Generalized Poisson distribution functions

Density, distribution, quantile, and random generation for the Consul-Jain generalized Poisson distribution in its constant-dispersion form: mean mu and dispersion lambda in (-1, 1), with \(P(Y = y) = \theta(\theta + \lambda y)^{y-1} e^{-\theta - \lambda y}/y!\) and \(\theta = \mu(1 - \lambda)\), so that the variance-to-mean ratio is \(1/(1-\lambda)^2\). lambda < 0 gives underdispersion, lambda = 0 the Poisson, lambda > 0 overdispersion. For lambda < 0 the support is finite (the terms are positive only while \(\theta + \lambda y > 0\)) and the pmf is renormalized on that support, so it is a proper distribution and the exact mean and variance (which then differ from mu and mu/(1-lambda)^2 only by the negligible mass beyond the support) are what count_reg(family = "genpois") reports through predict() and fitted().

Usage

dgenpois(x, mu, lambda, log = FALSE)

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

qgenpois(p, mu, lambda, lower.tail = TRUE, log.p = FALSE)

rgenpois(n, mu, lambda)

Arguments

x, q

Vector of quantiles (non-negative integers).

mu

Mean parameter (scalar or vector, recycled).

lambda

Dispersion parameter in (-1, 1) (scalar).

log, log.p

Return log probabilities.

lower.tail

If TRUE (default), \(P(X \le x)\).

p

Vector of probabilities.

n

Number of draws.

Value

dgenpois a density, pgenpois a CDF, qgenpois a quantile, rgenpois a numeric vector of count draws.

References

Consul, P. C. and Jain, G. C. (1973). A generalization of the Poisson distribution. Technometrics, 15(4), 791-799. Consul, P. C. and Famoye, F. (2006). Lagrangian Probability Distributions. Birkhauser.

See also

count_reg()

Examples

dgenpois(0:5, mu = 3, lambda = -0.3)
#> [1] 0.02024191 0.10656252 0.23734318 0.29125435 0.21495599 0.09781863
var(rgenpois(2000, mu = 3, lambda = -0.3)) / 3   # about 1/(1.3)^2
#> [1] 0.6026076