Probability distribution
Scaled inverse chi-squared
Probability density function
Cumulative distribution function
Parameters
ν
>
0
{\displaystyle \nu >0\,}
τ
2
>
0
{\displaystyle \tau ^{2}>0\,}
Support
x
∈
(
0
,
∞
)
{\displaystyle x\in (0,\infty )}
PDF
(
τ
2
ν
/
2
)
ν
/
2
Γ
(
ν
/
2
)
exp
[
−
ν
τ
2
2
x
]
x
1
+
ν
/
2
{\displaystyle {\frac {(\tau ^{2}\nu /2)^{\nu /2}}{\Gamma (\nu /2)}}~{\frac {\exp \left[{\frac {-\nu \tau ^{2}}{2x}}\right]}{x^{1+\nu /2}}}}
CDF
Γ
(
ν
2
,
τ
2
ν
2
x
)
/
Γ
(
ν
2
)
{\displaystyle \Gamma \left({\frac {\nu }{2}},{\frac {\tau ^{2}\nu }{2x}}\right)\left/\Gamma \left({\frac {\nu }{2}}\right)\right.}
Mean
ν
τ
2
ν
−
2
{\displaystyle {\frac {\nu \tau ^{2}}{\nu -2}}}
for
ν
>
2
{\displaystyle \nu >2\,}
Mode
ν
τ
2
ν
+
2
{\displaystyle {\frac {\nu \tau ^{2}}{\nu +2}}}
Variance
2
ν
2
τ
4
(
ν
−
2
)
2
(
ν
−
4
)
{\displaystyle {\frac {2\nu ^{2}\tau ^{4}}{(\nu -2)^{2}(\nu -4)}}}
for
ν
>
4
{\displaystyle \nu >4\,}
Skewness
4
ν
−
6
2
(
ν
−
4
)
{\displaystyle {\frac {4}{\nu -6}}{\sqrt {2(\nu -4)}}}
for
ν
>
6
{\displaystyle \nu >6\,}
Excess kurtosis
12
(
5
ν
−
22
)
(
ν
−
6
)
(
ν
−
8
)
{\displaystyle {\frac {12(5\nu -22)}{(\nu -6)(\nu -8)}}}
for
ν
>
8
{\displaystyle \nu >8\,}
Entropy
ν
2
+
ln
(
τ
2
ν
2
Γ
(
ν
2
)
)
{\displaystyle {\frac {\nu }{2}}\!+\!\ln \left({\frac {\tau ^{2}\nu }{2}}\Gamma \left({\frac {\nu }{2}}\right)\right)}
−
(
1
+
ν
2
)
ψ
(
ν
2
)
{\displaystyle \!-\!\left(1\!+\!{\frac {\nu }{2}}\right)\psi \left({\frac {\nu }{2}}\right)}
MGF
2
Γ
(
ν
2
)
(
−
τ
2
ν
t
2
)
ν
4
K
ν
2
(
−
2
τ
2
ν
t
)
{\displaystyle {\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-\tau ^{2}\nu t}{2}}\right)^{\!\!{\frac {\nu }{4}}}\!\!K_{\frac {\nu }{2}}\left({\sqrt {-2\tau ^{2}\nu t}}\right)}
CF
2
Γ
(
ν
2
)
(
−
i
τ
2
ν
t
2
)
ν
4
K
ν
2
(
−
2
i
τ
2
ν
t
)
{\displaystyle {\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-i\tau ^{2}\nu t}{2}}\right)^{\!\!{\frac {\nu }{4}}}\!\!K_{\frac {\nu }{2}}\left({\sqrt {-2i\tau ^{2}\nu t}}\right)}
The scaled inverse chi-squared distribution
ψ
inv-
χ
2
(
ν
)
{\displaystyle \psi \,{\mbox{inv-}}\chi ^{2}(\nu )}
, where
ψ
{\displaystyle \psi }
is the scale parameter, equals the univariate inverse Wishart distribution
W
−
1
(
ψ
,
ν
)
{\displaystyle {\mathcal {W}}^{-1}(\psi ,\nu )}
with degrees of freedom
ν
{\displaystyle \nu }
.
