Barndorff-Nielsen, O. E.; Jupp, P. E. Differential geometry, profile likelihood, L-sufficiency and composite transformation models. (English) Zbl 0702.62032 Ann. Stat. 16, No. 3, 1009-1043 (1988). Summary: Let \(\Omega\) denote the parameter space of a statistical model and let \({\mathcal K}\) 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 N. N. Chentsov [Statistical decision rules and optimal inference. (1982; Zbl 0484.62008)] and S. Amari [see “Differential-geometrical methods in statistics.” (1985; Zbl 0559.62001)], as well as the observed information metric and the observed \(\alpha\)-connections introduced by the first author in “Differential geometry in statistical inference”, 95-162 (1987; Zbl 0694.62001). Under certain conditions these geometric objects on \(\Omega\) can be transferred in a canonical purely differential-geometric way to \({\mathcal K}\). The transferred objects are related to structures on \({\mathcal K}\) 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. M. Rémon, Int. Stat. Rev. 52, 127-135 (1984; Zbl 0575.62007)] 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. Cited in 12 Documents MSC: 62F99 Parametric inference 53B20 Local Riemannian geometry 62B05 Sufficient statistics and fields 53B05 Linear and affine connections 53B21 Methods of local Riemannian geometry Keywords:alpha connections; ancillary; cuts; distributional shape; orthogonal parameters; parameter of interest; profile discrimination information; reproductive exponential models; statistical manifolds; submersion; tau- parallel foliations; tensors; expected information metric; observed information metric; derivatives of pseudolikelihood functions; profile likelihood; marginal likelihood; L-sufficient statistic; composite transformation models; Laplace approximation; integral representation Citations:Zbl 0484.62008; Zbl 0559.62001; Zbl 0694.62001; Zbl 0575.62007 PDFBibTeX XMLCite \textit{O. E. Barndorff-Nielsen} and \textit{P. E. Jupp}, Ann. Stat. 16, No. 3, 1009--1043 (1988; Zbl 0702.62032) Full Text: DOI