一般化ガンマ確率変数の指数変換の 分布の形状について

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1

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

（  ）１

−   −１４７

一般化ガンマ確率変数の指数変換の 分布の形状について

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Γ(α) :=

_∞

0 x^α−1e^−xdx, α >0. (1)

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g_∗(u|α) := 1

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=G_∗ v^1β

. 3

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

（  ）３

−   −１４９

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-GHI2O0< α, γ <∞PJ g₀(v|α, γ) = γ

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σ

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0 g₀(v|α, γ)dv

（  ）４4

−   −１５０

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g(w|α, γ, σ) = γ σΓ(α)

w

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5

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

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−   −１５１

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（  ）６6

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7

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

（  ）７

−   −１５３

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f(x|α, γ, δ) = γ δ^γαΓ(α)x

log1

x

_γα−1

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x

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（  ）８8

−   −１５４

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f(x|α,1, η) =η^αx^η−1 Γ(α)

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9

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

（  ）９

−   −１５５

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g0(v) :=g0(v|α, γ), f0(z) :=f0(z|α, γ).

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zg₀(−logz)

67*89&:2)*90;3<=#z6>?)*89&@A0B3CD EFGHI*-

f₀(z) =−1

z²{g0(−logz) +g₀(−logz)}. 8860123v∈(0,∞)&45/

g₀(v) = (γα−1)−γv^γ

v g0(v)

67*89&:2)*90

g0(v) +g₀(v) =(γα−1) +v−γv^γ

v g0(v)

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f₀(z)≶0 ⇐⇒ v−γv^γ ≷1−γα (9)

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（  ）１０10

−   −１５６

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1−γv^γ−1= 0 ⇐⇒ v=γ^1−γ1

11

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

（  ）１１

−   −１５７

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v−γv^γ = 0 ⇐⇒ v= 0567v=γ^1−γ1 . (10) 889'0 < γ <1):;'1/(1−γ)>1!<='0< γ^1/(1−γ)< γ <1 9>='γ >1):;'1/(1−γ)<0!<='0< γ^1/(1−γ)<1< γ9>

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v0:=γ^1−γ1 , 0< v0<min{1, γ}. (11) 56'v=−logz ⇐⇒ z =e^−vM%01!"#+,'DE!-)FG2 NO$%He⁻¹= 0.367879· · ·'I1JKL4

z₀:=e^−v0= exp

−γ^1−γ1

, max{e⁻¹, e^−γ}< z₀<1. (12)

0 1 2 3 4 5

0.00.20.40.60.81.00.30.70.9

O

I1: z=z0H(12)PL&z=z∗H(18)PL)QRSHTUz'VUγL4W X· · · z=z₀ = exp[−γ^1/(1−γ)]'YX· · · z=z_∗ = exp[−γ^2/(1−γ)]'ZW X· · · z= exp[−γ]'ZYX· · · z= exp[−1] = 0.367879. . .

（  ）１２12

−   −１５８

!"#(6)$%&'!()*+,-./0123456789-f0:2

;6<=>-α?γ@ABC)DE6FG?>-69H:#γ= 16IJ#

K@LM1%NOP@!(QDRS"#TU@VW":#γ = 16IJ@

;W"#γ >16IJXY3.1Z[?γ <16IJXY3.2Z[@4\"]

^C)DE6_%#`O(QabcdM?e"fgC)XY3.3Z[D

3.1 !"#$%& γ > 1 #'(

γ >16IJ#(9)$@V\)v−γv^γ6<h@;W"i69HjklC )m?@noC)Xp2qrDTU#m69Hc9H(∗)?e"stC)[u

1. v= 0 (⇐⇒ z=e^−v= 1)6v#v−γv^γ = 0.

2. 0< v < v₀6vX0< v₀=γ^1/(1−γ)<1[#0< v−γv^γ < v₀. 3. v=v0 (⇐⇒ z=z0=e^−v0)6v#v−γv^γ = 0.

4. v0< v <∞6v#v−γv^γ<0.

5. v→ ∞(⇐⇒ z→0)6v#v−γv^γ → −∞.