This family of scaled inverse chi-squared distributions is linked to the inverse-chi-squared distribution and to the chi-squared distribution :
If
X
∼
ψ
inv-
χ
2
(
ν
)
{\displaystyle X\sim \psi \,{\mbox{inv-}}\chi ^{2}(\nu )}
then
X
/
ψ
∼
inv-
χ
2
(
ν
)
{\displaystyle X/\psi \sim {\mbox{inv-}}\chi ^{2}(\nu )}
as well as
ψ
/
X
∼
χ
2
(
ν
)
{\displaystyle \psi /X\sim \chi ^{2}(\nu )}
and
1
/
X
∼
ψ
−
1
χ
2
(
ν
)
{\displaystyle 1/X\sim \psi ^{-1}\chi ^{2}(\nu )}
.
Instead of
ψ
{\displaystyle \psi }
, the scaled inverse chi-squared distribution is however most frequently
parametrized by the scale parameter
τ
2
=
ψ
/
ν
{\displaystyle \tau ^{2}=\psi /\nu }
and the distribution
ν
τ
2
inv-
χ
2
(
ν
)
{\displaystyle \nu \tau ^{2}\,{\mbox{inv-}}\chi ^{2}(\nu )}
is denoted by
Scale-inv-
χ
2
(
ν
,
τ
2
)
{\displaystyle {\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})}
.
In terms of
τ
2
{\displaystyle \tau ^{2}}
the above relations can be written as follows:
If
X
∼
Scale-inv-
χ
2
(
ν
,
τ
2
)
{\displaystyle X\sim {\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})}
then
X
ν
τ
2
∼
inv-
χ
2
(
ν
)
{\displaystyle {\frac {X}{\nu \tau ^{2}}}\sim {\mbox{inv-}}\chi ^{2}(\nu )}
as well as
ν
τ
2
X
∼
χ
2
(
ν
)
{\displaystyle {\frac {\nu \tau ^{2}}{X}}\sim \chi ^{2}(\nu )}
and
1
/
X
∼
1
ν
τ
2
χ
2
(
ν
)
{\displaystyle 1/X\sim {\frac {1}{\nu \tau ^{2}}}\chi ^{2}(\nu )}
.
This family of scaled inverse chi-squared distributions is a reparametrization of the inverse-gamma distribution .
Specifically, if
X
∼
ψ
inv-
χ
2
(
ν
)
=
Scale-inv-
χ
2
(
ν
,
τ
2
)
{\displaystyle X\sim \psi \,{\mbox{inv-}}\chi ^{2}(\nu )={\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})}
then
X
∼
Inv-Gamma
(
ν
2
,
ψ
2
)
=
Inv-Gamma
(
ν
2
,
ν
τ
2
2
)
{\displaystyle X\sim {\textrm {Inv-Gamma}}\left({\frac {\nu }{2}},{\frac {\psi }{2}}\right)={\textrm {Inv-Gamma}}\left({\frac {\nu }{2}},{\frac {\nu \tau ^{2}}{2}}\right)}
Either form may be used to represent the maximum entropy distribution for a fixed first inverse moment
(
E
(
1
/
X
)
)
{\displaystyle (E(1/X))}
and first logarithmic moment
(
E
(
ln
(
X
)
)
{\displaystyle (E(\ln(X))}
.
The scaled inverse chi-squared distribution also has a particular use in Bayesian statistics . Specifically, the scaled inverse chi-squared distribution can be used as a conjugate prior for the variance parameter of a normal distribution .
The same prior in alternative parametrization is given by
the inverse-gamma distribution .
