# R/dctp.R In cpd: Complex Pearson Distributions

#' The Complex Triparametric Pearson (CTP) Distribution
#'
#' @description
#' Probability mass function, distribution function, quantile function and random generation for the Complex Triparametric Pearson (CTP) distribution with parameters \eqn{a}, \eqn{b} and \eqn{\gamma}.
#'
#' @usage
#' dctp(x, a, b, gamma)
#'
#' pctp(q, a, b, gamma, lower.tail = TRUE)
#'
#' qctp(p, a, b, gamma, lower.tail = TRUE)
#'
#' rctp(n, a, b, gamma)
#'
#' @param x vector of (non-negative integer) quantiles.
#' @param 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 a parameter a (real)
#' @param b parameter b (real)
#' @param gamma parameter \eqn{\gamma} (positive)
#' @param lower.tail if TRUE (default), probabilities are \eqn{P(X<=x)}, otherwise, \eqn{P(X>x)}.
#'
#' @details
#' The CTP distribution with parameters \eqn{a}, \eqn{b} and \eqn{\gamma} has pmf
#' \deqn{f(x|a,b,\gamma) = C \Gamma(a+ib+x) \Gamma(a-ib+x) / (\Gamma(\gamma+x) x!), x=0,1,2,...}
#' where \eqn{i} is the imaginary unit, \eqn{\Gamma(·)} the gamma function and
#' \deqn{C = \Gamma(\gamma-a-ib) \Gamma(\gamma-a+ib) / (\Gamma(\gamma-2a) \Gamma(a+ib) \Gamma(a-ib))}
#' the normalizing constant.
#'
#' If \eqn{a=0} the CTP is a Complex Biparametric Pearson (CBP) distribution, so the pmf of the CBP distribution is obtained.
#'
#' The mean and the variance of the CTP distribution are
#' \eqn{E(X)=\mu=(a^2+b^2)/(\gamma-2a-1)} and \eqn{Var(X)=\mu(\mu+\gamma-1)/(\gamma-2a-2)}
#' so \eqn{\gamma > 2a + 2}.
#'
#' It is underdispersed if \eqn{a < - (\mu + 1) / 2}, equidispersed if \eqn{a = - (\mu + 1) / 2} or overdispersed
#' if \eqn{a > - (\mu + 1) / 2}. In particular, if \eqn{a >= 0} the CTP is always overdispersed.
#'
#' @return
#' \code{dctp} gives the pmf, \code{pctp} gives the distribution function, \code{qctp} gives the quantile function and \code{rctp} generates random values.
#'
#' If \eqn{a = 0} the probability mass function, distribution function, quantile function and random generation function for the CBP distribution arise.
#'
#' @references
#' \insertRef{RCS2003}{cpd}
#'
#' \insertRef{RCSO2004}{cpd}
#'
#' \insertRef{ROC2018}{cpd}
#'
#' @seealso
#' Functions for maximum-likelihood fitting of the CTP and CBP distributions: \code{\link{fitctp}} and \code{\link{fitcbp}}.
#'
#' @examples
#' # Examples for the function dctp
#' dctp(3,1,2,5)
#' dctp(c(3,4),1,2,5)
#'
#' @name ctp
#'
#' @rdname ctp
#' @importFrom fAsianOptions cgamma
#' @export
#'
dctp <- function(x, a, b, gamma) {

if ( mode(c(x,a,b,gamma)) != "numeric")
stop( "non-numeric argument to mathematical function" )

if( gamma <= 2 * a )
stop("gamma must be greater than 2a")

x <- as.vector(x)
errors<-c()

for ( i in 1:length(x) ){
if ( round(x[[i]]) != x[[i]] || x[[i]]<0){
warning( paste ("non-integer or negative x[[", i, "]]", sep = ""))
errors <- c(errors,i)
}

}

if (b==0){
lpmf<-2*(lgamma(gamma-a)+lgamma(x+a)-lgamma(a))-lgamma(gamma-2*a)-lgamma(gamma+x)-lgamma(x+1)
} else {
i <- sqrt(as.complex(-1))
lpmf<-Re(cgamma(gamma-a-b*i,log=TRUE)+cgamma(gamma-a+b*i,log=TRUE)+cgamma(x+a+b*i,log=TRUE)+cgamma(x+a-b*i,log=TRUE)) -lgamma(gamma-2*a)-Re(cgamma(a+b*i,log=TRUE)+cgamma(a-b*i,log=TRUE))-lgamma(gamma+x)-lgamma(x+1)
}
result <- exp(lpmf)
for (i in 1:length(errors)){
result[ errors[[i]] ] = 0
}
return(result)
}

