Generalized Waring Distribution

These functions provide density, distribution function, quantile function, and random number generation for the Generalized Waring Distribution.

Usage

dgwar(y, mu, k, rho, log = FALSE)

pgwar(q, mu, k, rho, lower.tail = TRUE, log.p = FALSE)

qgwar(p, mu, k, rho)

rgwar(n, mu, k, rho)

Arguments

y

non-negative integer vector of count outcomes.

mu

numeric vector of means of the distribution.

k

non-negative numeric parameter of the distribution.

rho

non-negative numeric parameter of the distribution.

log

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

q

non-negative integer 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

numeric vector of probabilities.

n

integer number of random numbers to generate.

Details

The Generalized Waring distribution is a 3-parameter count distribution that is used to model overdispersed count data.

dgwar computes the density (PMF) of the Generalized Waring Distribution.

pgwar computes the CDF of the Generalized Waring Distribution.

qwaring computes the quantile function of the Generalized Waring Distribution.

rwaring generates random numbers from the Generalized Waring Distribution.

The Probability Mass Function (PMF) for the Generalized Waring (GW) distribution is: $$f(y|a_x,k,\rho) = \frac{\Gamma(a_x+\rho)\Gamma(k+\rho)\left(a_x\right)_y(k)_y} {y!\Gamma(\rho)\Gamma(a_x+k+\rho)(a_x+k+\rho)_y}$$ Where \((\alpha)_r=\frac{\Gamma(\alpha+r)}{\Gamma(\alpha)}\), and \(a_x, \ k, \ \rho)>0\).

The mean value is: $$E[Y]=\frac{a_x K}{\rho-1}$$

Thus, we can use: $$a_x=\frac{\mu(\rho-1)}{k}$$

This results in a regression model where: $$\mu=e^{X\beta}$$ $$\sigma^2 = \mu \left(1-\frac{1}{\alpha+\rho+1} \right) + \mu^2\frac{(\alpha+\rho)^2}{\alpha\rho(\alpha+\rho+1)}$$

dgwar gives the density, pgwar gives the distribution function, qgwar gives the quantile function, and rgwar generates random deviates.

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

Examples

dgwar(0, mu=1, k=2, rho=3)
#> [1] 0.6
pgwar(c(0,1,2,3), mu=1, k=2, rho=3)
#> [1] 0.6000000 0.8000000 0.8857143 0.9285714
qgwar(0.8, mu=1, k=2, rho=3)
#> [1] 1
rgwar(10, mu=1, k=2, rho=3)
#>  [1] 0 0 1 4 0 0 1 0 1 1