- Generalized inverse Gaussian distribution
- Inverse Gaussian distribution
- Generalized normal distribution
- List of probability distributions
- Normal distribution
- Generalised hyperbolic distribution
- Normal-inverse Gaussian distribution
- Gamma distribution
- Student's t-distribution
- Chi-squared distribution
generalized inverse gaussian distribution
Generalized inverse Gaussian distribution GudangMovies21 Rebahinxxi LK21
In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function
f
(
x
)
=
(
a
/
b
)
p
/
2
2
K
p
(
a
b
)
x
(
p
−
1
)
e
−
(
a
x
+
b
/
x
)
/
2
,
x
>
0
,
{\displaystyle f(x)={\frac {(a/b)^{p/2}}{2K_{p}({\sqrt {ab}})}}x^{(p-1)}e^{-(ax+b/x)/2},\qquad x>0,}
where Kp is a modified Bessel function of the second kind, a > 0, b > 0 and p a real parameter. It is used extensively in geostatistics, statistical linguistics, finance, etc. This distribution was first proposed by Étienne Halphen.
It was rediscovered and popularised by Ole Barndorff-Nielsen, who called it the generalized inverse Gaussian distribution. Its statistical properties are discussed in Bent Jørgensen's lecture notes.
Properties
= Alternative parametrization
=By setting
θ
=
a
b
{\displaystyle \theta ={\sqrt {ab}}}
and
η
=
b
/
a
{\displaystyle \eta ={\sqrt {b/a}}}
, we can alternatively express the GIG distribution as
f
(
x
)
=
1
2
η
K
p
(
θ
)
(
x
η
)
p
−
1
e
−
θ
(
x
/
η
+
η
/
x
)
/
2
,
{\displaystyle f(x)={\frac {1}{2\eta K_{p}(\theta )}}\left({\frac {x}{\eta }}\right)^{p-1}e^{-\theta (x/\eta +\eta /x)/2},}
where
θ
{\displaystyle \theta }
is the concentration parameter while
η
{\displaystyle \eta }
is the scaling parameter.
= Summation
=Barndorff-Nielsen and Halgreen proved that the GIG distribution is infinitely divisible.
= Entropy
=The entropy of the generalized inverse Gaussian distribution is given as
H
=
1
2
log
(
b
a
)
+
log
(
2
K
p
(
a
b
)
)
−
(
p
−
1
)
[
d
d
ν
K
ν
(
a
b
)
]
ν
=
p
K
p
(
a
b
)
+
a
b
2
K
p
(
a
b
)
(
K
p
+
1
(
a
b
)
+
K
p
−
1
(
a
b
)
)
{\displaystyle {\begin{aligned}H={\frac {1}{2}}\log \left({\frac {b}{a}}\right)&{}+\log \left(2K_{p}\left({\sqrt {ab}}\right)\right)-(p-1){\frac {\left[{\frac {d}{d\nu }}K_{\nu }\left({\sqrt {ab}}\right)\right]_{\nu =p}}{K_{p}\left({\sqrt {ab}}\right)}}\\&{}+{\frac {\sqrt {ab}}{2K_{p}\left({\sqrt {ab}}\right)}}\left(K_{p+1}\left({\sqrt {ab}}\right)+K_{p-1}\left({\sqrt {ab}}\right)\right)\end{aligned}}}
where
[
d
d
ν
K
ν
(
a
b
)
]
ν
=
p
{\displaystyle \left[{\frac {d}{d\nu }}K_{\nu }\left({\sqrt {ab}}\right)\right]_{\nu =p}}
is a derivative of the modified Bessel function of the second kind with respect to the order
ν
{\displaystyle \nu }
evaluated at
ν
=
p
{\displaystyle \nu =p}
= Characteristic Function
=The characteristic of a random variable
X
∼
G
I
G
(
p
,
a
,
b
)
{\displaystyle X\sim GIG(p,a,b)}
is given as(for a derivation of the characteristic function, see supplementary materials of )
E
(
e
i
t
X
)
=
(
a
a
−
2
i
t
)
p
2
K
p
(
(
a
−
2
i
t
)
b
)
K
p
(
a
b
)
{\displaystyle E(e^{itX})=\left({\frac {a}{a-2it}}\right)^{\frac {p}{2}}{\frac {K_{p}\left({\sqrt {(a-2it)b}}\right)}{K_{p}\left({\sqrt {ab}}\right)}}}
for
t
∈
R
{\displaystyle t\in \mathbb {R} }
where
i
{\displaystyle i}
denotes the imaginary number.
