PowerCauchy {powdist}

The Power Cauchy Distribution

Description

Density, distribution function, quantile function and random generation for the power Cauchy distribution with parameters mu, sigma and lambda.

Usage

dpcauchy(x, lambda = 1, mu = 0, sigma = 1, log = FALSE)

ppcauchy(q, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

qpcauchy(p, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE,
  log.p = FALSE)

rpcauchy(n, lambda = 1, mu = 0, sigma = 1)

Arguments

x, q

vector of quantiles.

lambda

shape parameter.

mu, sigma

location and scale parameters.

log, log.p

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

lower.tail

logical; if TRUE (default), probabilities are P[X \le x ], otherwise, P[X > x].

p

vector of probabilities.

n

number of observations.

Details

The power Cauchy distribution has density

f(x)=\lambda\left [\frac{1}{\pi}\arctan\left ( \frac{x-\mu}{\sigma} \right )+\frac{1}{2} \right ]^{\lambda -1} \left[ \frac{1}{\pi\sigma\left( 1+\left (\frac{x-\mu}{\sigma} \right )^{2} \right)} \right],

where -\infty<\mu<\infty is the location paramether, \sigma^2>0 the scale parameter and \lambda>0 the shape parameter.

References

Anyosa, S. A. C. (2017) Binary regression using power and reversal power links. Master's thesis in Portuguese. Interinstitutional Graduate Program in Statistics. Universidade de São Paulo - Universidade Federal de São Carlos. Available in https://repositorio.ufscar.br/handle/ufscar/9016.

Bazán, J. L., Torres -Avilés, F., Suzuki, A. K. and Louzada, F. (2017) Power and reversal power links for binary regressions: An application for motor insurance policyholders. Applied Stochastic Models in Business and Industry, 33(1), 22-34.

Examples

dpcauchy(1, 1, 3, 4)
ppcauchy(1, 1, 3, 4)
qpcauchy(0.2, 1, 3, 4)
rpcauchy(5, 2, 3, 4)