The probability density function of the scaled inverse chi-squared distribution extends over the domain
x
>
0
{\displaystyle x>0}
and is
f
(
x
;
ν
,
τ
2
)
=
(
τ
2
ν
/
2
)
ν
/
2
Γ
(
ν
/
2
)
exp
[
−
ν
τ
2
2
x
]
x
1
+
ν
/
2
{\displaystyle f(x;\nu ,\tau ^{2})={\frac {(\tau ^{2}\nu /2)^{\nu /2}}{\Gamma (\nu /2)}}~{\frac {\exp \left[{\frac {-\nu \tau ^{2}}{2x}}\right]}{x^{1+\nu /2}}}}
where
ν
{\displaystyle \nu }
is the degrees of freedom parameter and
τ
2
{\displaystyle \tau ^{2}}
is the scale parameter . The cumulative distribution function is
F
(
x
;
ν
,
τ
2
)
=
Γ
(
ν
2
,
τ
2
ν
2
x
)
/
Γ
(
ν
2
)
{\displaystyle F(x;\nu ,\tau ^{2})=\Gamma \left({\frac {\nu }{2}},{\frac {\tau ^{2}\nu }{2x}}\right)\left/\Gamma \left({\frac {\nu }{2}}\right)\right.}
=
Q
(
ν
2
,
τ
2
ν
2
x
)
{\displaystyle =Q\left({\frac {\nu }{2}},{\frac {\tau ^{2}\nu }{2x}}\right)}
where
Γ
(
a
,
x
)
{\displaystyle \Gamma (a,x)}
is the incomplete gamma function ,
Γ
(
x
)
{\displaystyle \Gamma (x)}
is the gamma function and
Q
(
a
,
x
)
{\displaystyle Q(a,x)}
is a regularized gamma function . The characteristic function is
φ
(
t
;
ν
,
τ
2
)
=
{\displaystyle \varphi (t;\nu ,\tau ^{2})=}
2
Γ
(
ν
2
)
(
−
i
τ
2
ν
t
2
)
ν
4
K
ν
2
(
−
2
i
τ
2
ν
t
)
,
{\displaystyle {\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-i\tau ^{2}\nu t}{2}}\right)^{\!\!{\frac {\nu }{4}}}\!\!K_{\frac {\nu }{2}}\left({\sqrt {-2i\tau ^{2}\nu t}}\right),}
where
K
ν
2
(
z
)
{\displaystyle K_{\frac {\nu }{2}}(z)}
is the modified Bessel function of the second kind .
Parameter estimation [ edit ]
The maximum likelihood estimate of
τ
2
{\displaystyle \tau ^{2}}
is
τ
2
=
n
/
∑
i
=
1
n
1
x
i
.
{\displaystyle \tau ^{2}=n/\sum _{i=1}^{n}{\frac {1}{x_{i}}}.}
The maximum likelihood estimate of
ν
2
{\displaystyle {\frac {\nu }{2}}}
can be found using Newton's method on:
ln
(
ν
2
)
−
ψ
(
ν
2
)
=
1
n
∑
i
=
1
n
ln
(
x
i
)
−
ln
(
τ
2
)
,
{\displaystyle \ln \left({\frac {\nu }{2}}\right)-\psi \left({\frac {\nu }{2}}\right)={\frac {1}{n}}\sum _{i=1}^{n}\ln \left(x_{i}\right)-\ln \left(\tau ^{2}\right),}
where
ψ
(
x
)
{\displaystyle \psi (x)}
is the digamma function . An initial estimate can be found by taking the formula for mean and solving it for
ν
.
{\displaystyle \nu .}
Let
x
¯
=
1
n
∑
i
=
1
n
x
i
{\displaystyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}
be the sample mean. Then an initial estimate for
ν
{\displaystyle \nu }
is given by:
ν
2
=
x
¯
x
¯
−
τ
2
.
{\displaystyle {\frac {\nu }{2}}={\frac {\bar {x}}{{\bar {x}}-\tau ^{2}}}.}
Bayesian estimation of the variance of a normal distribution [ edit ]
The scaled inverse chi-squared distribution has a second important application, in the Bayesian estimation of the variance of a Normal distribution.
According to Bayes' theorem , the posterior probability distribution for quantities of interest is proportional to the product of a prior distribution for the quantities and a likelihood function :
p
(
σ
2
|
D
,
I
)
∝
p
(
σ
2
|
I
)
p
(
D
|
σ
2
)
{\displaystyle p(\sigma ^{2}|D,I)\propto p(\sigma ^{2}|I)\;p(D|\sigma ^{2})}
where D represents the data and I represents any initial information about σ2 that we may already have.
The simplest scenario arises if the mean μ is already known; or, alternatively, if it is the conditional distribution of σ2 that is sought, for a particular assumed value of μ.
