# PvfPoisson: Power Variance Function Poisson Distribution In rmutil: Utilities for Nonlinear Regression and Repeated Measurements Models

## Description

These functions provide information about the overdispersed power variance function Poisson distribution with parameters `m`, `s`, and `f`: density, cumulative distribution, quantiles, and random generation. This function is obtained from a Poisson distribution as a mixture with a power variance distribution. In the limit, for `f=0`, the mixing distribution is gamma so that it is a negative binomial distribution. For `f=0.5`, the mixing distribution is inverse Gaussian. For `f<0`, the mixing distribution is a compound distribution of the sum of a Poisson number of gamma distributions. For `f=1`, it is undefined.

The power variance function Poisson distribution with `m` = μ, the mean, `s` = θ, and `f` = α has density

p(y) = (exp(-m((s+1)^f/s^f-s)/f) / y!) sum_{i=1}^y c_{yi}(f) m^i (s+1)^{if-y} / s^{i(f-1)}

for y = 0, …, where `c_{yi}(f)` are coefficients obtained by recursion.

## Usage

 ```1 2 3 4``` ```dpvfpois(y, m, s, f, log=FALSE) ppvfpois(q, m, s, f) qpvfpois(p, m, s, f) rpvfpois(n, m, s, f) ```

## Arguments

 `y` vector of counts `q` vector of quantiles `p` vector of probabilities `n` number of values to generate `m` scalar or vector of means `s` scalar or vector of overdispersion parameters `f` scalar or vector of family parameters, all < 1 `log` if TRUE, log probabilities are supplied.

## Author(s)

J.K. Lindsey

`dpois` for the Poisson, `ddoublepois` for the double Poisson, `dmultpois` for the multiplicative Poisson, `dconsul` for the Consul generalized Poisson, `dgammacount` for the gamma count, and `dnbinom` for the negative binomial distribution.
 ```1 2 3 4``` ```dpvfpois(5,10,0.9,0.5) ppvfpois(5,10,0.9,0.5) qpvfpois(0.85,10,0.9,0.5) rpvfpois(10,10,0.9,0.5) ```