Poisson-inverse Gaussian distribution

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).

Value

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

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.

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/.

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)