Estimation of the two-parameter generalized Poisson distribution.

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

llambda, ltheta

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.

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`

.

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.

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.

```
# 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)
# }
```

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