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.
dpvfpois(y, m, s, f, log=FALSE)
ppvfpois(q, m, s, f)
qpvfpois(p, m, s, f)
rpvfpois(n, m, s, f)vector of counts
vector of quantiles
vector of probabilities
number of values to generate
scalar or vector of means
scalar or vector of overdispersion parameters
scalar or vector of family parameters, all < 1
if TRUE, log probabilities are supplied.
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.
# NOT RUN {
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)
# }
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