Third order asymptotic models: likelihood functions leading to accurate approximations for distribution functions

A very accurate saddlepoint approximation formula for the distribution function of the sample average was obtained by Lugannani and Rice (1980), and reformulated for exponential models in terms of likelihood by Daniels (1958, 1987). In this paper we obtain a simple third order asymptotic correspondence between cumulant generating functions and corresponding log density functions, leading to a correspondence between likelihood functions and distribution functions; a multivariate analog is also obtained. We use this correspondence to establish the third order accuracy of the invariant tail probability formula proposed by Fraser (1990) and of the conversion of conditional likelihood for an exponential model and marginal likelihood for a location model to tail probabilities for testing scalar parameters.