Saddlepoint approximations for nonlinear statistics

It is well known that saddlepoint expansions lead to accurate approximations to the cumulative distributions and densities of a sample mean and other simple linear statistics. The use of such expansions is explored in a broader situation. The saddlepoint formula for the tail probability of a certain type of nonlinear statistic is derived. The relative error of O(n –1 ), as in the linear case, is retained. A simple example is considered, to illustrate the great accuracy of the approximation.