Waring distribution

Probability mass function, distribution function, and random generation for the Waring distribution.

Usage

dwaring(x, mu = 2, sigma = 2, log = FALSE)

pwaring(q, mu = 2, sigma = 2, lower.tail = TRUE, log.p = FALSE)

rwaring(n, mu = 2, sigma = 2)

Arguments

x, q

vector of non-negative counts.

mu

mean parameter, must be positive.

sigma

dispersion parameter, must be positive. The variance is finite only for sigma < 1.

log, log.p

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

lower.tail

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

n

number of random values to return (for rwaring).

Value

dwaring gives the probability mass function, pwaring gives the distribution function, and rwaring generates random deviates.

Details

dwaring and pwaring allow for automatic differentiation with RTMB. The parameterisation follows the WARING family of the gamlss.dist package, in which \(\mu\) is exactly the mean.

$$P(X = k;\, \mu, \sigma) = \frac{B\bigl(k + \tfrac{\mu}{\sigma},\; \tfrac{1}{\sigma} + 2\bigr)}{B\bigl(\tfrac{\mu}{\sigma},\; \tfrac{1}{\sigma} + 1\bigr)}, \quad k = 0, 1, 2, \ldots$$

The Waring is the beta-geometric: a geometric distribution whose success probability carries a beta prior. It is the two-parameter long-tailed count law behind accident proneness and repeat-buying models, and generalises the Yule-Simon distribution, which is the case \(\sigma = \mu\) shifted to start at one. The variance $$\mathrm{Var}(X) = \frac{\mu (\sigma + 1)(\mu + 1)}{1 - \sigma}$$ is finite only for \(\sigma < 1\), and the tail is a power law throughout.

It is exactly the mean-parameterised beta-negative binomial with nu = 1. Fixing size at one is what gives it a closed-form distribution function, which the beta-negative binomial does not have in general, so one-step-ahead residuals are available here but not there.

References

Irwin, J. O. (1963) The place of mathematics in medical and biological statistics. Journal of the Royal Statistical Society A, 126, 1-45, doi:10.2307/2982445.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

See also

yules, bnbinom2, bnbinom

Examples

set.seed(123)
x <- rwaring(5, mu = 2, sigma = 0.5)
d <- dwaring(x, mu = 2, sigma = 0.5)
p <- pwaring(x, mu = 2, sigma = 0.5)