1 γ 1

γ

γ^1−γ1 γ^1−γ1

0 γ^1+γ1−γ

γ^1−γ2

p 2: u = γv^γ ?u = v6wxyXγ > 16IJuz{u#|{v[u}

~· · · u=γv^γ#•~· · · u=v

13

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

（  ）１３

−   −１５９

!"#$%(∗)&'()*+γ >1#,-(./*+(6)01234567 8$9f₀(z|α, γ)#:;(</*=>?6@A#B+α >1#,-Cα= 1

#,-+α <1#,-#3<#,-(DE*=>?6@

3.1.1 α >1!"#

α >1#,-+FG#γ >0(H)*1−γα <1−γ1I6@JK+LL 1M+N(γ >11I6LC(OG?6C+1−γα <1−γ <0&P6@QR

*+STUV(WX)K$%(∗)(OG?6C+YZ[v1> v0=γ^1−γ1 \]

^)*v₁−γv^γ₁ = 1−αγC1_6@LL1+z₁:=e^−v1C`a?5b+(9) 0(cd+(6)012345678$9f0(z|α, γ)#ef`M`31ghi5 6@LL1+zCvMj/(klfmno#$%z=e^−v ⇐⇒ v=−logz (I6LC(OG?6@L5(cd+γ >1p<α >1#,-+78$9 f0(z|α, γ)#:;M+

z=z₁=e^−v1

&qrstquvwC?6kxy1I6LC\z{?6@|)+v1M$%

0v₁−γv^γ₁ = 1−αγ&}K)+v₁> v₀ =γ^1/(1−γ)<11I6@L5(

cR*+quvz₁Mz₀cd~4[v1I6LC\Dp6•

z1=e^−v1 < e^−v0 =z0.

3.1.2 α= 1!"#

α= 1#,-+1−γα= 1−γ1Id+LL1+γ >11I6LC(O G?6C+1−γα = 1−γ < 0&P6@QR*+STUV(WX)K$

%(∗)(OG?6C+YZ[v₁> v₀\]^)*+v₁−γv^γ₁ = 1−γ ⇐⇒

(1−γv^γ−1₁ )v1 = 1−γ C1_6@LL1+v−γv^γ \v > v01klfm 1Id+p<+v₀<1()*v₀−γv^γ₀ = 01Id+1−γ <01I6LC (OG?6C+v1−γv^γ₁ = 1−γ ⇐⇒ v1= 11I6LC\Dp6@cR

*+z1 :=e^−v1 = 1/e&P*+(9)0(cd+(6)012345678$9 f₀(z|α, γ)#ef`M`41ghi56@|)+zCvMj/(klfmn

（  ）１４14

−   −１６０

!"#$z =e^−v ⇐⇒ v =−logz%&'()%*+,'-(.%/01 γ >123α= 1"45167#8f₀(z|1, γ)"9:;1

z=1

e = 0.367879· · ·

<=>?@=ABC),'DEFG&'()HIJ,'-KL1v0<1G

&'()%*+,')1=AB1/e;z₀/0MNOKBG&'()HP 2'Q

e⁻¹< e^−v0 =z0.

3.1.3 α <1!"#

α <1"451R+"γ >0%STU1−γα >1−γG&'-((G1 γ >1G&'()%*+,')1−γ <0<V'"G11−γα"WX;Y Z%;[\)K'-]^U1_`G;11−γα <0"45)1−γα= 0"

4511−γα >0"45"33"45%PaUbc,'-

dY%11−γα <0"45%3eUbc,'-((G1γ >1)α <1

%*+,')1f"gB#$<V'Q 1−γα <0 ⇐⇒ 0< 1

γ < α <1.

("h1ijkl%mnTo#$(∗)%*+,')1pqKv₁> v₀HrsT Uv1−γv₁^γ = 1−γα)Gt'-((G1z1:=e^−v1)uv,.w1(9)x%/

01(6)xG\yO.'67#8f₀(z|α, γ)"z{u;u5@|}~uCG•€

•.'-‚T1z)v;ƒe%D„{…†!"#$z=e^−v ⇐⇒ v=−logz

%&'()%*+,'-(.%/011< γ <∞231/γ < α <1"451 67#8f₀(z|α, γ)"9:;1

z=z1=e^−v1

<=>?@=ABC),'DEFG&'-‚T1v₁;#$xv₁−γv₁^γ = 1−αγ

<‡oT1v₁> v₀=γ^1/(1−γ)<1G&'-(.%/^U1=ABz₁;z₀

15

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

（  ）１５

−   −１６１

!"#$%&'()*+,-.)/

z1=e^−v1 < e^−v0 =z0.

01231−γα= 04562789:;<)=**'3γ >1+α <1 2>?<)+3@4A&BCDE)/

1−γα= 0 ⇐⇒ 0< 1

γ =α <1.

