E6B: Difference between revisions

Revision as of 20:52, 21 January 2014

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 and its specific importance is that it arises in the application of Bayesian inference to the normal distribution, where it can be used as the prior and posterior distribution for an unknown variance.

Definition

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. It is also often defined as the distribution of a random variable whose reciprocal divided by its degrees of freedom is a chi-squared distribution. That is, if $X$ has the chi-squared distribution with $ν$ degrees of freedom, then according to the first definition, $1 / X$ has the inverse-chi-squared distribution with $ν$ degrees of freedom; while according to the second definition, $ν / X$ has the inverse-chi-squared distribution with $ν$ degrees of freedom. Only the first definition will usually be covered in this article.

The first definition yields a probability density function given by

f_{1} (x; ν) = \frac{2^{- ν / 2}}{Γ (ν / 2)} x^{- ν / 2 - 1} e^{- 1 / (2 x)},

while the second definition yields the density function

f_{2} (x; ν) = \frac{(ν / 2)^{ν / 2}}{Γ (ν / 2)} x^{- ν / 2 - 1} e^{- ν / (2 x)} .

In both cases, $x > 0$ and $ν$ is the degrees of freedom parameter. Further, $Γ$ is the gamma function. Both definitions are special cases of the scaled-inverse-chi-squared distribution. For the first definition the variance of the distribution is $σ^{2} = 1 / ν,$ while for the second definition $σ^{2} = 1$ .

Related distributions

chi-squared: If $X \sim χ^{2} (ν)$ and $Y = \frac{1}{X}$ , then $Y \sim Inv- χ^{2} (ν)$
scaled-inverse chi-squared: If $X \sim Scale-inv- χ^{2} (ν, 1 / ν)$ , then $X \sim inv- χ^{2} (ν)$
Inverse gamma with $α = \frac{ν}{2}$ and $β = \frac{1}{2}$

References

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External links

InvChisquare in geoR package for the R Language.

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↑ ^1.0 ^1.1 Bernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory ,Wiley (pages 119, 431) ISBN 0-471-49464-X

[BS-1] 1.0 ^1.1 Bernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory ,Wiley (pages 119, 431) ISBN 0-471-49464-X

[1]

