Modern Likelihood‐Frequentist Inference

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.