*4F3GHIJ2KLMNBC(∗)2>?<)+3v= 0ONPv=v₀= γ^1/(1−γ)<12Q89v−γv^γ = 0'().R3(9)S2!"3(6)S'T U$V)WXBYf₀(z|α, γ)4Z[\P\5]^_`\a'bcRV)=d M3z+vPe82fg[hij4BCz=e^−v ⇐⇒ v=−logz2()

*+2>?<)=*V2!"3γ >1.7α= 1/γ <14563WXBY f0(z|α, γ)4klP3

z=z0=e^−v0=e^−γ1−γ1 >1

e = 0.367879· · · Dmno]mp&a+<)fqr'()=

0s231−γα >04562789:;<)=**'3γ >1+α <1 2>?<)+3@4A&BCDE)/

1−γα >0 ⇐⇒ 0< α < 1 γ <1.

$93t?4γ >12uM93v−γv^γ P3v4BY+M9vwxyz [0, v₀]'{|%4'3mn&D})]0< v₀=γ^1/(1−γ)<1a/

v∗:= arg max

v∈[0,v0]v−γv^γ. (13) ON3GHIJ2KLMNBC(∗)2!"3v∈(0, v0)2Q890< v−γv^γ <

v₀<1'("3.73v= 0ONPv=v₀2Q89v−γv^γ = 0'()*

+2>?<)+3mnov∗P~o•3€•3v∗∈(0, v0)2M9 0 < v∗−γv∗γ = max

v∈[0,v0]v−γv^γ < v0=γ^1−γ1 < 1

（  ）１６16

−   −１６２

!"#$%&'"()*+,-./γ >1012+,1−γα%α/34$

2+5602+789:;<=,'>,0< α <1/γ <1012+,

α→0lim1−γα= 1, lim

α→_γ¹1−γα= 0

;<"#$0?.@"$,AB!α_∗∈(0,1/γ)%C.0DE2+

1−γα∗=v∗−γv_∗^γ = max

v∈[0,v0]v−γv^γ (14)

$;F"(#G0H=,-./γ >1012+I/JK3LMN"O

v∗−γv∗γ ≶1−γα ⇐⇒ α≶α∗. (15)

P2,0< α <1/γ;<"#$0?.@"(

QR/ST0UVW+,XY/Z[\1−γα >0 ⇐⇒ α <1/γ/Z[]

0>W+^,γ >1!"_`/a,0< α < α∗/Z[$α=α∗/Z[, α_∗< α <1/γ/Z[/3>/Z[0&b+ST@"(

cd,0< α < α∗/Z[,(15)e0H=

1−γα > max

v∈[0,v0]v−γv^γ

;<=,'>,fghi0jk2l3L(∗)$(9)e0?.@"$,-./

v >0012+I/JK3LMN"O

1−γα > v−γv^γ ⇐⇒ f₀(z)>0, ∀z∈(0,1).

H*+,(6)e;mnoG"pq34f0(z|α, γ)/r9s^s5\atus]

;vwxG"(##;,z$v^yW0789:z{/3Lz=e^−v ⇐⇒

v=−logz0<"#$0?.@"()*+,pq34f₀(z|α, γ)/|}^

78!uQ=\78r~•];<"(

I0,α=α_∗/Z[,(15)e0H=

1−γα= max

v∈[0,v0]v−γv^γ =v∗−γv∗γ

17

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

（  ）１７

−   −１６３

!"#$

1−γα > v−γv^γ, ∀v=v_∗

%&'()*+$(9),!-./')$(6),%0123'4567f₀(z|α, γ) 89:;<;5=>?@;A%BC+3'D((%$z)v<EF!GH:

IJK86Lz =e^−v ⇐⇒ v =−logz!&M$z_∗ :=e^−v∗);N/'D OP$v∗8QRS.T!UF#<VW28XYZ[\23]FD(3!^

M$4567f₀(z|α, γ)8_`<GHOabM=GH9cdA)O'()e fY/'D

gh!$α∗< α <1/γ8ij$(15),!^M 1−γα < max

v∈[0,v0]v−γv^γ =v_∗−γv_∗^γ>0

%&M$*U$v∗eS.!0k'()$l"#$v= 0k]<v=v0!P F#v−γv^γ = 0%&'()!-./')$v−γv^γ ev867)"#m n%&'()!^M$0< v1< v∗< v2< v0<1Zo]/v1)v2epq

"#$

1−γα=v₁−γv^γ₁ =v₂−γv₂^γ )/'()e%r'D((%$

z₂:=e^−v2 < z_∗:=e^−v∗ < z₁:=e^−v1 (16) );N/')$stuv!wx"]6L(∗))(9),!^M$(6),%01 23'4567f₀(z|α, γ)89:;<;5=y?;A%BC+3'D((%$

z)v<EF!GH:IJK86Lz=e^−v ⇐⇒ v=−logz!&'()

!-./'DOP$z1)z2<l3z3v1)v28S.T8{|)"#S.

