Open Access
Translator Disclaimer
September, 1983 Exponential Models with Affine Dual Foliations
O. Barndorff-Nielsen, P. Blaesild
Ann. Statist. 11(3): 753-769 (September, 1983). DOI: 10.1214/aos/1176346243


Suppose an exponential model $\mathscr{M}$ is partitioned into submodels, all of the same parametric dimension. If each of these models corresponds to a linear hypothesis about the canonical parameter and also to a linear hypothesis about the mean parameter, we speak of the partitioning of $\mathscr{M}$ as an affine dual foliation. We study those cases where the parameter sets defining the hypotheses are parallel either in the canonical space or in the mean space, and obtain various characterisations and properties of these cases. It is shown, inter alia, that canonical parallelism and mean parallelism are related to likelihood independence of $\theta_1$ and $\tau_2$ (and hence to $S$-ancillarity and $S$-sufficiency), respectively to stochastic independence of $\hat{\theta}_1$ and $\hat{\tau}_2$. Here $(\theta_1, \tau_2)$ denotes a mixed parametrisation of $\mathscr{M}$ and $\hat{\theta}_1$ and $\hat{\tau}_2$ are the maximum likelihood estimators of $\theta_1$ and $\tau_2$. Also, the two types of parallelism are characterised in terms of observed and expected information. Mean parallelism is closely related to the concept of reproductivity of exponential models that forms the subject of a separate paper. A number of requisite general results for exponential families are established, and these are also of some independent interest.


Download Citation

O. Barndorff-Nielsen. P. Blaesild. "Exponential Models with Affine Dual Foliations." Ann. Statist. 11 (3) 753 - 769, September, 1983.


Published: September, 1983
First available in Project Euclid: 12 April 2007

zbMATH: 0529.62015
MathSciNet: MR707927
Digital Object Identifier: 10.1214/aos/1176346243

Primary: 62E15
Secondary: 62F99

Rights: Copyright © 1983 Institute of Mathematical Statistics


Vol.11 • No. 3 • September, 1983
Back to Top