The Inverse Gamma distribution

The Inverse Gamma distribution is commonly used as a prior distribution in Bayesian statistics, particularly for variance parameters.

dist_inverse_gamma(shape, rate = 1/scale, scale)

Arguments

shape, scale: parameters. Must be strictly positive.
rate: an alternative way to specify the scale.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_inverse_gamma.html

In the following, let $X$ be an Inverse Gamma random variable with shape parameter shape = $\alpha$ and rate parameter rate = $\beta$ (equivalently, scale = $1/\beta$).

Support: $x \in (0, \infty)$

Mean: $\frac{\beta}{\alpha - 1}$ for $\alpha > 1$, otherwise undefined

Variance: $\frac{\beta^2}{(\alpha - 1)^2 (\alpha - 2)}$ for $\alpha > 2$, otherwise undefined

Probability density function (p.d.f):

$$ f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} e^{-\beta/x} $$

Cumulative distribution function (c.d.f):

$$ F(x) = \frac{\Gamma(\alpha, \beta/x)}{\Gamma(\alpha)} = Q(\alpha, \beta/x) $$

where $\Gamma(\alpha, z)$ is the upper incomplete gamma function and $Q$ is the regularized incomplete gamma function.

Moment generating function (m.g.f):

$$ M_X(t) = \frac{2 (-\beta t)^{\alpha/2}}{\Gamma(\alpha)} K_\alpha\left(\sqrt{-4\beta t}\right) $$

for $t < 0$, where $K_\alpha$ is the modified Bessel function of the second kind. The MGF does not exist for $t \ge 0$.

Examples

dist <- dist_inverse_gamma(shape = c(1,2,3,3), rate = c(1,1,1,2))
dist
#> <distribution[4]>
#> [1] InvGamma(1, 1)   InvGamma(2, 1)   InvGamma(3, 1)   InvGamma(3, 0.5)

mean(dist)
#> [1]   NA 1.00 0.50 0.25
variance(dist)
#> [1]     NA     NA 0.2500 0.0625
support(dist)
#> <support_region[4]>
#> [1] (0,Inf) (0,Inf) (0,Inf) (0,Inf)
generate(dist, 10)
#> [[1]]
#>  [1]  0.9917914  3.1186729 47.1285069 19.4519820  1.1407050  2.1386756
#>  [7] 44.1410852  3.8318874  6.3278763 10.5052104
#> 
#> [[2]]
#>  [1] 0.3733395 0.7154693 0.3985912 0.4477348 2.0754918 0.8672564 0.7758725
#>  [8] 0.7153657 1.0945662 0.2585611
#> 
#> [[3]]
#>  [1] 0.4168679 0.3437469 0.2257971 1.1402373 0.1910216 0.3157200 0.3205713
#>  [8] 0.1664944 0.6499774 0.2423886
#> 
#> [[4]]
#>  [1] 0.26115203 0.41051559 0.09656743 0.21845522 0.12719092 0.26514069
#>  [7] 0.22362301 0.13507356 0.20343338 0.26233230
#> 

density(dist, 2)
#> [1] 0.151632665 0.075816332 0.018954083 0.003042191
density(dist, 2, log = TRUE)
#> [1] -1.886294 -2.579442 -3.965736 -5.795177

cdf(dist, 4)
#> [1] 0.7788008 0.9735010 0.9978385 0.9997035

quantile(dist, 0.7)
#> [1] 2.8036733 0.9112869 0.5225272 0.2612636

Arguments

Details

See also

Examples