Inverse-gamma distribution

Inverse-gamma
Probability density function
Cumulative distribution function
Parameters	$\alpha >0$ shape (real) $\beta >0$ rate (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	$\alpha \!+\!\ln(\beta \Gamma (\alpha ))\!-\!(1\!+\!\alpha )\Psi (\alpha )$ (see digamma function)
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

Probability density function

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 scale parameter $\beta$ .^[1] Here $\Gamma (\cdot )$ denotes the gamma function.

Unlike the Gamma distribution, which contains a somewhat similar exponential term, $\beta$ is a scale parameter as the distribution function satisfies:

f(x;\alpha ,\beta )={\frac {f({\frac {x}{\beta }};\alpha ,1)}{\beta }}

Cumulative distribution function

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

$K_{\alpha }(\cdot )$ in the expression of the characteristic function is the modified Bessel function of the 2nd kind.

Properties

For $\alpha >0$ and $\beta >0$ ,

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

and

\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

If $X\sim {\mbox{Inv-Gamma}}(\alpha ,\beta )$ then $kX\sim {\mbox{Inv-Gamma}}(\alpha ,k\beta )\,$
If $X\sim {\mbox{Inv-Gamma}}(\alpha ,{\tfrac {1}{2}})$ then $X\sim {\mbox{Inv-}}\chi ^{2}(2\alpha )\,$ (inverse-chi-squared distribution)
If $X\sim {\mbox{Inv-Gamma}}({\tfrac {\alpha }{2}},{\tfrac {1}{2}})$ then $X\sim {\mbox{Scaled Inv-}}\chi ^{2}(\alpha ,{\tfrac {1}{\alpha }})\,$ (scaled-inverse-chi-squared distribution)
If $X\sim {\textrm {Inv-Gamma}}({\tfrac {1}{2}},{\tfrac {c}{2}})$ then $X\sim {\textrm {Levy}}(0,c)\,$ (Lévy distribution)
If $X\sim {\mbox{Gamma}}(k,\theta )\,$ (Gamma distribution) then ${\tfrac {1}{X}}\sim {\mbox{Inv-Gamma}}(k,\theta )\,$ (see derivation in the next paragraph for details)
Inverse gamma distribution is a special case of type 5 Pearson distribution
A multivariate generalization of the inverse-gamma distribution is the inverse-Wishart distribution.
For the distribution of a sum of independent inverted Gamma variables see Witkovsky (2001)

Derivation from Gamma distribution

The pdf of the gamma distribution is

f(x)={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{\alpha -1}e^{-\beta x}

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

References

↑ http://reference.wolfram.com/language/ref/InverseGammaDistribution.html

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

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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