Inverse-chi-squared distribution
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Probability density function  | |||
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Cumulative distribution function  | |||
| Parameters | [math]\displaystyle{ \nu \gt 0\! }[/math] | ||
|---|---|---|---|
| Support | [math]\displaystyle{ x \in (0, \infty)\! }[/math] | ||
| [math]\displaystyle{ \frac{2^{-\nu/2}}{\Gamma(\nu/2)}\,x^{-\nu/2-1} e^{-1/(2 x)}\! }[/math] | |||
| CDF | [math]\displaystyle{ \Gamma\!\left(\frac{\nu}{2},\frac{1}{2x}\right) \bigg/\, \Gamma\!\left(\frac{\nu}{2}\right)\! }[/math] | ||
| Mean | [math]\displaystyle{ \frac{1}{\nu-2}\! }[/math] for [math]\displaystyle{ \nu \gt 2\! }[/math] | ||
| Median | [math]\displaystyle{ \approx \dfrac{1}{\nu\bigg(1-\dfrac{2}{9\nu}\bigg)^3} }[/math] | ||
| Mode | [math]\displaystyle{ \frac{1}{\nu+2}\! }[/math] | ||
| Variance | [math]\displaystyle{ \frac{2}{(\nu-2)^2 (\nu-4)}\! }[/math] for [math]\displaystyle{ \nu \gt 4\! }[/math] | ||
| Skewness | [math]\displaystyle{ \frac{4}{\nu-6}\sqrt{2(\nu-4)}\! }[/math] for [math]\displaystyle{ \nu \gt 6\! }[/math] | ||
| Kurtosis | [math]\displaystyle{ \frac{12(5\nu-22)}{(\nu-6)(\nu-8)}\! }[/math] for [math]\displaystyle{ \nu \gt 8\! }[/math] | ||
| Entropy |
[math]\displaystyle{ \frac{\nu}{2} \!+\!\ln\!\left(\frac{\nu}{2}\Gamma\!\left(\frac{\nu}{2}\right)\right) }[/math] [math]\displaystyle{ \!-\!\left(1\!+\!\frac{\nu}{2}\right)\psi\!\left(\frac{\nu}{2}\right) }[/math] | ||
| MGF | [math]\displaystyle{ \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 valued function | ||
| CF | [math]\displaystyle{ \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[1]) is a continuous probability distribution of a positive-valued random variable. It is closely related to the chi-squared distribution. It arises in Bayesian inference, where it can be used as the prior and posterior distribution for an unknown variance of the normal distribution.
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 [math]\displaystyle{ X }[/math] has the chi-squared distribution with [math]\displaystyle{ \nu }[/math] degrees of freedom, then according to the first definition, [math]\displaystyle{ 1/X }[/math] has the inverse-chi-squared distribution with [math]\displaystyle{ \nu }[/math] degrees of freedom; while according to the second definition, [math]\displaystyle{ \nu/X }[/math] has the inverse-chi-squared distribution with [math]\displaystyle{ \nu }[/math] degrees of freedom. Information associated with the first definition is depicted on the right side of the page.
The first definition yields a probability density function given by
- [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ x\gt 0 }[/math] and [math]\displaystyle{ \nu }[/math] is the degrees of freedom parameter. Further, [math]\displaystyle{ \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]\displaystyle{ \sigma^2=1/\nu , }[/math] while for the second definition [math]\displaystyle{ \sigma^2=1 }[/math].
Related distributions
- chi-squared: If [math]\displaystyle{ X \thicksim \chi^2(\nu) }[/math] and [math]\displaystyle{ Y = \frac{1}{X} }[/math], then [math]\displaystyle{ Y \thicksim \text{Inv-}\chi^2(\nu) }[/math]
- scaled-inverse chi-squared: If [math]\displaystyle{ X \thicksim \text{Scale-inv-}\chi^2(\nu, 1/\nu) \, }[/math], then [math]\displaystyle{ X \thicksim \text{inv-}\chi^2(\nu) }[/math]
- Inverse gamma with [math]\displaystyle{ \alpha = \frac{\nu}{2} }[/math] and [math]\displaystyle{ \beta = \frac{1}{2} }[/math]
- Inverse chi-squared distribution is a special case of type 5 Pearson distribution
See also
- Scaled-inverse-chi-squared distribution
- Inverse-Wishart distribution
References
External links
- InvChisquare in geoR package for the R Language.
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