Generalised Poisson distribution

Probability mass function, distribution function, and random generation for the generalised Poisson distribution.

Usage

dgenpois(x, lambda = 1, phi = 1, log = FALSE)

pgenpois(q, lambda = 1, phi = 1, lower.tail = TRUE, log.p = FALSE)

qgenpois(p, lambda = 1, phi = 1,
         lower.tail = TRUE, log.p = FALSE, max.value = 10000)

rgenpois(n, lambda = 1, phi = 1, max.value = 10000)

Arguments

x, q: integer vector of counts
lambda: vector of positive means
phi: vector of non-negative dispersion parameters
log, log.p: logical; return log-density if TRUE
lower.tail: logical; if TRUE, probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
p: vector of probabilities
max.value: a constant, set to the default value of 10000 for how far the algorithm should look for q.
n: number of random values to return.

Value

dgenpois gives the probability mass function, pgenpois gives the distribution function, qgenpois gives the quantile function, and rgenpois generates random deviates.

Details

This implementation of dgenpois allows for automatic differentiation with RTMB. The parameterisation follows the GPO family of the gamlss.dist package.

The distribution has mean $\lambda$ and variance $\lambda(1 + \phi \lambda)^2$. For $\phi = 0$ it reduces to the Poisson distribution, however $\phi$ must be strictly positive here.

$$P(X = x;\,\lambda,\phi) = \frac{\lambda\,(1+\phi x)^{x-1}\,e^{-\lambda(1+\phi x)/(1+\phi\lambda)}}{(1+\phi\lambda)^x\, x!}, \quad x = 0, 1, 2, \ldots$$

References

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 <- rgenpois(1, 2, 3)
d <- dgenpois(x, 2, 3)
p <- pgenpois(x, 2, 3)
q <- qgenpois(p, 2, 3)