Appendix A
An Inventory of Continuous
Distributions
A.1
Introduction
The incomplete gamma function is given by
1
Γ(α; x) =
Γ(α)
Z
x
tα−1 e−t dt,
α > 0, x > 0
0
with Γ(α) =
Z
∞
tα−1 e−t dt,
α > 0.
0
Also, define
G(α; x) =
Z
∞
tα−1 e−t dt,
x > 0.
x
At times we will need this integral for nonpositive values of α. Integration by parts produces the relationship
G(α; x) = −
xα e−x
1
+ G(α + 1; x)
α
α
This can be repeated until the first argument of G is α + k, a positive number. Then it can be evaluated
from
G(α + k; x) = Γ(α + k)[1 − Γ(α + k; x)].
The incomplete beta function is given by
Z
Γ(a + b) x a−1
t (1 − t)b−1 dt,
β(a, b; x) =
Γ(a)Γ(b) 0
1
a > 0, b > 0, 0 < x < 1.
APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS
A.2
Transformed beta family
A.2.2
Three-parameter distributions
A.2.2.1
Generalized Pareto (beta of the second kind)–α, θ, τ
f (x) =
Γ(α + τ ) θα xτ −1
Γ(α)Γ(τ ) (x + θ)α+τ
E[X k ] =
θk Γ(τ + k)Γ(α − k)
,
Γ(α)Γ(τ )
E[X k ] =
θk τ (τ + 1) · · · (τ + k − 1)
,
(α − 1) · · · (α − k)
2
F (x) = β(τ , α; u),
u=
x
x+θ
−τ < k < α
if k is an integer
θk Γ(τ + k)Γ(α − k)
β(τ + k, α − k; u) + xk [1 − F (x)],
Γ(α)Γ(τ )
τ −1
= θ
, τ > 1, else 0
α+1
E[(X ∧ x)k ] =
mode
A.2.2.2
k > −τ
Burr (Burr Type XII, Singh-Maddala)–α, θ, γ
f (x) =
E[X k ] =
αγ(x/θ)γ
x[1 + (x/θ)γ ]α+1
F (x) = 1 − uα ,
θk Γ(1 + k/γ)Γ(α − k/γ)
,
Γ(α)
u=
1
1 + (x/θ)γ
−γ < k < αγ
VaRp (X) = θ[(1 − p)−1/α − 1]1/γ
θk Γ(1 + k/γ)Γ(α − k/γ)
β(1 + k/γ, α − k/γ; 1 − u) + xk uα ,
Γ(α)
µ
¶1/γ
γ−1
= θ
, γ > 1, else 0
αγ + 1
E[(X ∧ x)k ] =
mode
A.2.2.3
k > −γ
Inverse Burr (Dagum)–τ , θ , γ
f (x) =
E[X k ] =
τ γ(x/θ)γτ
x[1 + (x/θ)γ ]τ +1
θk Γ(τ + k/γ)Γ(1 − k/γ)
,
Γ(τ )
F (x) = uτ ,
u=
(x/θ)γ
1 + (x/θ)γ
−τ γ < k < γ
VaRp (X) = θ(p−1/τ − 1)−1/γ
θk Γ(τ + k/γ)Γ(1 − k/γ)
β(τ + k/γ, 1 − k/γ; u) + xk [1 − uτ ],
Γ(τ )
µ
¶1/γ
τγ − 1
= θ
, τ γ > 1, else 0
γ+1
E[(X ∧ x)k ] =
mode
k > −τ γ
APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS
A.2.3
Two-parameter distributions
A.2.3.1
Pareto (Pareto Type II, Lomax)–α, θ
f (x) =
αθα
(x + θ)α+1
F (x) = 1 −
µ
θ
x+θ
E[X k ] =
θk Γ(k + 1)Γ(α − k)
,
Γ(α)
E[X k ] =
θk k!
