# Evaluation of the saddlepoint of the Esscher transform with 
# parameter 1 of the GAL distribution for given quantiles
# 
# Author: Francois Pelletier
#
# LGPL 3.0
###############################################################################


#' Evaluation of the saddlepoint of the Esscher transform with 
#' parameter 1 of the GAL distribution for given quantiles
#' @param x vector of quantiles
#' @param param Parameters of the underlying GAL distribution
#' @param eval.time Time of the process
#' @param type Choose between "mu" or "kappa" parametrization
#' @param log Logical for log-parameters
#' @return The value of the saddlepoint for each point of the vector of quantiles
#' 
#' @author Francois Pelletier
saddlepointEsscherGAL <- function(x,param,eval.time=1,type="mu",log=FALSE)
{
	if(type=="mu")
	{
		(-eval.time*param[1]*param[2]^2+eval.time*param[4]*param[2]^2+x*param[2]^2-
					eval.time*param[1]*param[3]+x*param[3]-
					sqrt(eval.time^2*param[1]^2*param[3]^2-2*eval.time*param[1]*param[3]^2*x+
									eval.time^2*param[4]^2*param[2]^4+x^2*param[3]^2+
									2*eval.time^2*param[1]^2*param[2]^2-
									4*eval.time*param[1]*param[2]^2*x+2*x^2*param[2]^2))/
				(param[2]^2*(eval.time*param[1]-x))
	}
	else if (type=="kappa")
	{
		
	}
}