Bounds for probabilities of small relative errors for empirical saddlepoint and bootstrap tail approximations

To obtain test probabilities based on empirical approximations to the distribution of a Studentized function of a mean, we need the approximations to be accurate with sufficiently high probability. In particular, when these test probabilities are small it is best to consider relative errors. Here we show that in the case of univariate standardized means and in the general case of tests based on smooth functions of means, the empirical approximations have asymptotically small relative errors on sets with probability differing from 1 by an exponentially small quantity and that these error rates hold for moderately large deviations. In particular, for standardized deviations of order n(1/6), the probabilities approximated are exponentially small with exponents of order n(1/3) and the corresponding relative errors tend to zero on sets whose complements have probabilities of the order of the probabilities being approximated.