#' Pareto type II distribution
#'
#' Provides density, distribution function, quantile function, random value generation
#' and raw moments of order \emph{r} for the Pareto type II distribution.
#'
#' @param x,q vector of quantiles.
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1}, the length is taken to be the number required.
#' @param r raw moment order
#' @param scale,shape scale and shape parameters; the shape argument cannot be a vector (must have length one).
#' @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, \eqn{P[X > x]}.
#'
#' @name ParetoII
#' @aliases ParetoII
#' @aliases dparetoII
#'
#' @keywords distribution
#' @concept Univariate
#' @concept Continuous
#'
#' @import stats ggplot2
#' @export
#'
#' @examples
#'
#' ## plot the density
#'
#' ggplot(data.frame(x = c(0, 20)),
#' aes(x)) +
#' stat_function(fun = dparetoII,
#' args = list(scale = 1,
#' shape = .3),
#' colour = 'royalblue4') +
#' labs(x = '',
#' y = 'Density') +
#' theme_classic()
#'
dparetoII <- function(x, scale, shape, log = FALSE) { ## density
if((scale <= 0) | (shape <= 0)) {
return(NaN)
} else {
d <- (1 + (shape*x)/scale)^(-1 - 1/shape)/scale
if (log) {d <- log(d)}
return(d)
}
}
#' @rdname ParetoII
#' @export
pparetoII <- function(q, scale, shape, lower.tail = TRUE, log.p = FALSE) { ## cdf
if((scale <= 0) | (shape <= 0)) {
return(NaN)
} else {
p <- 1 - (1 + (shape*q)/scale)^(-1/shape)
if (!lower.tail) {p <- 1 - p}
if (log.p) {p <- log(p)}
return(p)
}
}
#' @rdname ParetoII
#' @export
qparetoII <- function(p, scale, shape, lower.tail = TRUE, log.p = FALSE) { ## quantile function
if((scale <= 0) | (shape <= 0)) {
return(NaN)
} else {
if (!lower.tail) {p <- 1 - p}
if (log.p) {p <- log(p)}
q <- (scale*(-1 + (1 - p)^(-shape)))/shape
return(q)
}
}
#' @rdname ParetoII
#' @export
rparetoII <- function(n, scale, shape) { ## random numbers generator
if((scale <= 0) | (shape <= 0)) {
return(NaN)
} else {
return(qparetoII(pnorm(rnorm(n)), scale, shape))
}
}
#' @rdname ParetoII
#' @export
mparetoII <- function(r, scale, shape) {
if((scale <= 0) | (shape <= 0)) {
return(NaN)
} else {
return(((scale/shape)^r*gamma(1/shape - r)*gamma(1 + r))/gamma(1/shape))
}
}
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