genpoisson {VGAM} | R Documentation |
Estimation of the two parameters of a generalized Poisson distribution.
genpoisson(llambda="elogit", ltheta="loge", elambda=if(llambda=="elogit") list(min=-1,max=1) else list(), etheta=list(), ilambda=NULL, itheta=NULL, use.approx=TRUE, method.init=1, zero=1)
llambda, ltheta |
Parameter link functions for λ and theta.
See Links for more choices.
The λ parameter lies at least within the interval
[-1,1]; see below for more details.
The theta parameter is positive, therefore the default is the
log link.
|
elambda, etheta |
List. Extra argument for each of the links.
See earg in Links for general information.
|
ilambda, itheta |
Optional initial values for λ 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.
|
method.init |
An integer with value 1 or 2 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 .
|
zero |
An integer vector, containing the value 1 or 2.
If so, λ or theta respectively
are modelled as an intercept only.
If set to NULL then both linear/additive predictors are modelled
as functions of the explanatory variables.
|
The generalized Poisson distribution has density
f(y) = theta(theta+λ y)^{y-1} exp(-theta-λ y) / y!
for theta > 0 and y = 0,1,2,....
Now max(-1,-theta/m) <= λ <= 1
where m (>= 4) is the greatest positive
integer satisfying theta + 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
is not always appropriate.
An ordinary Poisson distribution corresponds to lambda=0. The mean (returned as the fitted values) is E(Y) = theta / (1 - λ) and the variance is theta / (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
.
This distribution is useful for dispersion modelling.
Convergence problems may occur when lambda
is very close to 0
or 1.
T. W. Yee
Consul, P. C. and Famoye, F. (2006) Lagrangian Probability Distributions, Boston: Birkhauser.
Jorgensen, B. (1997) The Theory of Dispersion Models. London: Chapman & Hall
Consul, P. C. (1989) Generalized Poisson Distributions: Properties and Applications. New York: Marcel Dekker.
n = 200 x = runif(n) y = rpois(n, lam=exp(2-x)) fit = vglm(y ~ x, genpoisson(zero=1), trace=TRUE) coef(fit, matrix=TRUE) summary(fit)