Empirical saddlepoint approximations of the Studentized mean under simple random sampling

We obtain a saddlepoint approximation for the Studentized mean of a simple random sample taken without replacement from a finite population. This is only possible if we know the entire population, so we also obtain an empirical saddlepoint approximation based on the sample alone. This empirical approximation can be used for tests of significance and confidence intervals for the population mean. We compare the empirical approximation to the true saddlepoint approximation, both theoretically and numerically. We also compare both approximations to values obtained in a large Monte Carlo simulation for a population of survival times. The comparisons show that good accuracy can be obtained from the empirical saddlepoint approximation. In addition, the approximations are compared numerically to the Edgeworth approximation of Sugden and Smith (Statist. Probab. Lett. 34 (3) (1997) 293-299; Statist. Probab. Lett. 37 (3) (1998) 317), with a correction in Sugden et al. (J. Roy. Statist. Soc. B 62 (2000) 787-794).