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+{{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 =<math>\nu > 0\!</math>|
+  support    =<math>x \in (0, \infty)\!</math>|
+  pdf        =<math>\frac{2^{-\nu/2}}{\Gamma(\nu/2)}\,x^{-\nu/2-1}  e^{-1/(2 x)}\!</math>|
+  cdf        =<math>\Gamma\!\left(\frac{\nu}{2},\frac{1}{2x}\right)
+\bigg/\, \Gamma\!\left(\frac{\nu}{2}\right)\!</math>|
+  mean       =<math>\frac{1}{\nu-2}\!</math> for <math>\nu >2\!</math>|
+  median     =|
+  mode       =<math>\frac{1}{\nu+2}\!</math>|
+  variance   =<math>\frac{2}{(\nu-2)^2 (\nu-4)}\!</math> for <math>\nu >4\!</math>|
+  skewness   =<math>\frac{4}{\nu-6}\sqrt{2(\nu-4)}\!</math> for <math>\nu >6\!</math>|
+  kurtosis   =<math>\frac{12(5\nu-22)}{(\nu-6)(\nu-8)}\!</math> for <math>\nu >8\!</math>|
+  entropy    =<math>\frac{\nu}{2}
+\!+\!\ln\!\left(\frac{1}{2}\Gamma\!\left(\frac{\nu}{2}\right)\right)</math>
+<math>\!-\!\left(1\!+\!\frac{\nu}{2}\right)\psi\!\left(\frac{\nu}{2}\right)</math>|
+  mgf        =<math>\frac{2}{\Gamma(\frac{\nu}{2})}
+\left(\frac{-t}{2i}\right)^{\!\!\frac{\nu}{4}}
+K_{\frac{\nu}{2}}\!\left(\sqrt{-2t}\right)</math>; does not exist as [[real number|real valued]] function|
+  char       =<math>\frac{2}{\Gamma(\frac{\nu}{2})}
+\left(\frac{-it}{2}\right)^{\!\!\frac{\nu}{4}}
+K_{\frac{\nu}{2}}\!\left(\sqrt{-2it}\right)</math>|
+}}
+In [[probability and statistics]], the '''inverse-chi-squared distribution''' (or '''inverted-chi-square distribution'''<ref name=BS>Bernardo, J.M.; Smith, A.F.M. (1993) ''Bayesian Theory'' ,Wiley (pages 119, 431) ISBN 0-471-49464-X</ref>)  is a [[continuous probability distribution]] of a positive-valued random variable. It is closely related to the [[chi-squared distribution]] and its specific importance is that it arises in the application of [[Bayesian inference]] to the [[normal distribution]], where it can be used as the [[prior distribution|prior]] and [[posterior distribution]] for an unknown [[variance]].
+==Definition==
+The inverse-chi-squared distribution (or inverted-chi-square distribution<ref name=BS/> )  is the [[probability distribution]] of a random variable whose [[multiplicative inverse]] (reciprocal) has a [[chi-squared distribution]]. It is also often defined as the distribution of a random variable whose reciprocal divided by its degrees of freedom is a chi-squared distribution. That is, if <math>X</math> has the chi-squared distribution with <math>\nu</math> [[degrees of freedom (statistics)|degrees of freedom]], then according to the first definition, <math>1/X</math> has the inverse-chi-squared distribution with <math>\nu</math> degrees of freedom; while according to the second definition, <math>\nu/X</math> has the inverse-chi-squared distribution with <math>\nu</math> degrees of freedom. Only the first definition will usually be covered in this article.
+The first definition yields a [[probability density function]] given by
+:<math>
+f_1(x; \nu) = \frac{2^{-\nu/2}}{\Gamma(\nu/2)}\,x^{-\nu/2-1}  e^{-1/(2 x)},
+</math>
+while the second definition yields the density function
+:<math>
+f_2(x; \nu) =
+\frac{(\nu/2)^{\nu/2}}{\Gamma(\nu/2)}  x^{-\nu/2-1}  e^{-\nu/(2 x)} .
+</math>
+In both cases, <math>x>0</math> and <math>\nu</math> is the [[degrees of freedom (statistics)|degrees of freedom]] parameter. Further, <math>\Gamma</math> is the [[gamma function]]. Both definitions are special cases of the [[scaled-inverse-chi-squared distribution]]. For the first definition the variance of the distribution is <math>\sigma^2=1/\nu ,</math> while for the second definition <math>\sigma^2=1</math>.
+==Related distributions==
+*[[chi-squared distribution|chi-squared]]: If <math>X \thicksim \chi^2(\nu)</math> and <math>Y = \frac{1}{X}</math>, then <math>Y \thicksim \text{Inv-}\chi^2(\nu)</math>
+*[[scaled-inverse-chi-squared distribution|scaled-inverse chi-squared]]: If <math>X \thicksim \text{Scale-inv-}\chi^2(\nu, 1/\nu) \, </math>, then <math>X \thicksim \text{inv-}\chi^2(\nu)</math>
+*[[Inverse-gamma distribution|Inverse gamma]] with <math>\alpha = \frac{\nu}{2}</math> and <math>\beta = \frac{1}{2}</math>
+==See also==
+*[[Scaled-inverse-chi-squared distribution]]
+*[[Inverse-Wishart distribution]]
+==References==
+{{reflist}}
+==External links==
+* [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]]
+[[Category:Probability distributions]]

E6B: Difference between revisions

Revision as of 20:52, 21 January 2014

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Related distributions

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E6B: Difference between revisions

Revision as of 20:52, 21 January 2014

Definition

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References

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