, if k is an integer
(α − 1) · · · (α − k)
3
¶α
−1 < k < α
VaRp (X) = θ[(1 − p)−1/α − 1]
θ(1 − p)−1/α
TVaRp (X) = VaRp (X) +
, α>1
α−1
"
µ
¶α−1 #
θ
θ
, α=
6 1
E[X ∧ x] =
1−
α−1
x+θ
µ
¶
θ
E[X ∧ x] = −θ ln
, α=1
x+θ
θk Γ(k + 1)Γ(α − k)
β[k + 1, α − k; x/(x + θ)] + xk
Γ(α)
= 0
E[(X ∧ x)k ] =
mode
A.2.3.2
µ
θ
x+θ
¶α
all k
Inverse Pareto–τ , θ
f (x) =
τ θxτ −1
(x + θ)τ +1
F (x) =
µ
x
x+θ
¶τ
E[X k ] =
θk Γ(τ + k)Γ(1 − k)
,
Γ(τ )
E[X k ] =
θk (−k)!
, if k is a negative integer
(τ − 1) · · · (τ + k)
−τ < k < 1
VaRp (X) = θ[p−1/τ − 1]−1
∙
µ
¶τ ¸
Z x/(x+θ)
x
E[(X ∧ x)k ] = θk τ
y τ +k−1 (1 − y)−k dy + xk 1 −
,
x+θ
0
τ −1
mode = θ
, τ > 1, else 0
2
A.2.3.3
,
k > −τ
Loglogistic (Fisk)–γ, θ
f (x) =
γ(x/θ)γ
x[1 + (x/θ)γ ]2
F (x) = u,
E[X k ] = θk Γ(1 + k/γ)Γ(1 − k/γ),
VaRp (X) = θ(p−1 − 1)−1/γ
u=
(x/θ)γ
1 + (x/θ)γ
−γ < k < γ
E[(X ∧ x)k ] = θk Γ(1 + k/γ)Γ(1 − k/γ)β(1 + k/γ, 1 − k/γ; u) + xk (1 − u),
¶1/γ
µ
γ−1
, γ > 1, else 0
mode = θ
γ+1
k > −γ
APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS
A.2.3.4
4
Paralogistic–α, θ
This is a Burr distribution with γ = α.
α2 (x/θ)α
x[1 + (x/θ)α ]α+1
f (x) =
F (x) = 1 − uα ,
θk Γ(1 + k/α)Γ(α − k/α)
,
Γ(α)
E[X k ] =
u=
1
1 + (x/θ)α
−α < k < α2
VaRp (X) = θ[(1 − p)−1/α − 1]1/α
θk Γ(1 + k/α)Γ(α − k/α)
β(1 + k/α, α − k/α; 1 − u) + xk uα ,
Γ(α)
µ
¶1/α
α−1
= θ
, α > 1, else 0
α2 + 1
E[(X ∧ x)k ] =
mode
A.2.3.5
k > −α
Inverse paralogistic–τ , θ
This is an inverse Burr distribution with γ = τ .
2
τ 2 (x/θ)τ
x[1 + (x/θ)τ ]τ +1
f (x) =
F (x) = uτ ,
θk Γ(τ + k/τ )Γ(1 − k/τ )
,
Γ(τ )
E[X k ] =
u=
(x/θ)τ
1 + (x/θ)τ
−τ 2 < k < τ
VaRp (X) = θ(p−1/τ − 1)−1/τ
θk Γ(τ + k/τ )Γ(1 − k/τ )
β(τ + k/τ , 1 − k/τ ; u) + xk [1 − uτ ],
Γ(τ )
E[(X ∧ x)k ] =
mode
A.3
= θ (τ − 1)1/τ ,
k > −τ 2
τ > 1, else 0
Transformed gamma family
A.3.2
Two-parameter distributions
A.3.2.1
Gamma–α, θ
f (x) =
(x/θ)α e−x/θ
xΓ(α)
M (t) = (1 − θt)−α ,
F (x) = Γ(α; x/θ)
t < 1/θ
E[X k ] = θk (α + k − 1) · · · α,
E[(X ∧ x)k ] =
mode
E[X k ] =
θk Γ(α + k)
,
Γ(α)
k > −α
if k is an integer
θk Γ(α + k)
Γ(α + k; x/θ) + xk [1 − Γ(α; x/θ)],
Γ(α)
k > −α
= α(α + 1) · · · (α + k − 1)θk Γ(α + k; x/θ) + xk [1 − Γ(α; x/θ)],
= θ(α − 1), α > 1, else 0
k an integer
APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS
A.3.2.2
5
