Higher Order Relations between Cornish-fisher Expansions and Inversions of Saddlepoint Approximations

In many numerical examples it has been demonstrated that the saddlepoint approximation for the cumulative distribution function of a general nor-malised statistic behaves better in comparison with the third order Edge-worth expansion. The eeect is especially pronounced in the tails. Here we are dealing with the inverse problem of quantile evaluation. The inversion 1 of the Lugannani-Rice approximation is compared with the Cornish-Fisher expansion both theoretically and numerically. We show in a very general setting that the expansion of the inversion of the Lugannani-Rice approximation up to third order coincides with the Cornish-Fisher expansion. Based on this, an explanation of the superiority of the former in comparison with the latter in the tails and for small samples is given. An explicit approximation of the inversion of the Lugannani-Rice formula is suggested that utilizes the information in the cumulant generating function and improves upon the Cornish-Fisher formula.