Generalized Waring Distribution

These functions provide the probability mass function, 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

positive numeric vector of distribution means.

k

positive numeric vector of shape parameters.

rho

numeric vector of shape parameters greater than 1. A finite variance requires \(\rho > 2\).

log

logical; if TRUE, log probabilities are returned.

q

non-negative integer vector of quantiles.

lower.tail

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

log.p

logical; if TRUE, probabilities are returned as log probabilities.

p

numeric vector of probabilities.

n

integer number of random numbers to generate.

Value

dgwar gives the probability mass function, 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.

Details

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

dgwar computes the probability mass function (PMF) of the Generalized Waring distribution.

pgwar computes the CDF of the Generalized Waring distribution.

qgwar computes the quantile function of the Generalized Waring distribution.

rgwar generates random numbers from the Generalized Waring distribution.

The probability mass function for the Generalized Waring distribution is: $$ f(y \mid a_x, k, \rho) = \frac{ \Gamma(a_x + \rho)\Gamma(k + \rho) (a_x)_y(k)_y }{ y!\Gamma(\rho)\Gamma(a_x + k + \rho) (a_x + k + \rho)_y } $$ where \((\alpha)_r = \frac{\Gamma(\alpha+r)}{\Gamma(\alpha)}\), and \(a_x > 0\), \(k > 0\), and \(\rho > 0\). Under the mean parameterization used by these functions, \(\rho > 1\) is required.

When \(\rho > 1\), the mean is: $$ E[Y] = \frac{a_x k}{\rho - 1}. $$

Therefore, the distribution can be parameterized in terms of its mean using: $$ a_x = \frac{\mu(\rho - 1)}{k}. $$

For a regression model: $$ \mu = \exp(X\beta). $$

When \(\rho > 2\), the variance under this mean parameterization is: $$ \mathrm{Var}(Y) = \frac{\mu(k+\mu)(k+\rho-1)}{k(\rho-2)}. $$

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 0 1 0 0 1 4 0 0