Modern Likelihood-Frequentist Inference

Summary We offer an exposition of modern higher order likelihood inference and introduce software to implement this in a quite general setting. The aim is to make more accessible an important development in statistical theory and practice. The software, implemented in an R package, requires only that the user provide code to compute the likelihood function and to specify extra-likelihood aspects of the model, such as stopping rule or censoring model, through a function generating a dataset under the model. The exposition charts a narrow course through the developments, intending thereby to make these more widely accessible. It includes the likelihood ratio approximation to the distribution of the maximum likelihood estimator, that is the p∗ formula, and the transformation of this yielding a second-order approximation to the distribution of the signed likelihood ratio test statistic, based on a modified signed likelihood ratio statistic r∗. This follows developments of Barndorff-Nielsen and others. The software utilises the approximation to required Jacobians as developed by Skovgaard, which is included in the exposition. Several examples of using the software are provided.