R: Power Variance Function Poisson Distribution

PvfPoisson {rmutil} | R Documentation

## Power Variance Function Poisson Distribution

### 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

    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

### See Also

`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.

### Examples

    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)

[Package rmutil version 1.1.3 Index]