An explicit large-deviation approximation to one-parameter tests

Abstract

An approximation is derived for tests of one-dimensional hypotheses in a general regular parametric model, possibly with nuisance parameters. The test statistic is most conveniently represented as a modified log-likelihood ratio statistic, just as the R^*-statistic from Barndorff-Nielsen (1986). In fact, the statistic is identical to a version of R^*, except that a certain approximation is used for the sample space derivatives required for the calculation of R^*. With this approximation the relative error for large-deviation tail probabilities still tends uniformly to zero for curved exponential models. The rate may, however, be O(n^-1/2) rather than O(n^-1) as for R^*. For general regular models asymptotic properties are less clear but still good compared to other general methods. The expression for the statistic is quite explicit, involving only likelihood quantities of a complexity comparable to an information matrix. A numerical example confirms the highly accurate tail probabilities. A sketch of the proof is given. This includes large parts which, despite technical differences, may be considered an overview of Barndorff-Nielsen's derivation of the formulae for p^* and R^*.