sichel: Sichel distribution

In RTMBdist: Distributions Compatible with Automatic Differentiation by 'RTMB'

Sichel distribution

Description

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

Usage

dsichel(x, mu, sigma, nu, log = FALSE)

psichel(q, mu, sigma, nu, lower.tail = TRUE, log.p = FALSE)

rsichel(n, mu, sigma, nu)

Arguments

x	vector of non-negative counts.
mu	mean parameter, must be positive.
sigma	dispersion parameter, must be positive.
nu	shape parameter, any real number.
log	logical; if TRUE, probabilities are returned on the log scale.
q	vector of quantiles.
lower.tail	logical; if TRUE (default), probabilities are P[X \le q], otherwise P[X > q].
log.p	logical; if TRUE, probabilities are returned on the log scale.
n	number of random values to return (for rsichel).

Details

This implementation allows for automatic differentiation with RTMB.

The Sichel distribution is a mixed Poisson distribution whose mean follows a generalised inverse Gaussian distribution. In the parameterisation of the SICHEL family of the gamlss.dist package, \mu is the mean and the probability mass function is

P(X = x) = \frac{(\mu/c)^x\, K_{x + \nu}(\alpha)}{x!\,(\alpha\sigma)^{x + \nu} K_\nu(1/\sigma)}, \qquad \alpha^2 = \frac{1}{\sigma^2} + \frac{2\mu}{c\,\sigma}, \quad c = \frac{K_{\nu + 1}(1/\sigma)}{K_\nu(1/\sigma)},

for x = 0, 1, 2, \ldots, where K_\nu is the modified Bessel function of the second kind. The variance is

\mathrm{Var}(X) = \mu + \mu^2\left(\frac{2\sigma(\nu + 1)}{c} + \frac{1}{c^2} - 1\right).

For \nu = -1/2 this is the Poisson-inverse Gaussian distribution, and as \sigma \to \infty with \nu > 0 it approaches the negative binomial.

The Bessel functions are evaluated with RTMB's besselK for orders up to 50. For larger orders, besselK overflows when its argument is small, so its uniform asymptotic expansion for large orders is used instead, which is accurate to about 10^{-10} on the log scale there. Both are differentiable in the order, so \nu can be estimated.

The distribution function has no closed form and is computed by summing the probability mass function over 0, \ldots, q, so its cost grows with the largest q. It is AD-compatible in the parameters, while q must be numeric data. This is also what one-step-ahead (OSA) residuals via RTMB::oneStepPredict need, so these are supported, e.g. with method = "cdf" and discrete = TRUE.

Value

dsichel gives the probability mass function, psichel gives the distribution function, and rsichel generates random deviates.

References

Sichel, H. S. (1982) Repeat-buying and the generalized inverse Gaussian-Poisson distribution. Journal of the Royal Statistical Society, Series C, 31, 193-204.

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/.

Olver, F. W. J. et al. (eds.) NIST Digital Library of Mathematical Functions, equation 10.41.4, https://dlmf.nist.gov/10.41.

See Also

pig, nbinom2, bnbinom2

Examples

set.seed(123)
x <- rsichel(5, mu = 4, sigma = 0.5, nu = 1)
d <- dsichel(x, mu = 4, sigma = 0.5, nu = 1)
p <- psichel(x, mu = 4, sigma = 0.5, nu = 1)

RTMBdist documentation built on Oct. 31, 2026, 5:07 p.m.