Then the likelihood term L (σ2 |D ) = p (D |σ2 ) has the familiar form
L
(
σ
2
|
D
,
μ
)
=
1
(
2
π
σ
)
n
exp
[
−
∑
i
n
(
x
i
−
μ
)
2
2
σ
2
]
{\displaystyle {\mathcal {L}}(\sigma ^{2}|D,\mu )={\frac {1}{\left({\sqrt {2\pi }}\sigma \right)^{n}}}\;\exp \left[-{\frac {\sum _{i}^{n}(x_{i}-\mu )^{2}}{2\sigma ^{2}}}\right]}
Combining this with the rescaling-invariant prior p(σ2 |I ) = 1/σ2 , which can be argued (e.g. following Jeffreys ) to be the least informative possible prior for σ2 in this problem, gives a combined posterior probability
p
(
σ
2
|
D
,
I
,
μ
)
∝
1
σ
n
+
2
exp
[
−
∑
i
n
(
x
i
−
μ
)
2
2
σ
2
]
{\displaystyle p(\sigma ^{2}|D,I,\mu )\propto {\frac {1}{\sigma ^{n+2}}}\;\exp \left[-{\frac {\sum _{i}^{n}(x_{i}-\mu )^{2}}{2\sigma ^{2}}}\right]}
This form can be recognised as that of a scaled inverse chi-squared distribution, with parameters ν = n and τ2 = s 2 = (1/n ) Σ (xi -μ)2
Gelman and co-authors remark that the re-appearance of this distribution, previously seen in a sampling context, may seem remarkable; but given the choice of prior "this result is not surprising."[ 1]
In particular, the choice of a rescaling-invariant prior for σ2 has the result that the probability for the ratio of σ2 / s 2 has the same form (independent of the conditioning variable) when conditioned on s 2 as when conditioned on σ2 :
p
(
σ
2
s
2
|
s
2
)
=
p
(
σ
2
s
2
|
σ
2
)
{\displaystyle p({\tfrac {\sigma ^{2}}{s^{2}}}|s^{2})=p({\tfrac {\sigma ^{2}}{s^{2}}}|\sigma ^{2})}
In the sampling-theory case, conditioned on σ2 , the probability distribution for (1/s2 ) is a scaled inverse chi-squared distribution; and so the probability distribution for σ2 conditioned on s 2 , given a scale-agnostic prior, is also a scaled inverse chi-squared distribution.
If more is known about the possible values of σ2 , a distribution from the scaled inverse chi-squared family, such as Scale-inv-χ2 (n 0 , s 0 2 ) can be a convenient form to represent a more informative prior for σ2 , as if from the result of n 0 previous observations (though n 0 need not necessarily be a whole number):
p
(
σ
2
|
I
′
,
μ
)
∝
1
σ
n
0
+
2
exp
[
−
n
0
s
0
2
2
σ
2
]
{\displaystyle p(\sigma ^{2}|I^{\prime },\mu )\propto {\frac {1}{\sigma ^{n_{0}+2}}}\;\exp \left[-{\frac {n_{0}s_{0}^{2}}{2\sigma ^{2}}}\right]}
Such a prior would lead to the posterior distribution
p
(
σ
2
|
D
,
I
′
,
μ
)
∝
1
σ
n
+
n
0
+
2
exp
[
−
n
s
2
+
n
0
s
0
2
2
σ
2
]
{\displaystyle p(\sigma ^{2}|D,I^{\prime },\mu )\propto {\frac {1}{\sigma ^{n+n_{0}+2}}}\;\exp \left[-{\frac {ns^{2}+n_{0}s_{0}^{2}}{2\sigma ^{2}}}\right]}
which is itself a scaled inverse chi-squared distribution. The scaled inverse chi-squared distributions are thus a convenient conjugate prior family for σ2 estimation.