Related distributions
= Special cases
=The inverse Gaussian and gamma distributions are special cases of the generalized inverse Gaussian distribution for p = −1/2 and b = 0, respectively. Specifically, an inverse Gaussian distribution of the form
f
(
x
;
μ
,
λ
)
=
[
λ
2
π
x
3
]
1
/
2
exp
(
−
λ
(
x
−
μ
)
2
2
μ
2
x
)
{\displaystyle f(x;\mu ,\lambda )=\left[{\frac {\lambda }{2\pi x^{3}}}\right]^{1/2}\exp {\left({\frac {-\lambda (x-\mu )^{2}}{2\mu ^{2}x}}\right)}}
is a GIG with
a
=
λ
/
μ
2
{\displaystyle a=\lambda /\mu ^{2}}
,
b
=
λ
{\displaystyle b=\lambda }
, and
p
=
−
1
/
2
{\displaystyle p=-1/2}
. A Gamma distribution of the form
g
(
x
;
α
,
β
)
=
β
α
1
Γ
(
α
)
x
α
−
1
e
−
β
x
{\displaystyle g(x;\alpha ,\beta )=\beta ^{\alpha }{\frac {1}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}}
is a GIG with
a
=
2
β
{\displaystyle a=2\beta }
,
b
=
0
{\displaystyle b=0}
, and
p
=
α
{\displaystyle p=\alpha }
.
Other special cases include the inverse-gamma distribution, for a = 0.
= Conjugate prior for Gaussian
=The GIG distribution is conjugate to the normal distribution when serving as the mixing distribution in a normal variance-mean mixture. Let the prior distribution for some hidden variable, say
z
{\displaystyle z}
, be GIG:
P
(
z
∣
a
,
b
,
p
)
=
GIG
(
z
∣
a
,
b
,
p
)
{\displaystyle P(z\mid a,b,p)=\operatorname {GIG} (z\mid a,b,p)}
and let there be
T
{\displaystyle T}
observed data points,
X
=
x
1
,
…
,
x
T
{\displaystyle X=x_{1},\ldots ,x_{T}}
, with normal likelihood function, conditioned on
z
:
{\displaystyle z:}
P
(
X
∣
z
,
α
,
β
)
=
∏
i
=
1
T
N
(
x
i
∣
α
+
β
z
,
z
)
{\displaystyle P(X\mid z,\alpha ,\beta )=\prod _{i=1}^{T}N(x_{i}\mid \alpha +\beta z,z)}
where
N
(
x
∣
μ
,
v
)
{\displaystyle N(x\mid \mu ,v)}
is the normal distribution, with mean
μ
{\displaystyle \mu }
and variance
v
{\displaystyle v}
. Then the posterior for
z
{\displaystyle z}
, given the data is also GIG:
P
(
z
∣
X
,
a
,
b
,
p
,
α
,
β
)
=
GIG
(
z
∣
a
+
T
β
2
,
b
+
S
,
p
−
T
2
)
{\displaystyle P(z\mid X,a,b,p,\alpha ,\beta )={\text{GIG}}\left(z\mid a+T\beta ^{2},b+S,p-{\frac {T}{2}}\right)}
where
S
=
∑
i
=
1
T
(
x
i
−
α
)
2
{\displaystyle \textstyle S=\sum _{i=1}^{T}(x_{i}-\alpha )^{2}}
.
= Sichel distribution
=The Sichel distribution results when the GIG is used as the mixing distribution for the Poisson parameter
λ
{\displaystyle \lambda }
.
Notes
References
See also
Inverse Gaussian distribution
Gamma distribution