Inverting a saddlepoint approximation

This paper is concerned with the inversion of a saddlepoint approximation for the tail probability of an asymptotically Normal statistic with cumulants expandable in powers of n-1/2. The inversion yields to an approximation for the quantile of the distribution of the statistic that is compared, both theoretically and numerically, with other well-known approximations, such as the normal one and the second-order Cornish-Fisher expansion.