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Differential geometry, profile likelihood, L-sufficiency and composite transformation models. (English) Zbl 0702.62032

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.

MSC:

62F99 Parametric inference
53B20 Local Riemannian geometry
62B05 Sufficient statistics and fields
53B05 Linear and affine connections
53B21 Methods of local Riemannian geometry
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