The Annals of Statistics

Exponential Models with Affine Dual Foliations

O. Barndorff-Nielsen and P. Blaesild

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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.

Article information

Ann. Statist., Volume 11, Number 3 (1983), 753-769.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62E15: Exact distribution theory
Secondary: 62F99: None of the above, but in this section

Cuts duality expected information independence Legendre transforms mixed parametrisations observed information pivots profile likelihood


Barndorff-Nielsen, O.; Blaesild, P. Exponential Models with Affine Dual Foliations. Ann. Statist. 11 (1983), no. 3, 753--769. doi:10.1214/aos/1176346243.

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