_m

) = γ(1 − 1/γ)

^1−1/γ

e

^{−(1−1/γ)}

ͱͳΔɽ·

ͨɼඪ४ Weibull ෼෍ (6) ࣜͷظ଴஋ͱ෼ࢄʹ͍ͭͯ͸ɼΓ(1) = 1 ʹ஫ҙ

͢Δͱɼ(10) ࣜʹΑΓ γ → ∞ ͷ࣌ʹͦΕͧΕ 1 ͱ 0 ʹऩଋ͢Δͱ෼͔Δɿ E(X) = Γ

1 + 1

γ

−−−−→

γ→∞

Γ(1) = 1,

V (X ) = Γ

1 + 2 γ

−

Γ

1 + 1 γ

₂

−−−−→

γ→∞

Γ(1) − [Γ(1)]

²

= 0.

4x→0ʹ͓͍ͯີ౓ؔ਺f0(x)͕0ʹऩଋ͢Δࡍɼͦͷ܏͖͸ɼγ <2ͷ৔߹ʹແݶେ

ʹൃࢄ͠ʢf₀(x)−−−→ ∞^x→0 ʣɼγ= 2ͷ৔߹ʹఆ਺ʹऩଋ͠ʢf₀(x)−−−→^x→0 2ʣɼγ >2ͷ৔

߹ʹ0ʹऩଋ͢Δʢf₀(x)−−−→^x→0 0ʣͱ͍͏ܗΛऔΔʢਤ13ɼਤ18ɼਤ19ࢀরʣɽ

(18)

͜ΕΒͷࣄ࣮ʹ͍ͭͯ͸ɼਤ 13 ΍ਤ 14 ͔Β΋ཧղ͞ΕΔʢิ࿦ B ࢀরʣɽ

࠷ޙʹɼඪ४ Weibull ෼෍ͷҐஔई౓෼෍଒ͷີ౓ؔ਺ (13) ࣜͷάϥϑ ͷܗঢ়ʹ͍ͭͯ͸ɼ໋୊ 2 ʹ͓͍ͯ α = 1 ͱஔ͚͹ɼͦͷ݁࿦͸ͦͷ··

ʹ੒ཱ͢Δɽͭ·ΓɼҐஔ฼਺ μ ͱई౓฼਺ σ ͕ີ౓ؔ਺ (13) ࣜͷάϥϑ ͷܗঢ়ʹٴ΅͢Өڹ͸໋୊ 2 ͷ݁࿦ͱશ͘ಉ͡Ͱ͋ΔɽΑͬͯɼҐஔ฼਺

͸ີ౓ؔ਺ͷάϥϑΛࠨӈʹฏߦҠಈͤ͞Δʢਤ 15ɼਤ 16ɼਤ 20 ࢀরʣɽ

·ͨɼई౓฼਺ʹ͍ͭͯ͸ɼσ < τ Ͱ͋Δ 2 ͭͷई౓฼਺ σ ͱ τ ʹରͯ͠ɼ y ͕Ұఆͷ஋ μ + y

_στ

ΑΓখ͍͞৔߹ʹ͸ f (y | 1, γ, μ, σ) > f(y | 1, γ, μ, τ )ɼ

େ͖͍৔߹ʹ͸ f (y | 1, γ, μ, σ) < f (y | 1, γ, μ, τ ) Ͱ͋Γɼy = μ + y

_στ

ͷ৔

߹ʹ͸ f (y | 1, γ, μ, σ) = f (y | 1, γ, μ, τ) Ͱ͋Δʢਤ 17ɼਤ 18ɼਤ 19ɼਤ 21

ࢀরʣ ɿ

y

_στ

:=

γσ

^γ

τ

^γ

τ

^γ

− σ

^γ

log τ

σ

_γ¹

.

