PvfPoisson {rmutil}

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 = \mu, the mean, s = \theta, and f = \alpha has density

p(y) = {\exp(-\mu((\theta+1)^\alpha/\theta^\alpha-\theta)/\alpha)\over y!} \sum_{i=1}^y c_{yi}(\alpha)\mu^i(\theta+1)^{i\alpha-y}/\theta^{i(\alpha-1)}

for y = 0, \ldots, 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)