The author deals with improvements of the classical asymptotic results for likelihood inference. The “improvements” refer to better asymptotic and numerical approximations and the results aim at parametric tests and confidence intervals. This paper is concerned with general results for well-behaved parametric models and with common types of models such as nonlinear regression or mixed effects models. The main results presented are test probabilities obtained by Barndorff-Nielsen’s statistic, \(r^{*}\), and its generalization for testing hypotheses involving several parameters. Examples illustrate the applicability and accuracy as well as the complexity of the required computations. The likelihood asymptotics are developed by merging two lines of research: asymptotic ancillarity is the basis of the statistical development, and saddlepoint approximations or Laplace-type approximations are simultaneously developed as the technical foundations.
Many other problems and ideas are presented. Among them are linear models with non-normal errors, nonparametric linear models obtained by estimation of the residual density in combination with the present results, and the generalization of the results to restricted maximum likelihood and similar structured models. The appendix of the paper contains a brief review of exponential family notation together with some technical points on the profile score and on relative errors in large deviation regions.