4 ^݁࿦

ແݶ۠ؒ (0, ∞ ) ্ʹఆٛ͞ΕΔҰൠԽΨϯϚ෼෍͸ɼͦͷಛघͳ৔߹ͱ

ͯ͠ɼΨϯϚ෼෍Λ࢝Ίͱͯ͠ΧΠ 2 ৐෼෍΍ Weibull ෼෍ɼࢦ਺෼෍Λ แؚ͢Δɽຊߘ͸ɼͦͷ෼෍ͷܗঢ়Λମܥతʹݕ౼͠ɼͦͷ݁Ռͱͯ͠ಘ ΒΕͨ஌ݟΛ໋୊ 1 ΍໋୊ 2ɼܥ 1ɼܥ 2 ͱͯ͠੔ཧͨ͠ɽଈͪɼҰൠԽ ΨϯϚ෼෍ͷີ౓ؔ਺ͷܗঢ়ʹ͍ͭͯɼ໋୊ 1 ʹ͓͍ͯ 2 ͭͷܗঢ়฼਺ͱ ͷؔ܎Λఏࣔ͠ɼ໋୊ 2 ʹ͓͍ͯҐஔ฼਺ͱई౓฼਺ͱͷؔ܎Λఏࣔͨ͠ɽ

·ͨɼͦͷಛघͳ৔߹ͱͯ͠ɼΨϯϚ෼෍ʹ͍ͭͯͷ஌ݟΛܥ 1ɼWeibull

෼෍ʹ͍ͭͯͷ஌ݟΛܥ 2 ͱͯ͠ఏࣔͨ͠ɽߋʹɼͦΕΒͷಛघͳ৔߹ͱ

ͯ͠ɼΧΠ 2 ৐෼෍΍ࢦ਺෼෍ͷܗঢ়ʹ͍ͭͯ΋ݴٴͨ͠ɽ

(19)

ࢀߟจݙ

ݘҪమ࿠ʢ1962ʣʰಛघവ਺ʱؠ೾ॻళɽ

ਿӜޫ෉ʢ1980ʣʰղੳೖ໳ Iʱ౦ژେֶग़൛ձɽ

ຸ୩ઍ႗඙ʢ2010ʣʰ౷ܭ෼෍ϋϯυϒοΫʢ૿ิ൛ʣʱே૔ॻళɽ Forbes, C., M. Evans, N. Hastings and B. Peacock (2011) Statistical Distributions, 4th edition , John Wiley & Sons.

Johnson, N. L. and S. Kotz (1972) “Power transformations of gamma variables”, Biometrika , 59, 226–229.

Johnson, N. L., S. Kotz and N. Balakrishnan (1994) Continuous Uni- variate Distributions, Volume 1, 2nd edition , John Wiley & Sons.

Khodabin, M. and A. Ahmadabadi (2010) “Some properties of general- ized gamma distribution”, Mathematical Sciences , 4, 9–28.

Stacy, E. W. (1962) “A generalization of the gamma distribution”, The

Annals of Mathematical Statistics , 33, 1187–1192.

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A ҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ͷ૿ݮද

ຊิ࿦͸ɼ(4) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ͷඪ४ܕͷີ౓ؔ਺

f

₀

(x | α, γ), x ∈ (0, ∞ ) ͷ૿ݮදΛఏࣔ͢Δɽ͜ΕʹΑΓɼ2 ͭͷܗঢ়฼਺

α, γ > 0 ͱີ౓ؔ਺ f

₀

ͷάϥϑͷܗঢ়ͱͷؔ܎͕໌Β͔ʹͳΔʢ໋୊ 1ɼ

ܥ 1ɼܥ 2ʣɽ

A.1 Ұൠͷ৔߹ɿҰൠԽΨϯϚ෼෍ͷඪ४ܕ

ද 1: ҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ f

₀

(x) ͷ૿ݮදʢγα < 1 ͷ৔߹ʣ x 0 · · · ∞

f

₀

−∞ − 0

f

₀

∞ 0

ද 2: ҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ f

₀

(x) ͷ૿ݮදʢγα = 1 ͷ৔߹ʣɿࠨ දʢγ < 1 ⇐⇒ α > 1 ͷ৔߹ʣɼதදʢγ = 1 ⇐⇒ α = 1 ͷ৔߹ʣɼӈද ʢγ > 1 ⇐⇒ α < 1 ͷ৔߹ʣ

x 0 · · · ∞

f

₀

−∞ − 0

f

₀

1/Γ(α + 1) 0

x 0 · · · ∞ f

₀

− 1 − 0

f

₀

1 0

x 0 · · · ∞

f

₀

0 − 0

f

₀

1/Γ(α + 1) 0

ද 3: ҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ f

₀

(x) ͷ૿ݮදʢγα > 1 ͷ৔߹ʣ x 0 · · · (α − 1/γ)

