Generalized Poisson Regression
Estimation of the two-parameter generalized Poisson distribution.
genpoisson(llambda = "rhobitlink", ltheta = "loglink",
ilambda = NULL, itheta = NULL, imethod = 1,
ishrinkage = 0.95, zero = "lambda")llambda, ltheta |
Parameter link functions for λ and θ.
See |
ilambda, itheta |
Optional initial values for λ and θ. The default is to choose values internally. |
imethod |
An integer with value |
ishrinkage, zero |
See |
This family function is not recommended for use;
instead try
genpoisson1 or
genpoisson2.
For underdispersion with respect to the Poisson
try the GTE (generally-truncated expansion) method
described by Yee and Ma (2020).
The generalized Poisson distribution has density
f(y)=θ(θ+λ * y)^(y-1) * exp(-θ-λ * y) / y!
for θ > 0 and y = 0,1,2,….
Now
max(-1,-θ/m) ≤ lambda ≤ 1
where m (≥ 4) is the greatest positive
integer satisfying θ + mλ > 0
when λ < 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 λ and m >= 4 are imposed to ensure that there are at least 5 classes with nonzero probability when λ is negative.
An ordinary Poisson distribution corresponds to lambda = 0. The mean (returned as the fitted values) is E(Y) = θ / (1 - λ) and the variance is θ / (1 - λ)^3.
For more information see Consul and Famoye (2006) for a summary and Consul (1989) for full details.
An object of class "vglmff"
(see vglmff-class).
The object is used by modelling functions such
as vglm,
and vgam.
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.
This family function handles multiple responses.
This distribution is potentially useful for dispersion modelling.
Convergence problems may occur when lambda is very close
to 0 or 1.
If a failure occurs then you might want to try something like
llambda = extlogitlink(min = -0.9, max = 1)
to handle the LHS complicated constraint,
and if that doesn't work, try
llambda = extlogitlink(min = -0.8, max = 1), etc.
T. W. Yee.
Easton Huch derived the EIM and it has been implemented
in the weights slot.
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.
Yee, T. W. and Ma, C. (2021). Generally–altered, –inflated and –truncated regression, with application to heaped and seeped counts. Under review.
## Not run: gdata <- data.frame(x2 = runif(nn <- 500)) # NBD data: gdata <- transform(gdata, y1 = rnbinom(nn, exp(1), mu = exp(2 - x2))) fit <- vglm(y1 ~ x2, genpoisson, data = gdata, trace = TRUE) coef(fit, matrix = TRUE) summary(fit) ## End(Not run)
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