A General Formulation for the Large-Sample Behaviour of a Class of Hypothesis Test Statistics

We bring together some strands of development concerning restricted likelihood ratio estimation and testing, including boundary hypothesis testing, going back to pioneering papers of Aitchison, Silvey and Chernoff for motivation. Thus, cases where the parameters are connected by a number of functional relationships, which may involve natural restrictions on the parameters and/or restrictions imposed by a null hypothesis, as well as situations where the null and alternate hypotheses place the true parameter at the boundary of disjoint subsets of the parameter space, are considered. Our asymptotic results are proved under clearly specified and minimal assumptions, which are probably close to the weakest possible. We illustrate with an example for distributions defined on the unit sphere in \(\mathbb {R}^{\varvec{d}}\) ${\mathbb{R}}^{\mathit{d}}$ .