^1/γ

· · · ∞

f

₀

∗ + 0 − 0

f

₀

0 0

∗ = ∞ ʢγα < 2 ͷ࣌ʣɼ 2/Γ(α + 1) ʢγα = 2 ͷ࣌ʣɼ 0 ʢγα > 2 ͷ࣌ʣ

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A.2 ز͔ͭͷಛघͳ৔߹

A.2.1 γ = 1 ͷ৔߹ɿඪ४ΨϯϚ෼෍

ද 4: ඪ४ΨϯϚ෼෍ͷ૿ݮදɿ্ஈࠨදʢα < 1 ͷ৔߹ʣɼ্ஈӈදʢα = 1 ͷ৔߹ɼඪ४ࢦ਺෼෍ʣɼԼஈදʢα > 1 ͷ৔߹ʣ

x 0 · · · ∞

f

₀

−∞ − 0

f

₀

∞ 0

x 0 · · · ∞ f

₀

− 1 − 0

f

₀

1 0

x 0 · · · α − 1 · · · ∞

f

₀

∗ + 0 − 0

f

₀

0 0

∗ = ∞ ʢα < 2 ͷ࣌ʣɼ 1 ʢα = 2 ͷ࣌ʣɼ 0 ʢα > 2 ͷ࣌ʣ

A.2.2 α = 1 ͷ৔߹ɿඪ४ Weibull ෼෍

ද 5: ඪ४ Weibull ෼෍ͷ૿ݮදɿ্ஈࠨදʢγ < 1 ͷ৔߹ʣɼ্ஈӈද ʢγ = 1 ͷ৔߹ɼඪ४ࢦ਺෼෍ʣɼԼஈදʢγ > 1 ͷ৔߹ʣ

x 0 · · · ∞

f

₀

−∞ − 0

f

₀

∞ 0

x 0 · · · ∞ f

₀

− 1 − 0

f

₀

1 0

x 0 · · · (1 − 1/γ)

^1/γ

· · · ∞

f

₀

∗ + 0 − 0

f

₀

0 0

∗ = ∞ ʢγ < 2 ͷ࣌ʣɼ 2 ʢγ = 2 ͷ࣌ʣɼ 0 ʢγ > 2 ͷ࣌ʣ

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B ҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ͷάϥϑ

ຊิ࿦Ͱఏࣔ͢Δਤ͸શͯ Maple 6 ʹΑΔ΋ͷͰ͋Δɽ

B.1 Ұൠͷ৔߹ɿҰൠԽΨϯϚ෼෍

ຊઅ͸ɼ (4) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ͷඪ४ܕͷີ౓ؔ਺ f

₀

(x | α, γ ) ͷάϥϑΛਤࣔ͢Δʢԣ࣠ɿxɼॎ࣠ɿf

₀

(x | α, γ)ʣɽ

0 0.5 1 1.5

2 2.5

0.5 1 1.5 2 2.5

ਤ 1: ҰൠԽΨϯϚ෼෍ͷඪ४ܕͷີ౓ؔ਺ʢαγ < 1 ͷ৔߹ʣ ɿ α = γ = 1/2 ʢଠ࣮ઢʣɼα = 1/2, γ = 1ʢଠ఺ઢʣɼα = 1/2, γ = 3/2ʢ࣮ઢʣɼ

α = 1, γ = 1/2ʢ఺ઢʣɼα = 3/2, γ = 1/2ʢࡉ࣮ઢʣ

0 0.2 0.4 0.6 0.8 1 1.2

0.5 1 1.5 2 2.5 3

ਤ 2: ҰൠԽΨϯϚ෼෍ͷඪ४ܕͷີ౓ؔ਺ʢαγ = 1 ͷ৔߹ʣ ɿ α = 1/3, γ =

3 ʢଠ࣮ઢʣɼ α = 1/2, γ = 2 ʢଠ఺ઢʣɼ α = γ = 1 ʢ࣮ઢʣɼ α = 2, γ = 1/2

ʢ఺ઢʣɼα = 3, γ = 1/3ʢࡉ࣮ઢʣ

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0 0.2 0.4 0.6 0.8 1

1 2 3 4 5

ਤ 3: ҰൠԽΨϯϚ෼෍ͷඪ४ܕͷີ౓ؔ਺ʢαγ > 1 ͷ৔߹ʣ ɿ α = 1/2, γ = 5/2ʢଠ࣮ઢʣɼα = 1, γ = 2ʢଠ఺ઢʣɼα = γ = 2ʢ࣮ઢʣɼα = 2, γ = 1 ʢ఺ઢʣɼα = 5/2, γ = 1/2ʢࡉ࣮ઢʣ

