An example in which the Lugannani-Rice saddlepoint formula fails

Abstract

In almost all documented applications, the normal-based Lugannani and Rice (1980) formula provides an extremely accurate approximation to tail probabilities. However, it would be wrong to conclude that this formula is always reliable. In the present paper, we consider an example, the first passage time of a random walk with drift, in which the overall performance of the normal-based Lugannani and Rice formula is rather poor. In contrast, a modified Lugannani and Rice formula, in which the normal base is replaced by an inverse Gaussian base, gives an excellent approximation. A similar modification of Barndorff-Nielsen's (1986) formula is also considered. The main focus of this paper is on a detailed study of the approximations in the extreme right tail of the distribution, and our theoretical results go some of the way towards explaining the observed numerical behaviour. However, on closer scrutiny, there is a disconcerting lack of correspondence between some of the theoretical limits and the numerical results, which appears to be due to convergence rates of some quantities being extremely slow.