The Annals of Statistics

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

O. E. Barndorff-Nielsen and P. E. Jupp

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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.

Article information

Ann. Statist. Volume 16, Number 3 (1988), 1009-1043.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F99: None of the above, but in this section
Secondary: 62B05: Sufficient statistics and fields

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


Barndorff-Nielsen, O. E.; Jupp, P. E. Differential Geometry, Profile Likelihood, $L$-Sufficiency and Composite Transformation Models. Ann. Statist. 16 (1988), no. 3, 1009--1043. doi:10.1214/aos/1176350946.

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