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0.5 1 1.5 2 2.5 3

ਤ 4: ҰൠԽΨϯϚ෼෍ͷඪ४ܕͷີ౓ؔ਺ʢαγ > 1 ͷ৔߹ʣ ɿα = γ = 2

ʢଠ࣮ઢʣɼα = 2, γ = 3ʢଠ఺ઢʣɼα = 3, γ = 2ʢ࣮ઢʣɼα = γ = 3

ʢ఺ઢʣɼα = γ = 5/2ʢࡉ࣮ઢʣ

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B.2 ^{ز͔ͭͷಛघͳ৔߹}

B.2.1 γ = 1 ͷ৔߹ɿΨϯϚ෼෍

ຊઅ͸ɼ(12) ࣜͰఆٛ͞ΕΔඪ४ΨϯϚ෼෍ͷҐஔई౓෼෍଒ͷີ౓ؔ

਺ f (y | α, 1, μ, σ) ͷάϥϑΛਤࣔ͢Δʢԣ࣠ɿyɼॎ࣠ɿf (y | α, 1, μ, σ)ʣɽ

0 0.2 0.4 0.6 0.8 1 1.2

2 4 6 8 10

ਤ 5: ΨϯϚ෼෍ͷີ౓ؔ਺ʢμ = 0, σ = 1ʣ ɿα = 1/2ʢଠ࣮ઢʣɼα = 1 ʢଠ఺ઢɼࢦ਺෼෍ʣɼα = 3/2ʢ࣮ઢʣɼα = 2ʢ఺ઢʣɼα = 3ʢࡉ࣮ઢʣɼ

α = 5ʢࡉ఺ઢʣ

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

10 20 30 40 50 60 70

ਤ 6: ΨϯϚ෼෍ͷີ౓ؔ਺ʢμ = 0, σ = 1ʣ ɿα = 5ʢ࣮ઢʣɼα = 10ʢ఺

ઢʣɼα = 20ʢࡉ࣮ઢʣɼα = 50ʢࡉ఺ઢʣ

(25)

0 0.2 0.4 0.6 0.8 1

–1 1 2 3 4 5

ਤ 7: ΨϯϚ෼෍ͷີ౓ؔ਺ʢα = 1/2, σ = 1ʣ ɿμ = − 1ʢ࣮ઢʣɼμ = 0 ʢ఺ઢʣɼμ = 1ʢࡉ࣮ઢʣ

0 0.1 0.2 0.3 0.4

2 4 6 8

ਤ 8: ΨϯϚ෼෍ͷີ౓ؔ਺ʢα = 2, σ = 1ʣ ɿμ = − 1ʢ࣮ઢʣɼμ = 0ʢ఺

ઢʣɼμ = 1ʢࡉ࣮ઢʣ

(26)

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

ਤ 9: ΨϯϚ෼෍ͷີ౓ؔ਺ʢα = 1/2, μ = 0ʣ ɿσ = 1/2ʢ࣮ઢʣɼσ = 1 ʢ఺ઢʣɼσ = 2ʢࡉ࣮ઢʣ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

2 4 6 8 10

ਤ 10: ΨϯϚ෼෍ͷີ౓ؔ਺ʢα = 2, μ = 0ʣ ɿσ = 1/2ʢ࣮ઢʣɼσ = 1

ʢ఺ઢʣɼσ = 2ʢࡉ࣮ઢʣ

(27)

0 0.1 0.2 0.3 0.4

2 4 6 8 10 12 14 16 18 20

ਤ 11: ΨϯϚ෼෍ͷີ౓ؔ਺ʢα = 5, μ = 0ʣ ɿσ = 1/2ʢ࣮ઢʣɼσ = 1 ʢ఺ઢʣɼσ = 2ʢࡉ࣮ઢʣ

0 0.1 0.2 0.3 0.4 0.5 0.6

2 4 6 8 10 12 14 16 18 20

ਤ 12: ΨϯϚ෼෍ͷີ౓ؔ਺ʢα = n/2, μ = 0, σ = 2ɼࣗ༝౓ n ͷΧΠ 2

৐෼෍ʣ ɿn = 1ʢଠ࣮ઢʣɼn = 2ʢଠ఺ઢʣɼn = 3ʢ࣮ઢʣɼn = 4ʢ఺

ઢʣɼn = 5ʢࡉ࣮ઢʣɼn = 10ʢࡉ఺ઢʣ

(28)

