Likelihood asymptotics

The paper gives an overview of modern likelihood asymptotics with emphasis on results and applicability. Only parametric inference in well-behaved mod els is considered and the theory discussed leads to highly accurate asympto tic tests for general smooth hypotheses. The tests are refinements of the u sual asymptotic likelihood ratio tests, and for one-dimensional hypotheses the test statistic is known as r*, introduced by Barndorff-Nielsen. Example s illustrate the applicability and accuracy as well as the complexity of th e required computations. Modern likelihood asymptotics has developed by mer ging two lines of research: asymptotic ancillarity is the basis of the stat istical development, and saddlepoint approximations or Laplace-type approxi mations have simultaneously developed as the technical foundation. The main results and techniques of these two lines will be reviewed, and a generali zation to multi-dimensional tests is developed. In the final part of the pa per further problems and ideas are presented. Among these are linear models with non-normal error, non-parametric linear models obtained by estimation of the residual density in combination with the present results, and the g eneralization of the results to restricted maximum likelihood and similar s tructured models.