VGAM (version 1.1-1)

genpoisson: Generalized Poisson distribution

Description

Estimation of the two-parameter generalized Poisson distribution.

Usage

genpoisson(llambda = "rhobitlink", ltheta = "loglink",
           ilambda = NULL, itheta = NULL,
           use.approx = TRUE, imethod = 1, ishrinkage = 0.95,
           zero = "lambda")

Arguments

llambda, ltheta

Parameter link functions for \(\lambda\) and \(\theta\). See Links for more choices. The \(\lambda\) parameter lies at least within the interval \([-1,1]\); see below for more details, and an alternative link is rhobitlink. The \(\theta\) parameter is positive, therefore the default is the log link.

ilambda, itheta

Optional initial values for \(\lambda\) and \(\theta\). The default is to choose values internally.

use.approx

Logical. If TRUE then an approximation to the expected information matrix is used, otherwise Newton-Raphson is used.

imethod

An integer with value 1 or 2 or 3 which specifies the initialization method for the parameters. If failure to converge occurs try another value and/or else specify a value for ilambda and/or itheta.

ishrinkage, zero

See CommonVGAMffArguments for information.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Warning

Monitor convergence! This family function is fragile. Don't get confused because theta (and not lambda) here really matches more closely with lambda of dpois.

Details

The generalized Poisson distribution has density $$f(y) = \theta(\theta+\lambda y)^{y-1} \exp(-\theta-\lambda y) / y!$$ for \(\theta > 0\) and \(y = 0,1,2,\ldots\). Now \(\max(-1,-\theta/m) \leq \lambda \leq 1\) where \(m (\geq 4)\) is the greatest positive integer satisfying \(\theta + m\lambda > 0\) when \(\lambda < 0\) [and then \(P(Y=y) = 0\) for \(y > m\)]. Note the complicated support for this distribution means, for some data sets, the default link for llambda will not always work, and some tinkering may be required to get it running.

As Consul and Famoye (2006) state on p.165, the lower limits on \(\lambda\) and \(m \ge 4\) are imposed to ensure that there are at least 5 classes with nonzero probability when \(\lambda\) is negative.

An ordinary Poisson distribution corresponds to \(\lambda = 0\). The mean (returned as the fitted values) is \(E(Y) = \theta / (1 - \lambda)\) and the variance is \(\theta / (1 - \lambda)^3\).

For more information see Consul and Famoye (2006) for a summary and Consul (1989) for full details.

References

Consul, P. C. and Famoye, F. (2006) Lagrangian Probability Distributions, Boston, USA: Birkhauser.

Jorgensen, B. (1997) The Theory of Dispersion Models. London: Chapman & Hall

Consul, P. C. (1989) Generalized Poisson Distributions: Properties and Applications. New York, USA: Marcel Dekker.

See Also

poissonff, dpois. dgenpois, rhobitlink, extlogitlink.

Examples

# NOT RUN {
gdata <- data.frame(x2 = runif(nn <- 200))
gdata <- transform(gdata, y1 = rpois(nn, exp(2 - x2)))  # Poisson data
fit <- vglm(y1 ~ x2, genpoisson, data = gdata, trace = TRUE)
coef(fit, matrix = TRUE)
summary(fit)
# }