!0kM$v₁)v₂8S.T!UF#<VW28XYZ[\23]FD;5

=y?;A!^M$4567f0(z|α, γ)8_`<$9c}:I}9c)F~3

•67€OabM8_`$•FKC3‚$GH%<OFƒ88abM8„

…Zx/()e†*'Ds‡%<$(3ZˆGH9cd)‰Š()!/'D

18^{（  ）１８}

−   −１６４

3.1.4 !"#$%

!"#$3.1"%&'()*+,-.&/012γ >1,342(6)56 789:);<=>?@ABCDEF,GHI>f0,JK&LM1N,O PQR)S

&'2 (;<=>?@ABCDEF,JKSγ >1,34). γ >1,342 (6)56789:)GHI>f0,TUV{(z, f0(z|α, γ))|z∈(0,1)},JK W2XY,Z>α&[\1]^,/_&7`)abc2

α∗= 1 γ−

1−1

γ

γ^1+γ1−γ, 0< α∗< 1 γ 6d)a`e2]^6Wf₀(z) :=f₀(z|α, γ)fghi)a 1. 0< α≤α_∗,34Sjklmn#o62o7pq%

GHI>f0(z)Wz,jklmI>6dr2s,TUVWjktu

*r,JKQvia

2. α∗< α <1/γ,34Swjklmn#o72o8pq%

GHI>f₀(z),TUVW2wjktu*r,JK#lmxyzx lmfM_JK2{5|}{pq%Qvia

3. 1/γ ≤α <∞,34Sj~n#o6pq%

GHI>f0(z),TUVW2z=z1Q•€•#•‚Y%fi)j~

n,JKQviS

z1=e^−v1.

bc2v₁WIƒ5v₁−γv^γ₁ = 1−αγQ„ec2v₁≥v₀=γ^1/(1−γ) 6d)a`e2…:&/r2z1=e^−v1 ≤z0=e^−v06d)a

v₁Wα,jklmI>fc17`r2s,OPfc12•‚Yz<sub>1</sub>W α,jkyzI>fc17`)a†&2α= 1/γ,‡2z₁ =e^−v0 = z0> e⁻¹6dr2α= 1,‡2z1=e⁻¹= 0.367879· · · 6d)a (). γ >1,‡2ˆ‰tα∗ ∈(0,1/γ)Š‹Œc1(6)56789:)G HI>f₀,JKŠ0 < α≤α_∗,34&jklmn2α_∗ < α <1/γ,3

19

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

（  ）１９

−   −１６５

!"#$%&'()1/γ < α <∞*+!"$,(-./0-"1234) 53.16"78/9:*;<"=>3?"@AB"CDEF=>3)GH I4)γ"JK3LM/α_∗*NOPQ/0-"R/F

α∗*N4(14)S"TU23LM/F=>3)α∗4v∗"VWX)v∗4 (13)S"=>3LYCD/FZ>3)GH"7234)v_∗OPQE9Iα_∗ O[\R/F]*9I)z∗*NOPQ/0-"R/F

^_)(13)S*`abcd"723)`aev_∗fgheIi/0-v_∗∈ (0, v0), v0=γ^1/(1−γ)-`ab*1jkl"=m)v∗4n*=o"pq"

LM/rs2tuvw

1−γ²v^γ−1_∗ = 0 ⇐⇒ v∗=γ^1−γ2 =v₀². (17) 00I)0< v₀ <1"xqR/-)0< v_∗ =v₀²< v₀<1"=m)yB"

v∗∈(0, v0)Ii/FME)`ab*2jkl4)γ >1"=m)(13)S*

`abcd"723yB"zECD32/0-"xqR/w

−γ²(γ−1)v^γ−2<0, ∀v∈[0, v0].