B.2.2 α = 1 ͷ৔߹ɿ Weibull ෼෍

ຊઅ͸ɼ(13) ࣜͰఆٛ͞ΕΔඪ४ Weibull ෼෍ͷҐஔई౓෼෍଒ͷີ౓

ؔ਺ f (y | 1, γ, μ, σ) ͷάϥϑΛਤࣔ͢Δʢԣ࣠ɿyɼॎ࣠ɿf(y | 1, γ, μ, σ)ʣɽ

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.5 1 1.5 2 2.5

ਤ 13: Weibull ෼෍ͷີ౓ؔ਺ʢμ = 0, σ = 1ʣ ɿγ = 1/2ʢଠ࣮ઢʣɼγ = 1 ʢଠ఺ઢɼࢦ਺෼෍ʣɼγ = 3/2ʢ࣮ઢʣɼγ = 2ʢ఺ઢʣɼγ = 3ʢࡉ࣮ઢʣɼ

γ = 5ʢࡉ఺ઢʣ

0 1 2 3 4 5 6 7

0.2 0.4 0.6 0.8 1 1.2 1.4

ਤ 14: Weibull ෼෍ͷີ౓ؔ਺ʢμ = 0, σ = 1ʣ ɿγ = 5ʢ࣮ઢʣɼγ = 10

ʢ఺ઢʣɼγ = 20ʢࡉ࣮ઢʣ

(29)

0 0.2 0.4 0.6 0.8 1

–1 1 2 3 4 5

ਤ 15: Weibull ෼෍ͷີ౓ؔ਺ʢγ = 1/2, σ = 1ʣ ɿμ = − 1ʢ࣮ઢʣɼ μ = 0 ʢ఺ઢʣɼμ = 1ʢࡉ࣮ઢʣ

0 0.2 0.4 0.6 0.8

–1 1 2 3 4

ਤ 16: Weibull ෼෍ͷີ౓ؔ਺ʢγ = 2, σ = 1ʣ ɿμ = − 1ʢ࣮ઢʣɼμ = 0

ʢ఺ઢʣɼμ = 1ʢࡉ࣮ઢʣ

(30)

0 0.2 0.4 0.6 0.8 1

0.5 1 1.5 2 2.5 3

ਤ 17: Weibull ෼෍ͷີ౓ؔ਺ʢγ = 1/2, μ = 0ʣ ɿσ = 1/2ʢ࣮ઢʣɼσ = 1 ʢ఺ઢʣɼσ = 2ʢࡉ࣮ઢʣ

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

1 2 3 4 5

ਤ 18: Weibull ෼෍ͷີ౓ؔ਺ʢγ = 2, μ = 0ʣ ɿσ = 1/2ʢ࣮ઢʣɼσ = 1

ʢ఺ઢʣɼσ = 2ʢࡉ࣮ઢʣ

(31)

0 1 2 3 4

0.5 1 1.5 2 2.5 3

ਤ 19: Weibull ෼෍ͷີ౓ؔ਺ʢγ = 5, μ = 0ʣ ɿσ = 1/2ʢ࣮ઢʣɼσ = 1 ʢ఺ઢʣɼσ = 2ʢࡉ࣮ઢʣ

B.2.3 α = γ = 1 ͷ৔߹ɿࢦ਺෼෍

ຊઅ͸ɼ(14) ࣜͰఆٛ͞ΕΔඪ४ࢦ਺෼෍ͷҐஔई౓෼෍଒ͷີ౓ؔ਺

f (y | 1, 1, μ, σ) ͷάϥϑΛਤࣔ͢Δʢԣ࣠ɿyɼॎ࣠ɿf (y | 1, 1, μ, σ)ʣɽ

0 0.2 0.4 0.6 0.8 1

–1 1 2 3 4 5

ਤ 20: ࢦ਺෼෍ͷີ౓ؔ਺ʢσ = 1ʣɿμ = − 1ʢ࣮ઢʣɼμ = 0ʢ఺ઢʣɼ

μ = 1ʢࡉ࣮ઢʣ

(32)

