Cauchy function

Cauchy distribution

Computes the pdf, cdf, value at risk and expected shortfall for the Cauchy distribution given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef (x) = \frac {1}{\pi} \frac {\sigma}{(x - \mu)^2 + \sigma^2},\\&\displaystyleF (x) = \frac {1}{2} + \frac {1}{\pi} \arctan \left( \frac {x - \mu}{\sigma} \right),\\&\displaystyle{\rm VaR}_p (X) = \mu + \sigma \tan \left( \pi \left( p - \frac {1}{2} \right) \right),\\&\displaystyle{\rm ES}_p (X) = \mu + \frac {\sigma}{p} \int_0^p \tan \left( \pi \left( v - \frac {1}{2} \right) \right) dv\end{array} [REMOVE_ME_2]$

for $-\infty < x < \infty$, $0 < p < 1$, $-\infty < \mu < \infty$, the location parameter, and $\sigma > 0$, the scale parameter.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Cauchy distribution given by

for $-\infty < x < \infty$, $0 < p < 1$, $-\infty < \mu < \infty$, the location parameter, and $\sigma > 0$, the scale parameter.

dCauchy(x, mu=0, sigma=1, log=FALSE)
pCauchy(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
varCauchy(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
esCauchy(p, mu=0, sigma=1)

Arguments

• x: scaler or vector of values at which the pdf or cdf needs to be computed
• p: scaler or vector of values at which the value at risk or expected shortfall needs to be computed
• mu: the value of the location parameter, can take any real value, the default is zero
• sigma: the value of the scale parameter, must be positive, the default is 1
• log: if TRUE then log(pdf) are returned
• log.p: if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
• lower.tail: if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Returns

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, tools:::Rd_expr_doi("10.1080/03610918.2014.944658")

Author(s)

Saralees Nadarajah

Examples

x=runif(10,min=0,max=1)
dCauchy(x)
pCauchy(x)
varCauchy(x)
esCauchy(x)