#' The Complex Biparametric Pearson (CBP) Distribution
#'
#' @description
#' Probability mass function, distribution function, quantile function and random generation for the Complex Biparametric Pearson (CBP) distribution with parameters \eqn{b} and \eqn{\gamma}.
#'
#' @usage
#' dcbp(x, b, gamma)
#'
#' pcbp(q, b, gamma, lower.tail = TRUE)
#'
#' qcbp(p, b, gamma, lower.tail = TRUE)
#'
#' rcbp(n, b, gamma)
#'
#' @param x vector of (non-negative integer) quantiles.
#' @param 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 b parameter b (real)
#' @param gamma parameter gamma (positive)
#' @param lower.tail if TRUE (default), probabilities are \eqn{P(X<=x)}, otherwise, \eqn{P(X>x)}.
#'
#' @details
#' The CBP distribution with parameters \eqn{b} and \eqn{\gamma} has pmf
#' \deqn{f(x|b,\gamma) = C \Gamma(ib+x) \Gamma(-ib+x) / (\Gamma(\gamma+x) x!), x=0,1,2,...}
#' where \eqn{i} is the imaginary unit, \eqn{\Gamma(·)} the gamma function and
#' \deqn{C = \Gamma(\gamma-ib) \Gamma(\gamma+ib) / (\Gamma(\gamma) \Gamma(ib) \Gamma(-ib))}
#' the normalizing constant.
#'
#' The CBP is a particular case of the CTP when \eqn{a=0}.
#'
#' The mean and the variance of the CBP distribution are
#' \eqn{E(X)=\mu=b^2/(\gamma-1)} and \eqn{Var(X)=\mu(\mu+\gamma-1)/(\gamma-2)}
#' so \eqn{\gamma > 2}.
#'
#' It is always overdispersed.
#'
#' @return
#' \code{dcbp} gives the pmf, \code{pcbp} gives the distribution function, \code{qcbp} gives the quantile function and \code{rcbp} generates random values.
#'
#' @references
#'
#' \insertRef{RCS2003}{cpd}
#'
#' @seealso
#' Probability mass function, distribution function, quantile function and random generation for the CTP distribution: \code{\link{dctp}}, \code{\link{pctp}}, \code{\link{qctp}} and \code{\link{rctp}}.
#' Functions for maximum-likelihood fitting of the CBP distribution: \code{\link{fitcbp}}.
#'
#' @examples
#' # Examples for the function dcbp
#' dcbp(3,2,5)
#' dcbp(c(3,4),2,5)
#'
#' @name cbp
#'
#' @rdname cbp
#' @importFrom fAsianOptions cgamma
#' @export
#'
#'

dcbp <- function(x, b, gamma) {

if ( mode(c(x,b,gamma)) != "numeric")
stop( "non-numeric argument to mathematical function" )

if( gamma <= 0)
stop("gamma must be greater than 0")

x <- as.vector(x)
errors<-c()

for ( i in 1:length(x) ){
if ( round(x[[i]]) != x[[i]] || x[[i]]<0){
warning( paste ("non-integer or negative x[[", i, "]]", sep = ""))
errors <- c(errors,i)
}

}

if (b==0){
lpmf<-2*(lgamma(gamma)+lgamma(x))-lgamma(gamma)-lgamma(gamma+x)-lgamma(x+1)
} else {
i <- sqrt(as.complex(-1))
lpmf<-Re(cgamma(gamma-b*i,log=TRUE)+cgamma(gamma+b*i,log=TRUE)+cgamma(x+b*i,log=TRUE)+cgamma(x-b*i,log=TRUE)) -lgamma(gamma)-Re(cgamma(b*i,log=TRUE)+cgamma(b*i,log=TRUE))-lgamma(gamma+x)-lgamma(x+1)
}
result <- exp(lpmf)
for (i in 1:length(errors)){
result[ errors[[i]] ] = 0
}
return(result)
}


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cpd documentation built on June 7, 2019, 9:02 a.m.