distributional (version 0.6.0)

dist_inverse_gamma: The Inverse Gamma distribution

Description

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

Usage

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 \(t \ge 0\).

See Also

actuar::InverseGamma

Examples

dist <- dist_inverse_gamma(shape = c(1,2,3,3), rate = c(1,1,1,2))
dist

if (FALSE) { # requireNamespace("actuar", quietly = TRUE)
mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)
}