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#' @title Beta-Binomial distribution
#' @description Return the Beta-Binomial distribution with parameters \eqn{m}, \eqn{csi} and \eqn{phi}.
#' @aliases betar
#' @usage betar(m,csi,phi)
#' @param m Number of ordinal categories
#' @param csi Feeling parameter of the Beta-Binomial distribution
#' @param phi Overdispersion parameter of the Beta-Binomial distribution
#' @export betar
#' @return The vector of length \eqn{m} of the Beta-Binomial distribution.
#' @seealso \code{\link{betabinomial}}
#' @references Iannario, M. (2014). Modelling Uncertainty and Overdispersion in Ordinal Data,
#' \emph{Communications in Statistics - Theory and Methods}, \bold{43}, 771--786
#' @keywords distribution
#' @examples
#' m<-9
#' csi<-0.8
#' phi<-0.2
#' pr<-betar(m,csi,phi)
#' plot(1:m,pr,type="h", main="Beta-Binomial distribution",xlab="Ordinal categories")
#' points(1:m,pr,pch=19)
betar <-function(m,csi,phi){
betar<-rep(NA,m)
km<-0:(m-2)
betar[1]<-prod(1-(1-csi)/(1+phi*km))
for(r in 1:(m-1)){
betar[r+1]<-betar[r]*((m-r)/r)*((1-csi+phi*(r-1))/(csi+phi*(m-r-1)))
}
return(betar)
}
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