Sichel Distribution

Density, distribution function, quantile function, and random generation for the Sichel distribution.

Usage

dsichel(x, mu = 1, sigma = 1, gamma = 1, log = FALSE)

psichel(q, mu = 1, sigma = 1, gamma = 1, lower.tail = TRUE, log.p = FALSE)

qsichel(p, mu = 1, sigma = 1, gamma = 1, lower.tail = TRUE, log.p = FALSE)

rsichel(n, mu = 1, sigma = 1, gamma = 1)

Arguments

x

numeric value or vector of non-negative integer values.

mu

numeric; mean of the distribution (mu > 0).

sigma

numeric; scale parameter (sigma > 0).

gamma

numeric; shape parameter (can be any real number).

log, log.p

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

q

quantile or vector of quantiles.

lower.tail

logical; if TRUE, probabilities are \(P[X <= x]\).

p

probability or vector of probabilities.

n

number of random values to generate.

Value

dsichel gives the density, psichel gives the distribution function, qsichel gives the quantile function, and rsichel generates random deviates.

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

Details

The Sichel distribution is a three-parameter discrete distribution that generalizes the Poisson-inverse Gaussian distribution. It is useful for modeling overdispersed count data.

The PMF is: $$f(y|\mu, \sigma, \gamma) = \frac{(\mu/c)^y K_{y+\gamma}(\alpha)}{K_\gamma(1/\sigma) y! (\alpha\sigma)^{y+\gamma}}$$

References

Rigby, R. A., Stasinopoulos, D. M., & Akantziliotou, C. (2008). A framework for modelling overdispersed count data, including the Poisson-shifted generalized inverse Gaussian distribution. Computational Statistics & Data Analysis, 53(2), 381-393.

Examples

# Basic usage
dsichel(0:10, mu = 5, sigma = 1, gamma = -0.5)
#>  [1] 0.09860584 0.14865390 0.14583707 0.12259787 0.09727854 0.07583610
#>  [7] 0.05907602 0.04630084 0.03659705 0.02918655 0.02347741

# Log-probabilities for numerical stability
dsichel(0:10, mu = 5, sigma = 1, gamma = -0.5, log = TRUE)
#>  [1] -2.316625 -1.906135 -1.925265 -2.098846 -2.330177 -2.579181 -2.828930
#>  [8] -3.072595 -3.307788 -3.534047 -3.751716

# CDF
psichel(5, mu = 5, sigma = 1, gamma = -0.5)
#> [1] 0.6888093