Saddlepoint approximation in exponential models with boundary points

Abstract

A saddlepoint approximation is developed for the likelihood ratio test statistic in non-regular exponential models. For the one-parameter case, it is proved that the saddlepoint correction has a relative error of order $O\left(n^{-1}\right)$, assuming that the canonical statistic has four finite moments. The results are applied in reliability theory and survival analysis, testing for exponentiality. Note that we are using the saddlepoint method for statistics without moment generating function.