R/dpcauchy.R

In anyosa/powdist: Power and Reversal Power Distributions

#' @title The Power Cauchy Distribution
#' @name PowerCauchy
#' @description Density, distribution function,
#' quantile function and random generation for
#' the power Cauchy distribution with parameters mu, sigma and lambda.
#' @param x,q vector of quantiles.
#' @param p vector of probabilities.
#' @param n number of observations.
#' @param lambda shape parameter.
#' @param mu,sigma location and scale parameters.
#' @param log,log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are \eqn{P[X \le x ]}, otherwise, P[X > x].
#' @references Anyosa, S. A. C. (2017) \emph{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 \url{https://repositorio.ufscar.br/handle/ufscar/9016}.
#' @references 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. \emph{Applied Stochastic Models in Business and Industry}, \strong{33}(1), 22-34.
#' @importFrom stats runif
#' @importFrom stats dcauchy
#' @importFrom stats pcauchy
#' @importFrom stats qcauchy
#' @details The power Cauchy distribution has density
#'
#' \eqn{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]}{f(x)=[\lambda/\sigma][1/(\pi(1+(x-\mu)/\sigma))][(1/\pi)arctan((x-\mu)/\sigma)+(1/2)]^(\lambda-1)},
#'
#' where \eqn{-\infty<\mu<\infty} is the location paramether, \eqn{\sigma^2>0} the scale parameter and \eqn{\lambda>0} the shape parameter.
#'
#' @examples
#' dpcauchy(1, 1, 3, 4)
#' @export
dpcauchy <- function(x, lambda = 1, mu = 0, sigma = 1, log = FALSE){
  d = (lambda/sigma) * dcauchy((x-mu)/sigma) * (pcauchy((x-mu)/sigma)**(lambda-1))
  if (log == TRUE) {
    d = log( (lambda/sigma) * dcauchy((x-mu)/sigma) * (pcauchy((x-mu)/sigma)**(lambda-1)) )
  }
  return(d)
}