0 0.5 1 1.5

2

1 2 3 4 5

ਤ 21: ࢦ਺෼෍ͷີ౓ؔ਺ʢμ = 0ʣɿσ = 1/2ʢ࣮ઢʣɼσ = 1ʢ఺ઢʣɼ σ = 2ʢࡉ࣮ઢʣ

B.2.4 ͦͷଞͷ৔߹

ຊઅ͸ɼ (4) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ͷඪ४ܕͷີ౓ؔ਺ f

₀

(x | α, γ) ͷάϥϑΛਤࣔ͢Δʢԣ࣠ɿxɼॎ࣠ɿf

₀

(x | α, γ)ʣɽ

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 1.2 1.4 1.6 1.8 2

ਤ 22: ҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ʢα = 1/2 ͷ৔߹ʣ ɿ γ = 1/2ʢଠ࣮ઢʣɼ

γ = 1ʢଠ఺ઢʣɼγ = 2ʢ࣮ઢʣɼγ = 3ʢ఺ઢʣɼγ = 4ʢࡉ࣮ઢʣ

(33)

0 0.2 0.4 0.6 0.8 1 1.2 1.4

1 2 3 4 5 6

ਤ 23: ҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ʢα = 2 ͷ৔߹ʣ ɿγ = 1/3ʢଠ࣮ઢʣɼ γ = 1/2ʢଠ఺ઢʣɼγ = 1ʢ࣮ઢʣɼγ = 2ʢ఺ઢʣɼγ = 3ʢࡉ࣮ઢʣ

0 0.2 0.4 0.6 0.8 1

1 2 3 4 5 6 7 8

ਤ 24: ҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺ʢγ = 1/2 ͷ৔߹ʣ ɿ α = 1/2ʢଠ࣮ઢʣɼ

α = 1ʢଠ఺ઢʣɼα = 2ʢ࣮ઢʣɼα = 3ʢ఺ઢʣɼα = 4ʢࡉ࣮ઢʣ

(34)

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0.5 1 1.5 2 2.5 3

ҰൠԽΨϯϚ෼෍͸ɼຸ୩ʢ2010ʣ΍ Stacy (1962), Johnson and Kotz (1972), Johnson, Kotz and Balakrishnan (1994), Khodabin and Ahmad- abadi (2010), Forbes, Evans, Hastings and Peacock (2011) ౳Ͱߟ࡯͞Ε

1 ং࿦

௨ৗͷΨϯϚ෼෍͸ɼແݶ۠ؒ (0, ∞ ) ্ʹఆٛ͞ΕΔ࿈ଓܕͷ֬཰෼෍

Ͱ͋Γɼܗঢ়฼਺ͱई౓฼਺ͷ 2 ͭͷ฼਺Λ࣋ͭɽΨϯϚ෼෍͸ɼͦͷಛ घͳ৔߹ͱͯ͠ɼࢦ਺෼෍΍ΧΠ 2 ৐෼෍Λแؚ͢Δɽ·ͨɼΨϯϚ෼෍

ʹै͏֬཰ม਺Λۊม׵͢Δ͜ͱʹΑͬͯಘΒΕΔ֬཰෼෍ΛҰൠԽΨϯ Ϛ෼෍ͱ͍͍ɼͦͷಛघͳ৔߹ͱͯ͠ɼΨϯϚ෼෍ͷଞʹ Weibull ෼෍Λ แؚ͢Δɽ

ҰൠԽΨϯϚ෼෍͸ɼຸ୩ʢ2010ʣ΍ Stacy (1962), Johnson and Kotz (1972), Johnson, Kotz and Balakrishnan (1994), Khodabin and Ahmad- abadi (2010), Forbes, Evans, Hastings and Peacock (2011) ౳Ͱߟ࡯͞Ε

͍ͯΔɽ͔͠͠ɼҰൠԽΨϯϚ෼෍ͷ฼਺ͱܗঢ়ͷؔ܎ʹؔ͢Δମܥతͳ ݴٴ͸ɼJohnson, Kotz and Balakrishnan (1994) ʹ͓͍ͯۇ͔ʹݟΒΕΔ ఔ౓Ͱ͋Δɽͦͯ͠ɼJohnson, Kotz and Balakrishnan (1994, p.389) ʹ͓

