dist_poisson_inverse_gaussian: The Poisson-Inverse Gaussian distribution

In distributional: Vectorised Probability Distributions

The Poisson-Inverse Gaussian distribution

Description

The Poisson-Inverse Gaussian distribution is a compound Poisson distribution where the rate parameter follows an Inverse Gaussian distribution. It is useful for modeling overdispersed count data.

Usage

dist_poisson_inverse_gaussian(mean, shape)

Arguments

mean, shape

parameters. Must be strictly positive. Infinite values are supported.

Details

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

In the following, let X be a Poisson-Inverse Gaussian random variable with parameters mean = \mu and shape = \phi.

Support: \{0, 1, 2, 3, ...\}

Mean: \mu

Variance: \frac{\mu}{\phi}(\mu^2 + \phi)

Probability mass function (p.m.f):

P(X = x) = \frac{e^{\phi}}{\sqrt{2\pi}} \left(\frac{\phi}{\mu^2}\right)^{x/2} \frac{1}{x!} \int_0^\infty u^{x-1/2} \exp\left(-\frac{\phi u}{2} - \frac{\phi}{2\mu^2 u}\right) du

for x = 0, 1, 2, \ldots

Cumulative distribution function (c.d.f):

P(X \le x) = \sum_{k=0}^{\lfloor x \rfloor} P(X = k)

The c.d.f does not have a closed form and is approximated numerically.

Moment generating function (m.g.f):

E(e^{tX}) = \exp\left\{\phi\left[1 - \sqrt{1 - \frac{2\mu^2}{\phi}(e^t - 1)}\right]\right\}

for t < -\log(1 + \phi/(2\mu^2))

See Also

actuar::PoissonInverseGaussian, actuar::dpoisinvgauss(), actuar::ppoisinvgauss(), actuar::qpoisinvgauss(), actuar::rpoisinvgauss()

Examples

dist <- dist_poisson_inverse_gaussian(mean = rep(0.1, 3), shape = c(0.4, 0.8, 1))
dist


mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

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

cdf(dist, 4)

quantile(dist, 0.7)

distributional documentation built on June 27, 2026, 5:06 p.m.