Estimation of the two-parameter generalized Poisson distribution (original parameterization).
genpoisson0(ltheta = "loglink", llambda = "logitlink",
itheta = NULL, ilambda = NULL, imethod = c(1, 1),
ishrinkage = 0.95, glambda = ppoints(5),
parallel = FALSE, zero = "lambda")An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
and vgam.
The generalized Poisson distribution (GPD) was proposed by Consul and Jain (1973), and it has PMF $$f(y)=\theta(\theta+\lambda y)^{y-1} \exp(-\theta-\lambda y) / y!$$ for \(0 < \theta\) and \(y = 0,1,2,\ldots\). Theoretically, \(\max(-1,-\theta/m) \leq \lambda \leq 1\) where \(m\) \((\geq 4)\) is the greatest positive integer satisfying \(\theta + m\lambda > 0\) when \(\lambda < 0\) [and then \(Pr(Y=y) = 0\) for \(y > m\)]. However, there are problems with a negative \(\lambda\) such as it not being normalized, so this family function restricts \(\lambda\) to \((0, 1)\).
This original parameterization is called the GP-0 by VGAM,
partly because there are two other common parameterizations
called the GP-1 and GP-2 (see Yang et al. (2009)),
genpoisson1
and genpoisson2)
that are more suitable for regression.
However, genpoisson() has been simplified to
genpoisson0 by only handling positive parameters,
hence only overdispersion relative to the Poisson is accommodated.
Some of the reasons for this are described in
Scollnik (1998), e.g., the probabilities do not
sum to unity when lambda is negative.
To simply things, VGAM 1.1-4 and later will only
handle positive lambda.
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 \(\theta / (1 - \lambda)^3\) so that the variance is proportional to the mean, just like the NB-1 and quasi-Poisson.
For more information see Consul and Famoye (2006) for a summary and Consul (1989) for more details.
Consul, P. C. and Jain, G. C. (1973). A generalization of the Poisson distribution. Technometrics, 15, 791--799.
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.
Yang, Z., Hardin, J. W., Addy, C. L. (2009). A score test for overdispersion in Poisson regression based on the generalized Poisson-2 model. J. Statist. Plann. Infer., 139, 1514--1521.
Yee, T. W. (2020). On generalized Poisson regression. In preparation.
gdata <- data.frame(x2 = runif(nn <- 500))
gdata <- transform(gdata, y1 = rgenpois0(nn, theta = exp(2 + x2),
logitlink(1, inverse = TRUE)))
gfit0 <- vglm(y1 ~ x2, genpoisson0, data = gdata, trace = TRUE)
coef(gfit0, matrix = TRUE)
summary(gfit0)
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