͍ͯ΋ɼຊߘͷ (11) ࣜͰఆٛ͞ΕΔҰൠԽΨϯϚ෼෍ʹ͍ͭͯɼͦͷ෼෍

ͷܗঢ়͸ γα > 1 ͳΒ͹௼৊ܕͰ͋Γɼͦ͏Ͱͳ͚Ε͹ٯ J ࣈܕͰ͋Δͱ

ݴٴ͞ΕΔͷΈͰ͋Δɽຊߘ͸ɼΑΓৄࡉͳܗͰɼҰൠԽΨϯϚ෼෍ͷີ

౓ؔ਺ͷάϥϑͷܗঢ়͕ͦͷ฼਺ʹԠͯ͡ͲͷΑ͏ʹมԽ͢Δ͔Λݕ౼͠ɼ

ͦͷ݁Ռͱͯ͠ಘΒΕͨ஌ݟΛ੔ཧ͢Δʢ໋୊ 1ɼ໋୊ 2 ࢀরʣɽ

一般化ガンマ分布の形状について

ຊߘ͸ҎԼͷΑ͏ʹߏ੒͞ΕΔɽઌͣɼୈ 2 અʹ͓͍ͯɼΨϯϚؔ਺Λਖ਼ نԽఆ਺ͱͯ͠ඪ४ΨϯϚ෼෍Λߏ੒্ͨ͠Ͱɼඪ४ΨϯϚ෼෍ʹै͏֬

཰ม਺Λۊม׵͢Δ͜ͱʹΑͬͯҰൠԽΨϯϚ෼෍Λಋೖ͢ΔɽߋʹɼҰ ൠԽΨϯϚ෼෍ʹै͏֬཰ม਺Λ 1 ࣍ม׵͢Δ͜ͱʹΑͬͯͦͷҐஔई౓

෼෍଒Λߏ੒͢Δɽ࣍ʹɼୈ 3 અʹ͓͍ͯɼҰൠԽΨϯϚ෼෍ͷີ౓ؔ਺

ͷάϥϑͷܗঢ়͕ͦͷ฼਺ʹԠͯ͡ͲͷΑ͏ʹมԽ͢Δ͔Λݕ౼͠ɼͦͷ

݁Ռͱͯ͠ಘΒΕͨ஌ݟΛ໋୊ 1ɼ໋୊ 2 ͱͯ͠੔ཧ͢Δɽ·ͨɼҰൠԽΨ ϯϚ෼෍ͷಛघͳ৔߹ͱͯ͠ɼΨϯϚ෼෍ͱ Weibull ෼෍ͷີ౓ؔ਺ʹͭ

͍ͯ΋ಉ༷ͷߟ࡯Λߦ͏ʢܥ 1ɼܥ 2 ࢀরʣɽҎ্ͷݕ౼ͷաఔͰಘΒΕΔ

ີ౓ؔ਺ͷ૿ݮද͸શͯิ࿦ A ʹఏࣔ͠ɼີ౓ؔ਺ͷάϥϑ͸શͯิ࿦ B ʹఏࣔ͢Δɽ࠷ޙʹɼୈ 4 અͰ݁࿦Λड़΂Δɽ

2 ҰൠԽΨϯϚ෼෍ͱͦͷҐஔई౓෼෍଒

࣮਺ α, x ʹରͯ͠ɼx ͷؔ਺ g(x) := x

e

Λߟ͑Δɽ͜͜Ͱɼ౳߸

:= ͸ͦͷࠨลΛͦͷӈลʹΑͬͯఆٛ͢Δ͜ͱΛҙຯ͢Δɽͯ͞ɼα > 1 ʹ ରͯ͠ g(0) = 0ɼ α = 1 ʹରͯ͠ g(0) = 1ɼα < 1 ʹରͯ͠ g(x) −−−−−→ ∞

