genpoisson | R Documentation |

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 |

`ilambda, itheta` |
Optional initial values for |

`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*.

`genpoisson1`

,
`genpoisson2`

,
`poissonff`

,
`dpois`

.
`dgenpois0`

,
`rhobitlink`

,
`extlogitlink`

.

## 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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