Inverse-chi-squared distribution

Probability distribution

Inverse-chi-squared
Probability density function
Cumulative distribution function
Parameters	$\nu >0\!$ {\displaystyle \nu >0\!}
Support	$x\in (0,\infty )\!$ {\displaystyle x\in (0,\infty )\!}
PDF	${\frac {2^{-\nu /2}}{\Gamma (\nu /2)}},円x^{-\nu /2-1}e^{-1/(2x)}\!$ {\displaystyle {\frac {2^{-\nu /2}}{\Gamma (\nu /2)}},円x^{-\nu /2-1}e^{-1/(2x)}\!}
CDF	$\Gamma \!\left({\frac {\nu }{2}},{\frac {1}{2x}}\right){\bigg /},円\Gamma \!\left({\frac {\nu }{2}}\right)\!$ {\displaystyle \Gamma \!\left({\frac {\nu }{2}},{\frac {1}{2x}}\right){\bigg /},円\Gamma \!\left({\frac {\nu }{2}}\right)\!}
Mean	${\frac {1}{\nu -2}}\!$ {\displaystyle {\frac {1}{\nu -2}}\!} for $\nu >2\!$ {\displaystyle \nu >2\!}
Median	$\approx {\dfrac {1}{\nu {\bigg (}1-{\dfrac {2}{9\nu }}{\bigg )}^{3}}}$ {\displaystyle \approx {\dfrac {1}{\nu {\bigg (}1-{\dfrac {2}{9\nu }}{\bigg )}^{3}}}}
Mode	${\frac {1}{\nu +2}}\!$ {\displaystyle {\frac {1}{\nu +2}}\!}
Variance	${\frac {2}{(\nu -2)^{2}(\nu -4)}}\!$ {\displaystyle {\frac {2}{(\nu -2)^{2}(\nu -4)}}\!} for $\nu >4\!$ {\displaystyle \nu >4\!}
Skewness	${\frac {4}{\nu -6}}{\sqrt {2(\nu -4)}}\!$ {\displaystyle {\frac {4}{\nu -6}}{\sqrt {2(\nu -4)}}\!} for $\nu >6\!$ {\displaystyle \nu >6\!}
Excess kurtosis	${\frac {12(5\nu -22)}{(\nu -6)(\nu -8)}}\!$ {\displaystyle {\frac {12(5\nu -22)}{(\nu -6)(\nu -8)}}\!} for $\nu >8\!$ {\displaystyle \nu >8\!}
Entropy	${\frac {\nu }{2}}\!+\!\ln \!\left({\frac {\nu }{2}}\Gamma \!\left({\frac {\nu }{2}}\right)\right)$ {\displaystyle {\frac {\nu }{2}}\!+\!\ln \!\left({\frac {\nu }{2}}\Gamma \!\left({\frac {\nu }{2}}\right)\right)} $\!-\!\left(1\!+\!{\frac {\nu }{2}}\right)\psi \!\left({\frac {\nu }{2}}\right)$ {\displaystyle \!-\!\left(1\!+\!{\frac {\nu }{2}}\right)\psi \!\left({\frac {\nu }{2}}\right)}
MGF	${\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-t}{2i}}\right)^{\!\!{\frac {\nu }{4}}}K_{\frac {\nu }{2}}\!\left({\sqrt {-2t}}\right)$ {\displaystyle {\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-t}{2i}}\right)^{\!\!{\frac {\nu }{4}}}K_{\frac {\nu }{2}}\!\left({\sqrt {-2t}}\right)}; does not exist as real valued function
CF	${\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-it}{2}}\right)^{\!\!{\frac {\nu }{4}}}K_{\frac {\nu }{2}}\!\left({\sqrt {-2it}}\right)$ {\displaystyle {\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-it}{2}}\right)^{\!\!{\frac {\nu }{4}}}K_{\frac {\nu }{2}}\!\left({\sqrt {-2it}}\right)}

In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distribution^[1]) is a continuous probability distribution of a positive-valued random variable. It is closely related to the chi-squared distribution. It is used in Bayesian inference as conjugate prior for the variance of the normal distribution.^[2]

Definition

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The inverse chi-squared distribution (or inverted-chi-square distribution^[1] ) is the probability distribution of a random variable whose multiplicative inverse (reciprocal) has a chi-squared distribution.

If $X$ {\displaystyle X} follows a chi-squared distribution with $\nu$ {\displaystyle \nu } degrees of freedom then $1/X$ {\displaystyle 1/X} follows the inverse chi-squared distribution with $\nu$ {\displaystyle \nu } degrees of freedom.

The probability density function of the inverse chi-squared distribution is given by

f(x;\nu )={\frac {2^{-\nu /2}}{\Gamma (\nu /2)}},円x^{-\nu /2-1}e^{-1/(2x)}

{\displaystyle f(x;\nu )={\frac {2^{-\nu /2}}{\Gamma (\nu /2)}},円x^{-\nu /2-1}e^{-1/(2x)}}

In the above $x>0$ {\displaystyle x>0} and $\nu$ {\displaystyle \nu } is the degrees of freedom parameter. Further, $\Gamma$ {\displaystyle \Gamma } is the gamma function.

The inverse chi-squared distribution is a special case of the inverse-gamma distribution. with shape parameter $\alpha ={\frac {\nu }{2}}$ {\displaystyle \alpha ={\frac {\nu }{2}}} and scale parameter $\beta ={\frac {1}{2}}$ {\displaystyle \beta ={\frac {1}{2}}}.

Related distributions

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chi-squared: If $X\thicksim \chi ^{2}(\nu )$ {\displaystyle X\thicksim \chi ^{2}(\nu )} and $Y={\frac {1}{X}}$ {\displaystyle Y={\frac {1}{X}}}, then $Y\thicksim {\text{Inv-}}\chi ^{2}(\nu )$ {\displaystyle Y\thicksim {\text{Inv-}}\chi ^{2}(\nu )}
scaled-inverse chi-squared: If $X\thicksim {\text{Scale-inv-}}\chi ^{2}(\nu ,1/\nu ),円$ {\displaystyle X\thicksim {\text{Scale-inv-}}\chi ^{2}(\nu ,1/\nu ),円}, then $X\thicksim {\text{inv-}}\chi ^{2}(\nu )$ {\displaystyle X\thicksim {\text{inv-}}\chi ^{2}(\nu )}
Inverse gamma with $\alpha ={\frac {\nu }{2}}$ {\displaystyle \alpha ={\frac {\nu }{2}}} and $\beta ={\frac {1}{2}}$ {\displaystyle \beta ={\frac {1}{2}}}
Inverse chi-squared distribution is a special case of type 5 Pearson distribution

References

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^ ^a ^b Bernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory, Wiley (pages 119, 431) ISBN 0-471-49464-X
^ Gelman, Andrew; et al. (2014). "Normal data with a conjugate prior distribution". Bayesian Data Analysis (Third ed.). Boca Raton: CRC Press. pp. 67–68. ISBN 978-1-4398-4095-5.

External links

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InvChisquare in geoR package for the R Language.

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Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
with infinite support	Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta

Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling's T-squared Hartman–Watson Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda
supported on the whole real line	Cauchy Exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q Generalized normal Generalized hyperbolic Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson's S_U Landau Laplace Asymmetric Logistic Noncentral t Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student's t Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda