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

Barndorff-Nielsen, O. E. ; Jupp, P. E.

Ann. Statist., Tome 16 (1988) no. 1, p. 1009-1043 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1988-09-14
Classification:  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,  62F99,  62B05