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Inverse-chi-squared distribution

{{short description|Probability distribution}}

{{Probability distribution|

name =Inverse-chi-squared|

type =density|

pdf_image =Image:Inverse chi squared density.png|

cdf_image =Image:Inverse chi squared distribution.png|

parameters =\nu > 0\!|

support =x \in (0, \infty)\!|

pdf =\frac{2^{-\nu/2}}{\Gamma(\nu/2)}\,x^{-\nu/2-1} e^{-1/(2 x)}\!|

cdf =\Gamma\!\left(\frac{\nu}{2},\frac{1}{2x}\right)

\bigg/\, \Gamma\!\left(\frac{\nu}{2}\right)\!|

mean =\frac{1}{\nu-2}\! for \nu >2\!|

median = \approx \dfrac{1}{\nu\bigg(1-\dfrac{2}{9\nu}\bigg)^3}|

mode =\frac{1}{\nu+2}\!|

variance =\frac{2}{(\nu-2)^2 (\nu-4)}\! for \nu >4\!|

skewness =\frac{4}{\nu-6}\sqrt{2(\nu-4)}\! for \nu >6\!|

kurtosis =\frac{12(5\nu-22)}{(\nu-6)(\nu-8)}\! for \nu >8\!|

entropy =\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)|

mgf =\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|

char =\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 distributionBernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory, Wiley (pages 119, 431) {{ISBN|0-471-49464-X}}) 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.{{cite book |first=Andrew |last=Gelman |first2=John B. |last2=Carlin |first3=Hal S. |last3=Stern |first4=David B. |last4=Dunson |first5=Aki |last5=Vehtari |first6=Donald B. |last6=Rubin |display-authors=1 |chapter=Normal data with a conjugate prior distribution |pages=67–68 |title=Bayesian Data Analysis |edition=Third |publisher=CRC Press |location=Boca Raton |year=2014 |isbn=978-1-4398-4095-5 }}

Definition

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

If X follows a chi-squared distribution with \nu degrees of freedom then 1/X follows the inverse chi-squared distribution with \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)}\,x^{-\nu/2-1} e^{-1/(2 x)}

In the above x>0 and \nu is the degrees of freedom parameter. Further, \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} and scale parameter \beta = \frac{1}{2}.

Related distributions

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

See also

References

{{reflist}}

External links

  • [https://web.archive.org/web/20091031132559/http://hosho.ees.hokudai.ac.jp/~kubo/Rdoc/library/geoR/html/InvChisquare.html InvChisquare] in geoR package for the R Language.

{{ProbDistributions|continuous-semi-infinite}}

Category:Continuous distributions

Category:Exponential family distributions

Category:Probability distributions with non-finite variance