pig: Poisson-inverse Gaussian distribution

In RTMBdist: Distributions Compatible with Automatic Differentiation by 'RTMB'

Poisson-inverse Gaussian distribution

Description

Probability mass function, distribution function and random generation for the Poisson-inverse Gaussian distribution.

Usage

dpig(x, mu, sigma, log = FALSE)

ppig(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)

rpig(n, mu, sigma)

Arguments

x	vector of non-negative counts.
mu	mean parameter, must be positive.
sigma	dispersion parameter, must be positive.
log	logical; if TRUE, probabilities are returned on the log scale.
q	vector of quantiles.
lower.tail	logical; if TRUE (default), probabilities are P[X \le q], otherwise P[X > q].
log.p	logical; if TRUE, probabilities are returned on the log scale.
n	number of random values to return (for rpig).

Details

This implementation allows for automatic differentiation with RTMB.

The Poisson-inverse Gaussian is a mixed Poisson distribution whose mean is drawn from an inverse Gaussian distribution with mean \mu and variance \sigma \mu^2. Its probability mass function is

P(X = x) = \left(\frac{2\alpha}{\pi}\right)^{1/2} \frac{\mu^x e^{1/\sigma} K_{x - 1/2}(\alpha)}{(\alpha\sigma)^x\, x!}, \qquad \alpha^2 = \frac{1}{\sigma^2} + \frac{2\mu}{\sigma}, \quad x = 0, 1, 2, \ldots

where K_\nu is the modified Bessel function of the second kind. The parameterisation follows the PIG family of the gamlss.dist package, with

E(X) = \mu, \qquad \mathrm{Var}(X) = \mu + \sigma \mu^2.

This is the same mean-variance relationship as the negative binomial with size = 1 / sigma, but the Poisson-inverse Gaussian has a heavier right tail and puts more mass at zero for the same mean and variance. It approaches the Poisson as \sigma \to 0.

It is the Sichel distribution with \nu = -1/2, and is computed as such; see there for how the Bessel functions are evaluated. The distribution function is summed from the probability mass function, so one-step-ahead (OSA) residuals via RTMB::oneStepPredict are supported, e.g. with method = "cdf" and discrete = TRUE.

Value

dpig gives the probability mass function, ppig gives the distribution function, and rpig generates random deviates.

References

Dean, C., Lawless, J. F., and Willmot, G. E. (1989) A mixed Poisson-inverse-Gaussian regression model. Canadian Journal of Statistics, 17, 171-181.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

See Also

sichel, nbinom2, invgauss

Examples

set.seed(123)
x <- rpig(5, mu = 4, sigma = 0.5)
d <- dpig(x, mu = 4, sigma = 0.5)
p <- ppig(x, mu = 4, sigma = 0.5)

RTMBdist documentation built on Oct. 31, 2026, 5:07 p.m.