.7)1−γ²v^γ−1 ≶0 ⇐⇒ v ≷γ^1−γ2 =v∗"=m)v−γv^γ4)v*{

|-X3)v < v_∗I$%&')v > v_∗I$%}~Ii/F0D"=>3) (16)S"•D/v₁-v₂fpq"LM/-€B/F

n")PQADEv∗"JK3)(14)S"Z>3α∗f•AD/w α_∗= 1

γ −

1

γv_∗−v^γ_∗

= 1

γ+γ^1−γ2γ −γ^1+γ1−γ = 1 γ−

1−1

γ

γ^1+γ1−γ. 00I)(13)S"123?"9:XE‚m)ƒq*v ∈(0, v0)"„X3) 0< v−γv^γ < v₀<1"xqR/-)

0< 1

γv−v^γ < 1 γ

Ii/BA)v∗=v₀²< v0"xqR/-)0< α∗<1/γO•/F

`…")z∗:=e^−v∗, v∗=v²₀4n*=o"†CD/rs1tuvw z∗=e^−v02= exp

−γ^1−γ2

. (18)

（  ）２０20

−   −１６６

3.2 !"#$%& γ < 1 #'(

γ <1!"#$(9)%&'()v−γv^γ!*+&,-./!012345) 67&895):;3<=>?@$6!01A01(∗∗)7B.CD5)EF

1. v= 0 (⇐⇒ z= 1)!G$v−γv^γ= 0.

2. 0< v < v0!G:0< v0=γ^1/(1−γ)< γE$−v0< v−γv^γ<0.

3. v=v₀ (⇐⇒ z=e^−v0)!G$v−γv^γ = 0.

4. v0< v <∞!G$v−γv^γ>0.

5. v→ ∞(⇐⇒ z→0)!G$v−γv^γ → ∞.

1 1

γ γ

γ^1−γ1 γ^1−γ1

0 γ^1+γ1−γ

γ^1−γ2

; 3: u = γv^γ 7u = v!HIJ:γ < 1!"#FKLu$MLvEFN O· · · u=γv^γ$PO· · · u=v

QR!01(∗∗)AS&B.$T3.1U7VW!XYAZ[\]^$(6)% _`abc)de0ff<sub>0</sub>!ghi7B.$γ <1!"#$i67i7$i 82jkc)>6c&lm.$γ <1!"#$(6)%_`abc)nopf qrstuvwx!de0ff₀!yz&,-./!{|Aj)F

21

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

（  ）２１

−   −１６７

!"3 (!"#$%&'()*+,-./0γ <1-12). γ <1-123 (6)456789:;<=$f₀->?@{(z, f₀(z|α, γ))|z∈(0,1)}-./

A3BC-D$αEFGHIJ-KLE6M:NOP3 α∗= 1

γ +

1

γ−1

γ^1+γ1−γ, α∗> 1 γ 5Q:NMR3IJ5Af0(z) :=f0(z|α, γ)STUV:N

1. 0< α≤1/γ-120UWXYZ9[\]

;<=$f0(z)->?@A3z=z1^_`aSV:UWX-./^

bV0

z1=e^−v1.

OP3v1A=c4v1−γv^γ₁ = 1−γα^dRP3v1≥v0=γ^1/(1−γ) 5Q:NMR3e9EKf3z₁=e^−v1 ≤z₀=e^−v05Q:N

gE3α= 1/γ-h3z1 =e^−v0 =z05Qf3α= 1-h3z1 = e⁻¹= 0.367879· · · 5Q:N

2. 1/γ < α < α∗-120ijklmXYZ10[\]

;<=$f0(z)->?@A3ijknoJf-./Ylmpqrp lmSsL./3t8uvt[\]^bVN

3. α∗≤α <∞-120jklmXYZ93Z10[\]

;<=$f₀(z)Az-jklm=$5Qf3w->?@Ajkno Jf-./^bVN

#$. xyz-{|E}sHA3yz2-{|S~•-€•^‚ƒ„…KsN OP3γ <1-12Eα_∗†1/γKf‡ˆ‰n:eSE}sHA30< γ <1 EŠ‹V:S31/γ−1>0Œ}γ^1+γ1−γ >05Q:Œ•3α∗>1/γ^Ž:N