Estimation of variance when mean is unknown [ edit ]
If the mean is not known, the most uninformative prior that can be taken for it is arguably the translation-invariant prior p (μ|I ) ∝ const., which gives the following joint posterior distribution for μ and σ2 ,
p
(
μ
,
σ
2
∣
D
,
I
)
∝
1
σ
n
+
2
exp
[
−
∑
i
n
(
x
i
−
μ
)
2
2
σ
2
]
=
1
σ
n
+
2
exp
[
−
∑
i
n
(
x
i
−
x
¯
)
2
2
σ
2
]
exp
[
−
n
(
μ
−
x
¯
)
2
2
σ
2
]
{\displaystyle {\begin{aligned}p(\mu ,\sigma ^{2}\mid D,I)&\propto {\frac {1}{\sigma ^{n+2}}}\exp \left[-{\frac {\sum _{i}^{n}(x_{i}-\mu )^{2}}{2\sigma ^{2}}}\right]\\&={\frac {1}{\sigma ^{n+2}}}\exp \left[-{\frac {\sum _{i}^{n}(x_{i}-{\bar {x}})^{2}}{2\sigma ^{2}}}\right]\exp \left[-{\frac {n(\mu -{\bar {x}})^{2}}{2\sigma ^{2}}}\right]\end{aligned}}}
The marginal posterior distribution for σ2 is obtained from the joint posterior distribution by integrating out over μ,
p
(
σ
2
|
D
,
I
)
∝
1
σ
n
+
2
exp
[
−
∑
i
n
(
x
i
−
x
¯
)
2
2
σ
2
]
∫
−
∞
∞
exp
[
−
n
(
μ
−
x
¯
)
2
2
σ
2
]
d
μ
=
1
σ
n
+
2
exp
[
−
∑
i
n
(
x
i
−
x
¯
)
2
2
σ
2
]
2
π
σ
2
/
n
∝
(
σ
2
)
−
(
n
+
1
)
/
2
exp
[
−
(
n
−
1
)
s
2
2
σ
2
]
{\displaystyle {\begin{aligned}p(\sigma ^{2}|D,I)\;\propto \;&{\frac {1}{\sigma ^{n+2}}}\;\exp \left[-{\frac {\sum _{i}^{n}(x_{i}-{\bar {x}})^{2}}{2\sigma ^{2}}}\right]\;\int _{-\infty }^{\infty }\exp \left[-{\frac {n(\mu -{\bar {x}})^{2}}{2\sigma ^{2}}}\right]d\mu \\=\;&{\frac {1}{\sigma ^{n+2}}}\;\exp \left[-{\frac {\sum _{i}^{n}(x_{i}-{\bar {x}})^{2}}{2\sigma ^{2}}}\right]\;{\sqrt {2\pi \sigma ^{2}/n}}\\\propto \;&(\sigma ^{2})^{-(n+1)/2}\;\exp \left[-{\frac {(n-1)s^{2}}{2\sigma ^{2}}}\right]\end{aligned}}}
This is again a scaled inverse chi-squared distribution, with parameters
n
−
1
{\displaystyle \scriptstyle {n-1}\;}
and
s
2
=
∑
(
x
i
−
x
¯
)
2
/
(
n
−
1
)
{\displaystyle \scriptstyle {s^{2}=\sum (x_{i}-{\bar {x}})^{2}/(n-1)}}
.
If
X
∼
Scale-inv-
χ
2
(
ν
,
τ
2
)
{\displaystyle X\sim {\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})}
then
k
X
∼
Scale-inv-
χ
2
(
ν
,
k
τ
2
)
{\displaystyle kX\sim {\mbox{Scale-inv-}}\chi ^{2}(\nu ,k\tau ^{2})\,}
If
X
∼
inv-
χ
2
(
ν
)
{\displaystyle X\sim {\mbox{inv-}}\chi ^{2}(\nu )\,}
(Inverse-chi-squared distribution ) then
X
∼
Scale-inv-
χ
2
(
ν
,
1
/
ν
)
{\displaystyle X\sim {\mbox{Scale-inv-}}\chi ^{2}(\nu ,1/\nu )\,}
If
X
∼
Scale-inv-
χ
2
(
ν
,
τ
2
)
{\displaystyle X\sim {\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})}
then
X
τ
2
ν
∼
inv-
χ
2
(
ν
)
{\displaystyle {\frac {X}{\tau ^{2}\nu }}\sim {\mbox{inv-}}\chi ^{2}(\nu )\,}
(Inverse-chi-squared distribution )
If
X
∼
Scale-inv-
χ
2
(
ν
,
τ
2
)
{\displaystyle X\sim {\mbox{Scale-inv-}}\chi ^{2}(\nu ,\tau ^{2})}
then
X
∼
Inv-Gamma
(
ν
2
,
ν
τ
2
2
)
{\displaystyle X\sim {\textrm {Inv-Gamma}}\left({\frac {\nu }{2}},{\frac {\nu \tau ^{2}}{2}}\right)}
(Inverse-gamma distribution )
Scaled inverse chi square distribution is a special case of type 5 Pearson distribution
Gelman, Andrew; et al. (2014). Bayesian Data Analysis (Third ed.). Boca Raton: CRC Press. p. 583. ISBN 978-1-4398-4095-5 .
^ Gelman, Andrew; et al. (2014). Bayesian Data Analysis (Third ed.). Boca Raton: CRC Press. p. 65. ISBN 978-1-4398-4095-5 .
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint) Directional Degenerate and singular Families