Inverse-gamma distribution

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Inverse-gamma
Probability density function
Inverse gamma pdf.png
Cumulative distribution function
Inverse gamma cdf.png
Parameters \alpha>0 shape (real)
\beta>0 scale (real)
Support x\in(0;\infty)\!
pdf \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)
CDF \frac{\Gamma(\alpha,\beta/x)}{\Gamma(\alpha)} \!
Mean \frac{\beta}{\alpha-1}\! for \alpha > 1
Mode \frac{\beta}{\alpha+1}\!
Variance \frac{\beta^2}{(\alpha-1)^2(\alpha-2)}\! for \alpha > 2
Skewness \frac{4\sqrt{\alpha-2}}{\alpha-3}\! for \alpha > 3
Ex. kurtosis \frac{30\,\alpha-66}{(\alpha-3)(\alpha-4)}\! for \alpha > 4
Entropy \alpha\!+\!\ln(\beta\Gamma(\alpha))\!-\!(1\!+\!\alpha)\Psi(\alpha)
MGF Does not exist.
CF \frac{2\left(-i\beta t\right)^{\!\!\frac{\alpha}{2}}}{\Gamma(\alpha)}K_{\alpha}\left(\sqrt{-4i\beta t}\right)

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution. Perhaps the chief use of the inverse gamma distribution is in Bayesian statistics, where the distribution arises as the marginal posterior distribution for the unknown variance of a normal distribution if an uninformative prior is used; and as an analytically tractable conjugate prior if an informative prior is required.

However, it is common among Bayesians to consider an alternative parametrization of the normal distribution in terms of the precision, defined as the reciprocal of the variance, which allows the gamma distribution to be used directly as a conjugate prior. Other Bayesians prefer to parametrize the inverse gamma distribution differently, as a scaled inverse chi-squared distribution.

Characterization[edit]

Probability density function[edit]

The inverse gamma distribution's probability density function is defined over the support x > 0


f(x; \alpha, \beta)
= \frac{\beta^\alpha}{\Gamma(\alpha)}
x^{-\alpha - 1}\exp\left(-\frac{\beta}{x}\right)

with shape parameter \alpha and rate parameter \beta.

Cumulative distribution function[edit]

The cumulative distribution function is the regularized gamma function

F(x; \alpha, \beta) = \frac{\Gamma\left(\alpha,\frac{\beta}{x}\right)}{\Gamma(\alpha)} = Q\left(\alpha, \frac{\beta}{x}\right)\!

where the numerator is the upper incomplete gamma function and the denominator is the gamma function. Many math packages allow you to compute Q, the regularized gamma function, directly.

Characteristic function[edit]

K_{\alpha}(\cdot) in the expression of the characteristic function is the modified Bessel function of II kind.

Properties[edit]

For \alpha>0 and \beta>0

\mathbb{E}[\ln(X)] = \ln(\beta) - \psi(\alpha).\,
\mathbb{E}[X^{-1}] = \frac{\alpha}{\beta}.\,

where \psi(\alpha) is the digamma function.


Differential equation


\left\{x^2 f'(x)+f(x) (-\beta +\alpha  x+x)=0,f(1)=\frac{e^{-\beta }
   \beta ^{\alpha }}{\Gamma (\alpha )}\right\}

Related distributions[edit]

Derivation from Gamma distribution[edit]

The pdf of the gamma distribution is

 f(x) = x^{k-1} \frac{e^{-x/\theta}}{\theta^k \, \Gamma(k)}

and define the transformation Y = g(X) = \frac{1}{X} then the resulting transformation is


f_Y(y) = f_X \left( g^{-1}(y) \right) \left| \frac{d}{dy} g^{-1}(y) \right|

=
\frac{1}{\theta^k \Gamma(k)}
\left(
 \frac{1}{y}
\right)^{k-1}
\exp
 \left(
  \frac{-1}{\theta y}
 \right)
\frac{1}{y^2}

=
\frac{1}{\theta^k \Gamma(k)}
\left(
 \frac{1}{y}
\right)^{k+1}
\exp
 \left(
  \frac{-1}{\theta y}
 \right)

=
\frac{1}{\theta^k \Gamma(k)}
y^{-k-1}
\exp
 \left(
  \frac{-1}{\theta y}
 \right).

Replacing k with \alpha; \theta^{-1} with \beta; and y with x results in the inverse-gamma pdf shown above


f(x)
=
\frac{\beta^\alpha}{\Gamma(\alpha)}
x^{-\alpha-1}
\exp
 \left(
  \frac{-\beta}{x}
 \right).

Occurrence[edit]

  • The first hitting time of a standard Wiener process has an inverse-gamma distribution with parameters \alpha = \frac{1}{2} and \beta = \frac{x^2}{2} where x is the value to hit.

See also[edit]

References[edit]

  • V. Witkovsky (2001) Computing the distribution of a linear combination of inverted gamma variables, Kybernetika 37(1), 79-90