Preben Blæsild

The Fascination of Sand

The physics of wind-blown sand poses a variety of intriguing problems whose proper solution seems to require statistical ideas and methods, partly new. The main traits of the processes of transport and sorting of sand particles by wind or by water are described, and a review is given of results obtained and of subjects for further study. Various aspects of general statistical interest are also discussed; in particular, properties and applications of hyperbolic and certain related distributions are outlined. Other, nontechnical ramifications are considered, and this part of the discussion includes a brief biography of R. A. Bagnold, the founder of the physics of blown sands as a scientific field.

Exponential transformation models

The class of exponential transformation models, i. e. transformation models that are also exponential families, is investigated. It is shown, by using standard exponential family theory, that the group of transformations on the sample space generating the model induces affine transformation groups on the parameter space and on the range space of the minimal sufficient statistic, the induced groups being in fact representations of the original transformation group. The model function may be explicitly expressed in terms of these representations and it possesses a number of important properties. The possibility of extending exponential transformation models to larger (composite) exponential transformation models is considered, such extensions serving, for instance, the purpose of model control. The inference for (composite) exponential transformation models is described in general terms and examples of such models are discussed. The paper starts with an introduction to transformation models and a unified treatment of their distribution theory, required in the subsequent discussion of the exponential transformation models.

Multivariate Distributions of Hyperbolic Type

The family of generalized d-dimensional hyperbolic distributions is introduced and shown to be closed under margining, conditioning and affine transformation and to contain as well multivariate location-scale submodels as exponential submodels. Two members of this family, the d-dimensional hyperbolic distribution, which describes a specific form of non-normal variation, and the d-dimensional hyperboloid distribution, an analogue of the von Mises-Fisher distribution, are discussed in more detail and applications of these distributions are given.

Decomposition and Invariance of Measures and Statistical Transformation Models

1. Introduction.- 2. Topological groups and actions.- 3. Matrix Lie groups.- 4. Invariant, relatively invariant, and quasi-invariant measures.- 5. Decomposition and factorization of measures.- 6. Construction of invariant measures.- 7. Exterior calculus.- 8. Statistical transformation models.- Further results and exercises.- References, with author index.- Notation index.

Multivariate distributions with generalized inverse gaussian marginals, and associated poisson mixtures

Several types of multivariate extensions of the inverse Gaussian (IG) distribution and the reciprocal inverse Gaussian (RIG) distribution are proposed. Some of these types are obtained as random-additive-effect models by means of well-known convolution properties of the IG and RIG distributions, and they have one-dimensional IG or RIG marginals. They are used to define a flexible class of multivariate Poisson mixtures.

Coordinate-free definition of structurally symmetric derivative strings

On donne une definition independante des coordonnees des cordes derivees structurellement symetriques

Yokes and tensors derived from yokes

A yoke on a differentiable manifold M gives rise to a whole family of derivative strings. Various elemental properties of a yoke are discussed in terms of these strings. In particular, using the concept of intertwining from the theory of derivative strings it is shown that a yoke induces a family of tensors on M. Finally, the expected and observed α-geometries of a statistical model and related tensors are shown to be derivable from particular yokes.A yoke on a differentiable manifold M gives rise to a whole family of derivative strings. Various elemental properties of a yoke are discussed in terms of these strings. In particular, using the concept of intertwining from the theory of derivative strings it is shown that a yoke induces a family of tensors on M. Finally, the expected and observed α-geometries of a statistical model and related tensors are shown to be derivable from particular yokes.

Derivative strings: contravariant aspect

A concept termed strings, which generalizes those of derivatives of scalars, tensors and affine connections, was introduced and studied in two previous papers by the authors. The results obtained were, essentially, of a ‘covariant nature’. In this paper a corresponding contravariant theory is developed and combined with the previous results. A fairly complete theoretical framework is thereby established. In particular, the previously defined operations of intertwining and of covariant differentiation for strings are generalized to arbitrary strings of mixed covariant-contravariant type and further properties of these operations are derived. To distinguish strings in the sense of this and the two foregoing papers from various other concepts that are also termed strings and that are now of great interest in physics and cosmology, we shall occasionally refer to the present concept as derivative strings, as has already been done in the title.

Maximum likelihood estimation in exponential orthogeodesic models

An orthogeodesic statistical model is defined in terms of five conditions of differential geometric nature. These conditions are reviewed together with a characterization theorem for exponential orthogeodesic models. Orthogonal projections, relevant for maximum likelihood estimation in exponential orthogeodesic models, are described in a simple way in terms of some of the quantities in the characterization theorem. A unified procedure for performing maximum likelihood estimation in exponential orthogenodesic models is given and the use of this procedure is illustrated for some of the most important models of this kind such as ϑ-parallel models, τ-parallel models and certain transformation models.An orthogeodesic statistical model is defined in terms of five conditions of differential geometric nature. These conditions are reviewed together with a characterization theorem for exponential orthogeodesic models. Orthogonal projections, relevant for maximum likelihood estimation in exponential orthogeodesic models, are described in a simple way in terms of some of the quantities in the characterization theorem. A unified procedure for performing maximum likelihood estimation in exponential orthogenodesic models is given and the use of this procedure is illustrated for some of the most important models of this kind such as ϑ-parallel models, τ-parallel models and certain transformation models.

Construction of invariant measures

We shall discuss here methods of constructing a G-invariant measure µ on the space χ, in the sense of expressing µ on the form \(\chi \left( {k,x} \right) = 1,x \in \chi ,k \in {G_x}.\)