VGAM (version 1.1-14)

genpoisson2: Generalized Poisson Regression (GP-2 Parameterization)

Description

Estimation of the two-parameter generalized Poisson distribution (GP-2 parameterization) which has the variance as a cubic function of the mean.

Usage

genpoisson2(lmeanpar = "loglink", ldisppar = "loglink",
    parallel = FALSE, zero = "disppar",
    vfl = FALSE, oparallel = FALSE,
    imeanpar = NULL, idisppar = NULL, imethod = c(1, 1),
    ishrinkage = 0.95, gdisppar = exp(1:5))

Value

An object of class "vglmff" (see

vglmff-class). The object is used by modelling functions such as

vglm, and vgam.

Details

This is a variant of the generalized Poisson distribution (GPD) and called GP-2 by some writers such as Yang, et al. (2009). Compared to the original GP-0 (see genpoisson0) the GP-2 has \(\theta = \mu / (1 + \alpha \mu)\) and \(\lambda = \alpha \mu / (1 + \alpha \mu)\) so that the variance is \(\mu (1 + \alpha \mu)^2\). The first linear predictor by default is \(\eta_1 = \log \mu\) so that the GP-2 is more suitable for regression than the GP-0.

This family function can handle only overdispersion relative to the Poisson. An ordinary Poisson distribution corresponds to \(\alpha = 0\). The mean (returned as the fitted values) is \(E(Y) = \mu\).

References

Letac, G. and Mora, M. (1990). Natural real exponential familes with cubic variance functions. Annals of Statistics 18, 1--37.

See Also

Genpois2, genpoisson0, genpoisson1, poissonff, negbinomial, Poisson, quasipoisson.

Examples

gdata <- data.frame(x2 = runif(nn <- 500))
gdata <- transform(gdata, y1 = rgenpois2(nn, exp(2 + x2),
                               loglink(-1, inverse = TRUE)))
gfit2 <- vglm(y1 ~ x2, genpoisson2, gdata, trace = TRUE)
coef(gfit2, matrix = TRUE)
summary(gfit2)