#' @title Unconstrained parameters
#'
#' @description This function computes the unconstrainted parameter alpha for a given Kendall's tau
#'
#' @param family 'Gaussian' , 't' , 'Clayton' , 'Frank' , 'Gumbel'
#' @param tau Kendall's tau of the copula family
#'
#' @author Bouchra R. Nasri, August 14, 2019
#' @return \item{alpha}{Unconstrainted parameter}
#'
#' @examples ParamTau('Clayton',0.5)
#'
#' @export
ParamTau<- function(family,tau){
switch(family,
"Gaussian" = {
theta = copula::iTau(copula::normalCopula(), tau) ;
alpha = log( (1+theta) / (1-theta) );
},
"t" = {
theta = copula::iTau(copula::normalCopula(), tau) ;
alpha = log( (1+theta) / (1-theta) );
},
"Clayton" = {
theta = copula::iTau(copula::claytonCopula(), tau);
alpha = log(theta);
},
"Frank" = {
theta = copula::iTau(copula::frankCopula(), tau);
alpha=theta;
},
"Gumbel" = {
theta = copula::iTau(copula::gumbelCopula(), tau );
alpha = log(theta-1);
}
)
return(alpha)
}