Saddlepoint approximation for the distribution function of the mean of random variables (Q1195587)

When we want to find an approximation for the distribution function, several saddlepoint methods are available. Most simple method is to integrate the approximated probability density function obtained by the saddlepoint method. However this integration may not be easily carried out. Another method is based on the inversion formula from the cumulant generating function to the distribution function. Recall that the saddlepoint is defined by the solution \(T\) of the equation \(\kappa'(T)=\bar x\), where \(\kappa(T)\) is a cumulant generating function. We consider the equation \(\kappa'(T)=\bar x+1/(nT)\) of \(T\). Its solution will be called the quasi-saddlepoint. Using the quasi-saddlepoint, we propose an alternate approximation formula for the distribution function.