Differential Geometry, Profile Likelihood, $L$-Sufficiency and Composite Transformation Models

Abstract

Let $\Omega$ denote the parameter space of a statistical model and let $\mathscr{K}$ be the domain of variation of the parameter of interest. Various differential-geometric structures on $\Omega$ are considered, including the expected information metric and the $\alpha$-connections studied by Chentsov and Amari, as well as the observed information metric and the observed $\alpha$-connections introduced by Barndorff-Nielsen. Under certain conditions these geometric objects on $\Omega$ can be transferred in a canonical purely differential-geometric way to $\mathscr{K}$. The transferred objects are related to structures on $\mathscr{K}$ obtained from derivatives of pseudolikelihood functions such as the profile likelihood, the modified profile likelihood and the marginal likelihood based on an $L$-sufficient statistic (cf. Remon) when such a statistic exists. For composite transformation models it is shown that the modified profile likelihood is very close to the Laplace approximation to a certain integral representation of the marginal likelihood.