genpoisson {VGAM} | R Documentation

## Generalized Poisson distribution

### Description

Estimation of the two parameters of a generalized Poisson distribution.

### Usage

    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)

### Arguments

`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.

### Details

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.

### Value

An object of class `"vglmff"` (see `vglmff-class`). The object is used by modelling functions such as `vglm`, and `vgam`.

### Note

This distribution is useful for dispersion modelling. Convergence problems may occur when `lambda` is very close to 0 or 1.

### Author(s)

T. W. Yee

### References

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.

### See Also

`poissonff`.

### Examples

    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)

[Package VGAM version 0.7-9 Index]