Conway-Maxwell-Poisson (COM) Distribution

These functions provide the density function, distribution function, quantile function, and random number generation for the Conway-Maxwell-Poisson (COM) Distribution

Usage

dcom(x, mu = NULL, lambda = 1, nu = 1, log = FALSE)

pcom(q, mu = NULL, lambda = 1, nu = 1, lower.tail = TRUE, log.p = FALSE)

qcom(p, mu = NULL, lambda = 1, nu = 1)

rcom(n, mu = NULL, lambda = 1, nu = 1)

Arguments

x

numeric value or a vector of values.

mu

optional. Numeric value or vector of mean values for the distribution (the values have to be greater than 0).

lambda

optional. Numeric value or vector of values for the rate parameter of the distribution (the values have to be greater than 0). If `mu` is provided, `lambda` is ignored.

nu

optional. Numeric value or vector of values for the decay parameter of the distribution ((the values have to be greater than 0).

log

logical; if TRUE, probabilities p are given as log(p).

q

quantile or a vector of quantiles.

lower.tail

logical; if TRUE, probabilities p are \(P[X\leq x]\) otherwise, \(P[X>x]\).

log.p

logical; if TRUE, probabilities p are given as log(p).

p

probability or a vector of probabilities.

n

the number of random numbers to generate.

Details

dcom computes the density (PDF) of the COM Distribution.

pcom computes the CDF of the COM Distribution.

qcom computes the quantile function of the COM Distribution.

rcom generates random numbers from the COM Distribution.

The Probability Mass Function (PMF) for the Conway-Maxwell-Poisson distribution is: $$f(x|\lambda, \nu) = \frac{\lambda^x}{(x!)^\nu Z(\lambda,\nu)}$$

Where \(\lambda\) and \(\nu\) are distribution parameters with \(\lambda>0\) and \(\nu>0\), and \(Z(\lambda,\nu)\) is the normalizing constant.

The normalizing constant is given by: $$Z(\lambda,\nu)=\sum_{n=0}^{\infty}\frac{\lambda^n}{(n!)^\nu}$$

The mean and variance of the distribution are given by: $$E[x]=\mu=\lambda \frac{\delta}{\delta \lambda} \log(Z(\lambda,\nu))$$ $$Var(x)=\lambda \frac{\delta}{\delta \lambda} \mu$$

When the mean value is given, the rate parameter (\(\lambda\)) is computed using the mean and the decay parameter (\(\nu\)). This is useful to allow the calculation of the rate parameter when the mean is known (e.g., in regression))

dcom gives the density, pcom gives the distribution function, qcom gives the quantile function, and rcom generates random deviates.

The length of the result is determined by n for rcom, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Examples

dcom(1, mu=0.75, nu=3)
#> [1] 0.5347848
pcom(c(0,1,2,3,5,7,9,10), lambda=0.75, nu=0.75)
#> [1] 0.4480730 0.7841277 0.9339922 0.9833004 0.9993079 0.9999816 0.9999997
#> [8] 1.0000000
qcom(c(0.1,0.3,0.5,0.9,0.95), mu=0.75, nu=0.75)
#> [1] 0 0 1 2 2
rcom(30, mu=0.75, nu=0.5)
#>  [1] 1 0 0 0 0 0 1 1 2 0 0 0 0 0 0 0 0 3 0 1 1 0 3 1 0 1 1 1 0 0