Inverse-gamma distribution

From Wikipedia, the free encyclopedia

Inverse-gamma
Probability density function
Cumulative distribution function
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	${\frac {2\left(-\beta t\right)^{{\!\!{\frac {\alpha }{2}}}}}{\Gamma (\alpha )}}K_{{\alpha }}\left({\sqrt {-4\beta t}}\right)$ ; does not exist as real valued function
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 $\beta$ .

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 II kind.

Properties

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.

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 ${\tfrac {1}{X}}\sim {\mbox{Inv-Gamma}}(k,\theta ^{{-1}})\,$ (see derivation in the next paragraph for details)
If $X\sim {\mbox{Gamma}}(\alpha ,\beta ^{{-1}})\,$ (Gamma distribution) then ${\tfrac {1}{X}}\sim {\mbox{Inv-Gamma}}(\alpha ,\beta )\,$
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

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

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

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.

References

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

Probability distributions

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, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Triangular U-quadratic uniform Wigner semicircle Xenakis

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Burr
chi-squared
chi
Coxian
Dagum
Davis
EL
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
Maxwell–Boltzmann
Maxwell–Jüttner
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Poly-Weibull
Rayleigh
relativistic Breit–Wigner
Rice
Rosin–Rammler
shifted Gompertz
truncated normal
type-2 Gumbel
Weibull
Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson SU Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled 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 Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

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

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.