LaplacesDemon (version 16.1.1)

# dist.Generalized.Poisson: Generalized Poisson Distribution

## Description

The density function is provided for the univariate, discrete, generalized Poisson distribution with location parameter $$\lambda$$ and scale parameter $$\omega$$.

## Usage

dgpois(x, lambda=0, omega=0, log=FALSE)

## Arguments

x

This is a vector of quantiles.

lambda

This is the parameter $$\lambda$$.

omega

This is the parameter $$\omega$$, which should be in the interval [0,1) for positive counts.

log

Logical. If log=TRUE, then the logarithm of the density is returned.

## Value

dgpois gives the density.

## Details

• Application: Discrete Univariate

• Density: $$p(\theta) = (1 - \omega) \lambda \frac{[(1 - \omega) \lambda + \omega \theta]^{\theta - 1}}{\theta!} \exp{-[(1 - \omega) \lambda + \omega \theta]}$$

• Inventor: Consul (1989) and Ntzoufras et al. (2005)

• Notation 1: $$\theta \sim \mathrm{GP}(\lambda,\omega)$$

• Notation 2: $$p(\theta) = \mathrm{GP}(\theta | \lambda, \omega)$$

• Parameter 1: location parameter $$\lambda$$

• Parameter 2: scale parameter $$\omega \in [0,1)$$

• Mean: $$E(\theta) = \lambda$$

• Variance: $$var(\theta) = \lambda(1 - \omega)^{-2}$$

The generalized Poisson distribution (Consul, 1989) is also called the Lagrangian Poisson distribution. The simple Poisson distribution is a special case of the generalized Poisson distribution. The generalized Poisson distribution is used in generalized Poisson regression as an extension of Poisson regression that accounts for overdispersion.

The dgpois function is parameterized according to Ntzoufras et al. (2005), which is easier to interpret and estimates better with MCMC.

Valid values for omega are in the interval [0,1) for positive counts. For $$\omega = 0$$, the generalized Poisson reduces to a simple Poisson with mean $$\lambda$$. Note that it is possible for $$\omega < 0$$, but this implies underdispersion in count data, which is uncommon. The dgpois function returns warnings or errors, so $$\omega$$ should be non-negative here.

The dispersion index (DI) is a variance-to-mean ratio, and is $$DI = (1 - \omega)^{-2}$$. A simple Poisson has DI=1. When DI is far from one, the assumption that the variance equals the mean of a simple Poisson is violated.

## References

Consul, P. (1989). "Generalized Poisson Distribution: Properties and Applications". Marcel Decker: New York, NY.

Ntzoufras, I., Katsis, A., and Karlis, D. (2005). "Bayesian Assessment of the Distribution of Insurance Claim Counts using Reversible Jump MCMC", North American Actuarial Journal, 9, p. 90--108.

dnbinom and dpois.

## Examples

Run this code
# NOT RUN {
library(LaplacesDemon)
y <- rpois(100, 5)
lambda <- rpois(100, 5)
x <- dgpois(y, lambda, 0.5)

#Plot Probability Functions
x <- seq(from=0, to=20, by=1)
plot(x, dgpois(x,1,0.5), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dlaplace(x,1,0.6), type="l", col="green")
lines(x, dlaplace(x,1,0.7), type="l", col="blue")
legend(2, 0.9, expression(paste(lambda==1, ", ", omega==0.5),
paste(lambda==1, ", ", omega==0.6), paste(lambda==1, ", ", omega==0.7)),
lty=c(1,1,1), col=c("red","green","blue"))
# }
`

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