e-a3•Eα_∗<2/γ5Q:eS•‘’89:Nw-“3 α_∗= 1

γ +

1

γ −1

γ^1−γ1+γ = 1 γ −

1

γv_∗−v_∗^γ

（  ）２２22

−   −１６８

!"#$%&'()#*+,-v_∗ =v²₀ < v₀, v₀ =γ^1/(1−γ) <1!"#*

$$!-./01&23,456(∗∗)&78-9(:v∈(0, v0)&;,<

−1<−v₀< v−γv^γ <0!"#$%&'()#%-

−1 γ < 1

γv−v^γ <0

!"#=>-v∗ =v²₀ < v0&'()#%-?=&1/γ < α∗ <2/γ!"#

$%@A=#*

3.3 !"#$%&'()*+,*)-+./+01

BC1%BC2-BC3:DEFGH)#$%&7I<-(6)J!KLM N#OP;QRSTUVWAX:YZ5Qf0:[\&]^<_:`abc C1dFe#*fg-cC1Fh%,<23,4:@h1!"8-i%,<

23,4:@i4!"#*

23 1 (OP;QRSTUVWAX:[\). (6)J!KLMN#YZ5Q f0:jkl{(z, f0(z|α, γ))| z∈(0,1)}:[\m-no:pQα%γ&q r<st:7u&Kv#bh1-i4-BwBxyd*+,-

α_∗= 1 γ−

1−1

γ

γ^1+γ1−γ = 1 γ+

1

γ −1

γ^1+γ1−γ (19)

!"#bγ= 1d*v4-st!mf₀(z) :=f₀(z|α, γ)%z{)#*

1. 0< γ <1:|Hb}~&]^<mBC3xyd•α∗>1/γFe#*

(a) 0< α≤1/γ:|H•

YZ5Qf₀:jklm-U€•:[\F3)*

(b) 1/γ < α < α∗:|H•

YZ5Qf₀:jklm-‚ƒ„…†•b‡ˆ‰fŠt8d:

[\F3)*

(c) α∗≤α <∞:|H•

YZ5Qf₀:jklm-ƒ„…†•bŠt8d:[\F3)*

23

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

（  ）２３

−   −１６９

2. γ= 1!"#$%&'()*+,-1./01 (a) 0< α <1/γ!"#1

2345f₀!678+9:;<=>$?@A0!BCDEFG (b) α= 1/γ!"#$HIJK01

2345f₀!678+9LM>!BCDEFG (c) 1/γ < α <∞!"#1

2345f₀!678+9:;NO>$?@A0!BCDEFG

3. 1< γ <∞!"#$%&'()*+,-2./01α_∗<1/γDPQ. (a) 0< α≤α∗!"#1

2345f₀!678+9:;<=>$?@A0!BCDEFG (b) α∗< α <1/γ!"#1

2345f₀!678+9R:;<=>$STUV?@A0!

BCDEFG

(c) 1/γ≤α <∞!"#1

2345f₀!678+9:W>!BCDEFG

X 1: 2345f₀(z|α, γ)!BC!JY$0< α, γ <∞9Z-1./0

0 · · · α∗ · · · 1/γα · · · α∗ · · · ∞

γ <1 ... U ^{[> R:;NO> :;NO>} ...

γ= 1 ... :;<=> HIJK :;NO> ...

γ >1 ... :;<=> R:;<=> :W> ...

∗γ <1^!"#+α∗>1/γ >1^\VA, γ >1^!"#+α∗<1/γ <1^\VQ.

（  ）２４24

−   −１７０

!4: "#$%f₀(z|α, γ)&'(&)*+,-α./-γ.01123456 7· · · α=α∗= 1/γ−(1−1/γ)γ(1+γ)/(1−γ).87· · · α= 1/γ

4 ^!"

9:;<(0,∞)=>?@ABCDEF%)G>HIJKL%>MNOP

&F%LQRSTUV>WXOYZBCJKL%&)G[.\];<(0,1)

=>?@ABCDE^_)G`aCbcd[.U&e6Rf>NO.F%

)GRghVNOijk)GlWeibull)GRmnopqVNOrsTC

^tuijk)G>HIJKL%>MNOP&F%LQRSTUV>WX O.^_)GRghVNOM%ijk)GlM%Weibull)GRmnop qVNOrsTCWv^twoJK)Gx+M%^tuijk)G4Ry zN{b|&"#$%[(8)}.|NO.|&DE~&"#$%[(6)}>

•€ZBCbm>.DE~&pq+(6)}4>•‚O[.|&ƒ%V"#&

25

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

（  ）２５

−   −１７１

!"#$%&'()*+,-./#012-3456789&:;12- 3<=-7>[email protected]#CDE(8)FG*HI3J./#KLM&NOC

*P=QRS./#TU2VW#!"#$%*HI3#'()?+,JX

;2-3YZ63IR>

[\*@I3]^Z67VW$U(6)F&_H`abcEdefUAB ghijbcGJ.2H#TU&_k.lH.Ambc#ABg2?n3 IR>Ambc#ABg2-3op)?PJqrsbctuvw>qrs bcxy72H#TU&_k./#2H#TU*z{3qrsbcJ/#

VW#!"&|}*~gZ•R>€•‚')*J.qrsbc#VW#!

