#' Burr Type III distribution
#'
#' Provides density, distribution function, quantile function, random value generation,
#' and raw moments of order \emph{r} for the Burr Type III 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,shape1,shape2 scale and shape parameters; the shape arguments cannot be a vectors (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 BurrIII
#' @aliases BurrIII
#' @aliases dburrIII
#'
#' @keywords distribution
#' @concept Univariate
#' @concept Continuous
#'
#' @import stats ggplot2
#' @export
#'
#' @examples
#'
#' ## plot the density
#'
#' ggplot(data.frame(x = c(1, 15)),
#' aes(x)) +
#' stat_function(fun = dburrIII,
#' args = list(scale = 5,
#' shape1 = .25,
#' shape2 = .75),
#' colour = 'royalblue4') +
#' labs(x = '',
#' y = 'Density') +
#' theme_classic()
dburrIII <- function(x, scale, shape1, shape2, log = FALSE) { ## density
if((shape1 <= 0) | (shape2 <= 0)) {
return(NaN)
} else {
d <- ((x/scale)^(-1 - shape2^(-1))*(1 + 1/(shape1*(x/scale)^shape2^(-1)))^(-1 - shape1*shape2))/scale
if (log) {d <- log(d)}
return(d)
}
}
#' @rdname BurrIII
#' @export
pburrIII <- function(q, scale, shape1, shape2, lower.tail = TRUE, log.p = FALSE) { ## cdf
if((shape1 <= 0) | (shape2 <= 0)) {
return(NaN)
} else {
p <- (1 + 1/(shape1*(q/scale)^shape2^(-1)))^(-(shape1*shape2))
if (!lower.tail) {p <- 1 - p}
if (log.p) {p <- log(p)}
return(p)
}
}
#' @rdname BurrIII
#' @export
qburrIII <- function(p, scale, shape1, shape2, lower.tail = TRUE, log.p = FALSE) { ## quntile
if((shape1 <= 0) | (shape2 <= 0)) {
return(NaN)
} else {
if (!lower.tail) {p <- 1 - p}
if (log.p) {p <- log(p)}
q <- scale*(shape1*(p^(-1/(shape1*shape2)) - 1))^(-shape2)
return(q)
}
}
#' @rdname BurrIII
#' @export
rburrIII <- function(n, scale, shape1, shape2) {
if((shape1 <= 0) | (shape2 <= 0)) {
return(NaN)
} else {
return(qburrIII(runif(n), scale, shape1, shape2))
}
}
#' @rdname BurrIII
#' @export
mburrIII <- function(r, scale, shape1, shape2) {
if((shape1 <= 0) | (shape2 <= 0)) {
return(NaN)
} else {
return((scale^r*gamma((r + shape1)*shape2)*gamma(1 - r*shape2))/(shape1^(r*shape2)*gamma(shape1*shape2)))
}
}
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