# dsc: Skew-Cauchy Distribution In sn: The Skew-Normal and Related Distributions Such as the Skew-t

## Description

Density function, distribution function, quantiles and random number generation for the skew-Cauchy (SC) distribution.

## Usage

 ```1 2 3 4``` ```dsc(x, xi = 0, omega = 1, alpha = 0, dp = NULL, log = FALSE) psc(x, xi = 0, omega = 1, alpha = 0, dp = NULL) qsc(p, xi = 0, omega = 1, alpha = 0, dp = NULL) rsc(n = 1, xi = 0, omega = 1, alpha = 0, dp = NULL) ```

## Arguments

 `x` vector of quantiles. Missing values (`NA`s) and `Inf`'s are allowed. `p` vector of probabilities. Missing values (`NA`s) are allowed. `xi` vector of location parameters. `omega` vector of (positive) scale parameters. `alpha` vector of slant parameters. `dp` a vector of length 3 whose elements represent the parameters described above. If `dp` is specified, the individual parameters cannot be set. `n` sample size. `log` logical flag used in `dsc` (default `FALSE`). When `TRUE`, the logarithm of the density values is returned.

## Value

density (`dsc`), probability (`psc`), quantile (`qsc`) or random sample (`rsc`) from the skew-Cauchy distribution with given `xi`, `omega` and `alpha` parameters or from the extended skew-normal if `tau!=0`

## Details

Typical usages are

 ```1 2 3 4 5 6 7 8``` ```dsc(x, xi=0, omega=1, alpha=0, log=FALSE) dsc(x, dp=, log=FALSE) psc(x, xi=0, omega=1, alpha=0) psc(x, dp= ) qsc(p, xi=0, omega=1, alpha=0) qsc(x, dp=) rsc(n=1, xi=0, omega=1, alpha=0) rsc(x, dp=) ```

## Background

The skew-Cauchy distribution can be thought as a skew-t with tail-weight parameter `nu=1`. In this case, closed-form expressions of the distribution function and the quantile function have been obtained by Behboodian et al. (2006). The key facts are summarized in Complement 4.2 of Azzalini and Capitanio (2014). A multivariate version of the distribution exists.

## References

Azzalini, A. with the collaboration of Capitanio, A. (2014). The Skew-normal and Related Families. Cambridge University Press, IMS Monographs series.

Behboodian, J., Jamalizadeh, A., and Balakrishnan, N. (2006). A new class of skew-Cauchy distributions. Statist. Probab. Lett. 76, 1488–1493.

`dst`, `dmsc`
 ```1 2 3 4 5``` ```pdf <- dsc(seq(-5,5,by=0.1), alpha=3) cdf <- psc(seq(-5,5,by=0.1), alpha=3) q <- qsc(seq(0.1,0.9,by=0.1), alpha=-2) p <- psc(q, alpha=-2) rn <- rsc(100, 5, 2, 5) ```