Yule-Simon distribution

Probability mass function, distribution function, and random generation for the Yule-Simon distribution.

Usage

dyules(x, shape = 1, log = FALSE)

pyules(q, shape = 1, lower.tail = TRUE, log.p = FALSE)

ryules(n, shape = 1)

Arguments

x, q: vector of quantiles.
shape: positive shape parameter $\rho$.
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 ryules).

Value

dyules gives the probability mass function, pyules gives the distribution function, and ryules generates random deviates.

Details

dyules and pyules allow for automatic differentiation with RTMB.

$$P(X = k;\, \rho) = \rho\, B(k,\, \rho + 1), \quad k = 1, 2, \ldots$$

The Yule-Simon distribution is the classical long-tailed frequency law, used for word counts, city sizes, citation counts and species-per-genus data. Its tail is a power law, $P(X = k) \sim \rho\,\Gamma(\rho + 1) k^{-(\rho + 1)}$, so the mean $\rho/(\rho - 1)$ exists only for $\rho > 1$ and the variance $\rho^2 / \{(\rho - 1)^2 (\rho - 2)\}$ only for $\rho > 2$.

It is the beta-negative binomial with size = 1 and shape2 = 1, shifted to start at one, and equivalently the Waring distribution with sigma = mu, shifted the same way. That special case is what gives it a closed-form distribution function, which the beta-negative binomial does not have in general.

The support here starts at one, as in VGAM. The YULE family of gamlss.dist instead starts at zero and is parameterised by its mean $\mu$, which corresponds to $\rho = (\mu + 1)/\mu$; use dwaring(x, mu, mu) for that version.

References

Simon, H. A. (1955) On a class of skew distribution functions. Biometrika, 42, 425-440, doi:10.1093/biomet/42.3-4.425.

Examples

set.seed(123)
x <- ryules(5, shape = 2)
d <- dyules(x, shape = 2)
p <- pyules(x, shape = 2)