On the empirical approximation to quantiles from Lugannani–Rice saddlepoint formula

Lugannani-Rice saddlepoint formula approximates the tail probability and the cumulative distribution function of the sample mean of independent and equally distributed variables. This note revisits Lugannani-Rice formula with a proposal for inverting it to approximate the quantile of the distribution empirically. The asymptotic behavior of the empirical approximation is assessed theoretically and its numerical accuracy for finite samples is studied and compared with the normal approximation and the second order Cornish-Fisher expansion by means of a simulation study. The outcomes of the simulation experiment shed light on the limitations of the empirical inversions of saddlepoint formulae to approximate quantiles.