Inverse gamma (Vinci)–α, θ
f (x) =
(θ/x)α e−θ/x
xΓ(α)
E[X k ] =
θk Γ(α − k)
,
Γ(α)
E[(X ∧ x)k ] =
F (x) = 1 − Γ(α; θ/x)
E[X k ] =
k<α
θk
,
(α − 1) · · · (α − k)
if k is an integer
θk Γ(α − k)
[1 − Γ(α − k; θ/x)] + xk Γ(α; θ/x)
Γ(α)
θk Γ(α − k)
G(α − k; θ/x) + xk Γ(α; θ/x), all k
Γ(α)
= θ/(α + 1)
=
mode
A.3.2.3
Weibull–θ, τ
τ
τ
τ (x/θ)τ e−(x/θ)
f (x) =
F (x) = 1 − e−(x/θ)
x
E[X k ] = θk Γ(1 + k/τ ), k > −τ
VaRp (X) = θ[− ln(1 − p)]1/τ
τ
E[(X ∧ x)k ] = θk Γ(1 + k/τ )Γ[1 + k/τ ; (x/θ)τ ] + xk e−(x/θ) ,
µ
¶1/τ
τ −1
, τ > 1, else 0
mode = θ
τ
A.3.2.4
k > −τ
Inverse Weibull (log Gompertz)–θ, τ
τ
τ (θ/x)τ e−(θ/x)
x
E[X k ] = θk Γ(1 − k/τ ), k < τ
VaRp (X) = θ(− ln p)−1/τ
f (x) =
τ
F (x) = e−(θ/x)
h
i
τ
E[(X ∧ x)k ] = θk Γ(1 − k/τ ){1 − Γ[1 − k/τ ; (θ/x)τ ]} + xk 1 − e−(θ/x) ,
h
i
τ
= θk Γ(1 − k/τ )G[1 − k/τ ; (θ/x)τ ] + xk 1 − e−(θ/x)
µ
¶1/τ
τ
mode = θ
τ +1
all k
APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS
A.3.3
One-parameter distributions
A.3.3.1
Exponential–θ
f (x) =
M (t)
E[X k ]
VaRp (X)
TVaRp (X)
=
=
=
=
e−x/θ
F (x) = 1 − e−x/θ
θ
(1 − θt)−1
E[X k ] = θk Γ(k + 1),
k
θ k!, if k is an integer
−θ ln(1 − p)
−θ ln(1 − p) + θ
6
k > −1
E[X ∧ x] = θ(1 − e−x/θ )
E[(X ∧ x)k ] = θk Γ(k + 1)Γ(k + 1; x/θ) + xk e−x/θ , k > −1
= θk k!Γ(k + 1; x/θ) + xk e−x/θ , k an integer
mode = 0
A.3.3.2
Inverse exponential–θ
θe−θ/x
x2
k
k
E[X ] = θ Γ(1 − k),
VaRp (X) = θ(− ln p)−1
f (x) =
F (x) = e−θ/x
k<1
E[(X ∧ x)k ] = θk G(1 − k; θ/x) + xk (1 − e−θ/x ),
mode = θ/2
A.5
A.5.1.1
all k
Other distributions
Lognormal–μ,σ (μ can be negative)
1
ln x − μ
√ exp(−z 2 /2) = φ(z)/(σx), z =
σ
xσ 2π
k
2 2
E[X ] = exp(kμ + k σ /2)
µ
¶
ln x − μ − kσ 2
k
2 2
E[(X ∧ x) ] = exp(kμ + k σ /2)Φ
+ xk [1 − F (x)]
σ
f (x) =
mode = exp(μ − σ2 )
F (x) = Φ(z)
APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS
A.5.1.2
Inverse Gaussian–μ, θ
f (x) =
F (x) =
M (t) =
E[X ∧ x] =
A.5.1.3
7
µ
µ
¶1/2
¶
θ
θz 2
x−μ
exp −
, z=
2πx3
2x
μ
" µ ¶ #
µ ¶ " µ ¶1/2 #
1/2
2θ
x+μ
θ
θ
+ exp
, y=
Φ z
Φ −y
x
μ
x
μ
" Ã
!#
r
2tμ2
θ
θ
exp
E[X] = μ,
1− 1−
, t < 2,
μ
θ
2μ
" µ ¶ #
µ ¶ " µ ¶1/2 #
1/2
θ
2θ
θ
x − μzΦ z
− μy exp
Φ −y
x
μ
x
Var[X] = μ3 /θ
log-t–r, μ, σ (μ can be negative)
Let Y have a t distribution with r degrees of freedom. Then X = exp(σY + μ) has the log-t distribution.
