dgenpois: The Generalized Poisson Distribution

Description

Density, cumulative density, random-deviate functions for the generalized Poisson distribution.

Usage

dgenpois(x, lambda1, lambda2, log = FALSE)
pgenpois(q, lambda1, lambda2, lower.tail = TRUE, log.p = FALSE)
rgenpois(n, lambda1, lambda2)

Value

dgenpois gives the density, pgenpois gives the cumulative distribution function, and rgenpois generates random deviates.

Arguments

x: a vector of (non-negative integer) quantiles
lambda1: a single numeric value for parameter lambda1 with $lambda1 > 0$
lambda2: a single numeric value for parameter lambda2 with $-1 \le lambda2 < 1$. When lambda2=0, the generalized Poisson distribution reduces to the Poisson distribution
log: logical; if TRUE, the log-density is returned
q: a numeric vector of quantiles
lower.tail: logical; if TRUE (default), probabilities are $P(X \le q)$.
log.p: logical; if TRUE, probabilities are returned on the log scale.
n: number of random deviates to draw

Author

Based on Joe and Zhu (2005). Original implementation by Vitali Witowski (2013).

Details

The generalized Poisson distribution has the density $$ p(x) = \lambda_1 (\lambda_1 + \lambda_2 \cdot x)^{x-1} \frac{ \exp(-\lambda_1-\lambda_2 \cdot x) )}{x!}$$ for $x = 0,1,2,\ldots$, with $\mbox{E}(X)= \frac{\lambda_1}{1-\lambda_2}$ and variance $\mbox{var}(X)=\frac{\lambda_1}{(1-\lambda_2)^3}$.

References

Joe, H., Zhu, R. (2005). Generalized Poisson distribution: the property of mixture of Poisson and comparison with negative binomial distribution. Biometrical Journal 47(2):219--229.

Examples

Run this code

dgenpois(x = seq(0, 20), lambda1 = 10, lambda2 = 0.5)
pgenpois(q = 5, lambda1 = 10, lambda2 = 0.5)
set.seed(101)
hist(rgenpois(n = 1000, lambda1 = 10, lambda2 = 0.5))

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