| Abstract: | ABSTRACT. In this paper, we develop an approximation for the most powerful invariant test of one location-scale family against another one. The approach is based on the Laplace method for integrals and yields a very accurate approximation of the density of a maximal invariant. Moreover, it can be applied to a much wider set of pairs of densities than previously possible. Many examples are worked out. The resulting test is easy to compute and its power is shown to be very close to that of the best test. By using versions of the Laplace method, the approach is extended to goodness-of-fit tests for residuals in regression and to some multivariate distributions. A small simulation study confirms the theoretical results. An example concludes the paper. |