Positive moments do not exist for this distribution. Just as the t distribution has a heavier tail than the
normal distribution, this distribution has a heavier tail than the lognormal distribution.
µ
¶
r+1
Γ
2
f (x) =
,
"
µ
¶2 #(r+1)/2
³r´
√
1 ln x − μ
xσ πrΓ
1+
2
r
σ
µ
¶
ln x − μ
F (x) = Fr
with Fr (t) the cdf of a t distribution with r d.f.,
σ
⎧
⎤
⎡
⎪
⎪
⎪
⎥
⎪
⎪
1 ⎢
r
⎥
⎢r 1
⎪
⎪
,
0 < x ≤ eμ ,
β
,
;
⎢
⎪
µ
¶
2⎥
⎪ 2 ⎣2 2
⎦
⎪
ln
x
−
μ
⎪
⎪
r+
⎨
σ
⎤
⎡
F (x) =
⎪
⎪
⎪
⎪
⎥
⎪
1 ⎢
r
⎪
⎥
⎢r 1
⎪
1
−
β
,
;
⎪
⎢
µ
¶2 ⎥ , x ≥ eμ .
⎪
⎪
⎦
2
⎣
2
2
ln
x
−
μ
⎪
⎪
r+
⎩
σ
A.5.1.4
Single-parameter Pareto–α, θ
f (x) =
αθα
,
xα+1
x>θ
VaRp (X) = θ(1 − p)−1/α
αθk
,
α−k
= θ
E[X k ] =
mode
k<α
F (x) = 1 − (θ/x)α ,
x>θ
αθ(1 − p)−1/α
, α>1
α−1
αθk
kθα
, x≥θ
E[(X ∧ x)k ] =
−
α − k (α − k)xα−k
TVaRp (X) =
Note: Although there appears to be two parameters, only α is a true parameter. The value of θ must be
set in advance.
APPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS
A.6
Distributions with finite support
For these two distributions, the scale parameter θ is assumed known.
A.6.1.1
Generalized beta–a, b, θ, τ
Γ(a + b) a
τ
u (1 − u)b−1 ,
Γ(a)Γ(b)
x
F (x) = β(a, b; u)
f (x) =
E[X k ] =
E[(X ∧ x)k ] =
A.6.1.2
θk Γ(a + b)Γ(a + k/τ )
,
Γ(a)Γ(a + b + k/τ )
0 < x < θ,
u = (x/θ)τ
k > −aτ
θk Γ(a + b)Γ(a + k/τ )
β(a + k/τ , b; u) + xk [1 − β(a, b; u)]
Γ(a)Γ(a + b + k/τ )
beta–a, b, θ
Γ(a + b) a
1
u (1 − u)b−1 ,
Γ(a)Γ(b)
x
F (x) = β(a, b; u)
f (x) =
0 < x < θ,
E[X k ] =
θk Γ(a + b)Γ(a + k)
,
Γ(a)Γ(a + b + k)
E[X k ] =
θk a(a + 1) · · · (a + k − 1)
,
(a + b)(a + b + 1) · · · (a + b + k − 1)
E[(X ∧ x)k ] =
u = x/θ
k > −a
if k is an integer
θk a(a + 1) · · · (a + k − 1)
β(a + k, b; u)
(a + b)(a + b + 1) · · · (a + b + k − 1)
+xk [1 − β(a, b; u)]
8