- Scale-inverse-chi-square distribution
Probability distribution
name =Scale-inverse-chi-square
type =density
pdf_
cdf_
parameters =u > 0,
sigma^2 > 0,
support =x in (0, infty)
pdf =frac{(sigma^2 u/2)^{ u/2{Gamma( u/2)}~frac{expleft [ frac{- u sigma^2}{2 x} ight] }{x^{1+ u/2
cdf =Gammaleft(frac{ u}{2},frac{sigma^2 u}{2x} ight)left/Gammaleft(frac{ u}{2} ight) ight.
mean =frac{ u sigma^2}{ u-2} for u >2,
median =
mode =frac{ u sigma^2}{ u+2}
variance =frac{2 u^2 sigma^4}{( u-2)^2 ( u-4)}for u >4,
skewness =frac{4}{ u-6}sqrt{2( u-4)}for u >6,
kurtosis =frac{12(5 u-22)}{( u-6)( u-8)}for u >8,
entropy =frac{ u}{2}!+!lnleft(frac{sigma^2 u}{2}Gammaleft(frac{ u}{2} ight) ight)left(1!+!frac{ u}{2} ight)psileft(frac{ u}{2} ight)
mgf =frac{2}{Gamma(frac{ u}{2})}left(frac{-sigma^2 u t}{2} ight)^{!!frac{ u}{4!!K_{frac{ u}{2left(sqrt{-2sigma^2 u t} ight)
char =frac{2}{Gamma(frac{ u}{2})}left(frac{-isigma^2 u t}{2} ight)^{!!frac{ u}{4!!K_{frac{ u}{2left(sqrt{-2isigma^2 u t} ight)
The scaled inverse chi-square distribution arises inBayesian statistics. It is a more general distribution than theinverse-chi-square distribution . Itsprobability density function over the domain x>0 is:f(x; u, sigma^2)=frac{(sigma^2 u/2)^{ u/2{Gamma( u/2)}~frac{expleft [ frac{- u sigma^2}{2 x} ight] }{x^{1+ u/2
where u is the degrees of freedom parameter and sigma^2 is the
scale parameter . The cumulative distribution function is:F(x; u, sigma^2)=Gammaleft(frac{ u}{2},frac{sigma^2 u}{2x} ight)left/Gammaleft(frac{ u}{2} ight) ight.:Qleft(frac{ u}{2},frac{sigma^2 u}{2x} ight)
where Gamma(a,x) is the
incomplete Gamma function , Gamma(x) is theGamma function and Q(a,x) is a regularized Gamma function. The characteristic function is:varphi(t; u,sigma^2)=:frac{2}{Gamma(frac{ u}{2})}left(frac{-isigma^2 u t}{2} ight)^{!!frac{ u}{4!!K_{frac{ u}{2left(sqrt{-2isigma^2 u t} ight)
where K_{frac{ u}{2(z) is the modified
Bessel function of the second kind.Parameter estimation
The
maximum likelihood estimate of sigma^2 is:sigma^2 = n/sum_{i=1}^N frac{1}{x_i}.
The
maximum likelihood estimate of frac{ u}{2} can be found usingNewton's method on::ln(frac{ u}{2}) + psi(frac{ u}{2}) = sum_{i=1}^N ln(x_i) - n ln(sigma^2)
where psi(x) is the
digamma function . An initial estimate can be found by taking the formula for mean and solving it for u. Let ar{x} = frac{1}{n}sum_{i=1}^N x_i be the sample mean. Then an initial estimate for u is given by::frac{ u}{2} = frac{ar{x{ar{x} - sigma^2}.
Related distributions
* Relation to
chi-square distribution : If X sim chi^2( u) and Y = frac{sigma^2 u}{X} then Y sim mbox{Scale-inv-}chi^2( u, sigma^2)
* Relation to theinverse gamma distribution : If X sim extrm{Inv-Gamma}left(frac{ u}{2}, frac{ usigma^2}{2} ight) then X sim mbox{Scale-inv-}chi^2( u, sigma^2).
* The scale-inverse-chi-square distribution is aconjugate prior for thevariance parameter of anormal distribution .ee also
*
Inverse chi-square distribution
*Chi-square distribution
*Bayesian probability
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