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]   2.6213179   2.2634405   5.2080916   0.4762990   2.1961705   1.4460740
#>  [7]   0.6605219 256.3582586   0.5001634   1.2191401
#> 
#> [[2]]
#>  [1] 0.5031377 0.2279797 1.2797352 0.3565268 0.9475674 1.7909942 0.3664776
#>  [8] 0.7476891 0.3780903 0.9671384
#> 
#> [[3]]
#>  [1] 0.4472460 0.2701471 0.4068668 0.5246646 1.2266273 0.2482667 0.6233011
#>  [8] 0.6373356 0.2007577 0.2092655
#> 
#> [[4]]
#>  [1] 0.06733771 0.13384022 0.30555479 0.21132656 0.38265102 0.05609552
#>  [7] 0.23928793 0.16327412 0.45018199 0.15228056
#> 

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