"J./#2H#TU*z{3.ƒ„…5†‡EAmbcˆ‰Š‡ˆU‹

‡ˆ‰Œ•Ž‡ˆ‰Œ••‡G*b†Z6REP‘’.Johnson, Kotz and Balakrishnan 1995, p.219“”•2018–—G>/6*f-3.[\*€n 3]^Z67defUABghijbc#VW$U(6)F#!"J./#2 H#TU*z{3.ƒ„…7†‡EAmbcˆ‰Š‡ˆU‹‡ˆ‰Œ•Ž

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-7E:;1.p1–—G>

œ•*€•.[\*@I3]^Z67defUABghijbcJ.qr sbctJpžZ64?I˜‰Œ•Ž‡2˜‰Œ••‡2Iw!"&=- 4R™2S›5l*?n7>™#˜‰Œ‡#!"&=Q#J.(6)F*ž6 RTUα2γ#Ÿ567 ¡.¢k.γ <1lH1/γ < α < α∗#CDE˜

‰Œ••‡G2γ >1lHα∗< α <1/γ#CDE˜‰Œ•Ž‡G*£¤

?IEp1.¥4–—G>-l-.•?…2x™#¦*@I3.[\*€n 3]^Z67defUABghijbc#VWJqrsbc#VW€•x

|}?!"&=Q™2S›5l*Z67>

（  ）２６26

−   −１７２

!"#$

[1] !"#$%1962&'()*+,-./01

[2] 2345%2018&6789:;<=>?@ABC'DEFGHIGJK, L62ML1N2OPpp.51–76.

[3] 2345%2019&6789QRS+<T+SU<:;<=>?@ABC 'DEFGHIGJK,L63ML2OPpp.195–224.

[4] VWXY%1980&'Z[\]I,^_FG`ab1

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

[6] Consul, P. C. and G. C. Jain (1971) “On the log-gamma distribution and its properties”,Statistisch Hefte, 12, pp.100–106.

[7] Crooks, G. E. (2019) Field Guide to Continuous Probability Distri- butions, Berkeley Institute for Theoretical Sciences.

[8] Grassia A. (1977) “On a family of distributions with argument be- tween 0 and 1 obtained by transformation of the gamma and de- rived compound distributions”, Australian Journal of Statistics, 19, pp.108–114.

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

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

[11] Leemis, M. L. and J. T. McQueston (2008) “Univariate distribution relationships”,The American Statistician, 62, 45–53.

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一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

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[12] Mazucheli, J., A. F. B. Menezes, L. B. Fernandes, R. P. de Oliveira and M. E. Ghitany (2019) “The unit-Weibull distribution as an alter- native to the Kumaraswamy distribution for the modeling of quan- tiles conditional on covariates”,Journal of Applied Statistics, (to be appeared), DOI: 10.1080/02664763.2019.1657813.

[13] McDonald, J.B. and Y.J. Xu (1995) “A generalization of the beta distribution with applications”,Journal of Econometrics, 66, pp.133–

152.

[14] Nadarajah, S. and A. Gupta (2004) “Generalizations and related uni- variate distributions”, InHandbook of Beta Distribution and Its Ap- plications, edited by A.K. Gupta and S. Nadarajah, Marcel Dekker, pp.97–163.

[15] Rinne, H. (2009) The Weibull Distribution: A Handbook, Taylor &

Francis Group.

