## The Annals of Statistics

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

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

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

#### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176350946

**Digital Object Identifier**

doi:10.1214/aos/1176350946

**Mathematical Reviews number (MathSciNet)**

MR959187

**Zentralblatt MATH identifier**

0702.62032

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F99: None of the above, but in this section

Secondary: 62B05: Sufficient statistics and fields

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1176350946