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`

*= theta*, 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 1.0

Index]