（  ）２８28

−   −１７４

A !"#$%&'()*+,*)-+./+012 345*+678

!"#$%&'()*+,-./012345678(6)9:;<)*=

>?@/f₀(z|α, γ), z∈(0,1)ABCD!EF8G>?@/f₀(z|α, γ), z∈ (0,1)HIJKALMN=DO*#PQG2CHRST/α, γ >0U>?@

/f₀HVWXHRSUH@YZ[\]#^=D_H`aUb&GEc 1U Ec2GEc3Zd\*Ge%&8Gfc1Zd\*=D^$GghHK

#$%&Gz0Uz∗, α∗H;<#C%&8_*i*(12)9U(18)9G(19) 9:jk\*=lm1Gm4nopDq+Gz₁Uz₂H;<#C%&8G_

*i*Hrs#tu&!vwHxFyEcAno)*+%D

A.1 γ = 1 +9:

K2: >?@/f0(z| α, γ)HIJKz{Klα <1HrspGwKlα= 1 Hrsz0|67pG}Klα >1Hrsp

z 0 · · · 1

f₀ +

f₀

z 0 · · · 1

f₀ 0

f₀ →

z 0 · · · 1

f₀ −

f₀

29

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

（  ）２９

−   −１７５

A.2 γ > 1 !"#

! 3: "#$%f0(z| α, γ)&'(!)α >1&*+, z 0 · · · z₁ · · · 1

f₀ + 0 −

f0

! 4: "#$%f0(z| α, γ)&'(!)α= 1&*+, z 0 · · · 1/e · · · 1

f₀ + 0 −

f0

!5:"#$%f0(z|α, γ)&'(!)α <1&*+,-./0!)1/γ < α <1

&*+,1./2!)α= 1/γ&*+,13/!)α_∗ < α <1/γ&*+,1 4/0!)α=α∗&*+,14/2!)0< α < α∗&*+,

z 0 · · · z1 · · · 1

f₀ + 0 −

f0

z 0 · · · z0 · · · 1

f₀ + 0 − 0

f0

z 0 · · · z₂ · · · z₁ · · · 1

f₀ + 0 − 0 +

f0 0

z 0 · · · z_∗ · · · 1

f₀ + 0 +

f0

z 0 · · · 1

f₀ +

f0

（  ）３０30

−   −１７６

A.3 γ < 1 !"#

! 6: "#$%f0(z| α, γ)&'(!)α <1&*+, z 0 · · · z₁ · · · 1

f₀ − 0 +

f0

! 7: "#$%f0(z| α, γ)&'(!)α= 1&*+, z 0 · · · 1/e · · · 1

f₀ − 0 +

f0

!8:"#$%f0(z|α, γ)&'(!)α >1&*+,-./0!)1< α <1/γ

&*+,1./2!)α= 1/γ&*+,13/!)1/γ < α < α_∗&*+,1 4/0!)α=α∗&*+,14/2!)α∗< α <∞&*+,

z 0 · · · z1 · · · 1

f₀ − 0 +

f0

z 0 · · · z0 · · · 1

f₀ − 0 + 0

f0

z 0 · · · z₂ · · · z₁ · · · 1

f₀ − 0 + 0 −

f0 0

z 0 · · · z_∗ · · · 1

f₀ − 0 −

f0

z 0 · · · 1

f₀ −

f0

31

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

（  ）３１

−   −１７７

B !"#$%&'()*+,*)-+./+012 345*+678

!"#$%(6)&'()*+,-./0f0(z|α, γ), z∈(0,1)12345 678,9:;<f₀(z|α, γ)%=;<z>?

B.1 γ = 1 +9:

0 1 2 3 4 5

0.2 0.4 0.6 0.8 1

6 5: -./0f₀(z|α, γ)12349γ = 11@A><α= 0.39BCD>%

α= 0.69BED>%α= 19CD>%α= 29ED>%α= 39FCD>

B.2 γ > 1 +9:

0 0.5 1 1.5 2 2.5 3 3.5

0.2 0.4 0.6 0.8 1

6 6: -./0f₀(z|α, γ)12349γ = 21@A><α= 0.19BCD>%

α= 0.259BED>%α= 1/γ= 0.59CD>%α= 0.759ED>%α= 19F CD>%α= 29FED>

（  ）３２32

−   −１７８

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0.2 0.4 0.6 0.8 1

! 7: "#$%f₀(z|α, γ)&'()*γ = 2&+,-.α= 0.4*/01-2 α = α∗ = 0.4375*/31-2α = 0.45*01-2α = 0.47*31-2 α= 0.49*401-2α= 1/γ = 0.5*431-

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

!8: "#$%f₀(z|α, γ)&'()*γ= 5&+,-.α= 0.12< α_∗0.128

*/01-2α = 0.13*/31-2α = 0.15*01-2α = 0.17*31-2 α= 0.19*401-2α= 1/γ = 0.2*431-

33

一般化ガンマ確率変数の指数変換の分布の形状について（鍵原）

（  ）３３

−   −１７９