dist.Generalized.Poisson: Generalized Poisson Distribution
In LaplacesDemonR/LaplacesDemonCpp: C++ Extension for LaplacesDemon

Description Usage Arguments Details Value References See Also Examples

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

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

`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.

Application: Discrete Univariate
Density: (1 - omega) lambda (((1 - omega) lambda + omega theta)^(y-1) / theta!) e(-((1 - omega) lambda + omega theta))
Inventor: Consul (1989) and Ntzoufras et al. (2005)
Notation 1: theta ~ GP(lambda, omega)
Notation 2: p(theta) = 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.

dgpois gives the density.

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.

library(LaplacesDemonCpp)
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"))