Differential Geometry, Profile Likelihood, $L$-Sufficiency and Composite Transformation Models
O. E. Barndorff-Nielsen, P. E. Jupp
The Annals of Statistics · 1988 · Volume 16 · Issue 3
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.
Keywords
62B05, 62F99, Alpha connections, ancillary, composite transformation models, connections, Cuts, distributional shape, expected information metric, likelihood, marginal likelihood, modified profile likelihood, observed alpha-connections, observed information metric, orthogonal parameters, parameter of interest, profile discrimination information, reproductive exponential models, statistical manifolds, submersion, tau-parallel foliations, tensors
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