Ͱ͋Γɼ·ͨɼ೚ҙͷ α ∈ ( −∞ , ∞ ) ʹରͯ͠ g(x) −−−−→

0 Ͱ͋Δ͜ͱʹ

஫ҙ͢Δɽ·ͣɼα ≥ 1 ͷ৔߹ɼ೚ҙͷਖ਼ͷ࣮਺ M ʹରͯ͠ɼؔ਺ g(x) ͸ ด۠ؒ [0, M ] Ͱ࿈ଓͳͷͰఆੵ෼

x

e

dx ͸ଘࡏ͠ɼ͔ͭɼ޿ٛੵ

෼

x

e

dx := lim

x

e

dx ΋ଘࡏ͢Δʢྫ͑͹ɼݘҪ 1962ɼ p.1 ΍ਿӜ 1980ɼ pp.295–296 ࢀরʣɽ࣍ʹɼ α < 1 ͷ৔߹ɼ 0 < < M ͳΔ೚ҙͷ࣮਺ ʹରͯ͠ɼؔ਺ g(x) ͸ด۠ؒ [, M ] Ͱ࿈ଓͳͷͰఆੵ෼

x

e

dx ͸ଘࡏ͢Δɽ͜͜Ͱɼߋʹɼα > 0 Ͱ͋ΔͳΒ͹ɼ޿ٛੵ

෼

x

e

dx := lim

x

e

dx ͸ଘࡏ͢ΔʢݘҪ 1962ɼ p.1ɼਿӜ 1980ɼpp.295–296 ࢀরʣɽैͬͯɼ೚ҙͷ α > 0 ʹରͯ͠ɼੵ

෼

x

e

dx ͕ఆٛ͞ΕΔɽ͜ͷੵ෼Λਖ਼ͷ࣮਺ α ͷؔ਺ͱݟ၏ͨ͠

΋ͷΛΨϯϚؔ਺ͱ͍͍ɼΓ(α) ͱදه͢Δɿ

Γ(α) :=

x

e

dx, α > 0. (1)

೚ҙͷਖ਼ͷ࣮਺ x ∈ (0, ∞ ) ʹରͯ͠ x

e

> 0 Ͱ͋ΔͷͰɼ೚ҙͷ α > 0 ʹରͯ͠ɼΨϯϚؔ਺͸ৗʹਖ਼஋ΛऔΔɽଈͪɼ೚ҙͷਖ਼ͷ࣮਺ α > 0 ʹ ରͯ͠ Γ(α) > 0 Ͱ͋Δɽ

ҎԼɼୈ 2.1 અʹ͓͍ͯɼΨϯϚؔ਺Λਖ਼نԽఆ਺ͱͯ͠ඪ४ΨϯϚ෼෍

Λߏ੒্ͨ͠Ͱɼඪ४ΨϯϚ෼෍ʹै͏֬཰ม਺Λۊม׵͢Δ͜ͱʹΑͬ

ͯҰൠԽΨϯϚ෼෍Λಋೖ͢Δɽଓ͍ͯɼୈ 2.2 અʹ͓͍ͯɼҰൠԽΨϯ Ϛ෼෍ʹै͏֬཰ม਺Λ 1 ࣍ม׵͢Δ͜ͱʹΑͬͯͦͷҐஔई౓෼෍଒Λ ߏ੒͢Δɽ

2.1 ඪ४ΨϯϚ෼෍ͱҰൠԽΨϯϚ෼෍

ΨϯϚ෼෍ͱ͸ɼΨϯϚؔ਺Λਖ਼نԽఆ਺ͱͯ͠ߏ੒͞ΕΔ֬཰෼෍Ͱ

͋ΔɽଈͪɼҎԼͰఆٛ͞ΕΔີ౓ؔ਺ f

: (0, ∞ ) → ( −∞ , ∞ ) Λ࣋ͭ֬

཰෼෍ΛΨϯϚ෼෍ʢಛʹɼඪ४ΨϯϚ෼෍ʣͱ͍͍ɼα Λܗঢ়฼਺ͱ͍

͏ʢα > 0ʣ ɿ

f

(z | α) := 1

Γ(α) z

e

, z > 0. (2)

࣮ࡍɼ೚ҙͷਖ਼ͷ࣮਺ α, z > 0 ʹରͯ͠ɼz

e

> 0 ʹͯ͠ Γ(α) > 0 Ͱ

͋Δ͜ͱʹ஫ҙ͢Δͱɼf

(z | α) > 0 ΛಘΔɽ·ͨɼΨϯϚؔ਺ͷఆٛʹ஫

ҙ͢Δͱɼ

1 ^ং࿦