Decomposition and Invariance of Measures, and Statistical Transformation Models

Barndorff–Nielsen, Ole E.; Blæsild, P.; Eriksen, Poul Svante

doi:10.1007/978-1-4612-3682-5

Cited by 36 publications

(21 citation statements)

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“…Thus the measure µ is decomposed (disintegrated) in the sense of [12,Section 5] or [14] into the marginal measure π * µ on X /K and the conditional measures ν [x] on the fibres π −1 ([x]). Given a density f on X , definef 0 andΓ bỹ…”

Section: A General Setting For Symmetrymentioning

confidence: 99%

A general setting for symmetric distributions and their relationship to general distributions

Jupp

Regoli

Azzalini

2016

Journal of Multivariate Analysis

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Section: A General Setting For Symmetrymentioning

confidence: 99%

A general setting for symmetric distributions and their relationship to general distributions

Jupp

Regoli

Azzalini

2016

Journal of Multivariate Analysis

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“…For group invariance in statistics, the reader is referred to Eaton [7], Barndorff-Nielsen et al [2] and Wijsman [26]. For global cross sections and orbital decompositions in particular, see Wijsman [24,25], Koehn [19], Bondar [3] and Kamiya [15].…”

Section: Orbital Decomposition and Decomposable Distributionmentioning

confidence: 99%

Hierarchical orbital decompositions and extended decomposable distributions

Kamiya

Takemura

2008

Journal of Multivariate Analysis

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Elliptically contoured distributions can be considered to be the distributions for which the contours of the density functions are proportional ellipsoids. Kamiya, Takemura and Kuriki [Star-shaped distributions and their generalizations, J. Statist. Plann. Inference, 2006, available at http://arxiv.org/abs/math.ST/0605600 , to appear] generalized the elliptically contoured distributions to star-shaped distributions, for which the contours are allowed to be arbitrary proportional star-shaped sets. This was achieved by considering the so-called orbital decomposition of the sample space in the general framework of group invariance. In the present paper, we extend their results by conducting the orbital decompositions in steps and obtaining a further, hierarchical decomposition of the sample space. This allows us to construct probability models and distributions with further independence structures. The general results are applied to the star-shaped distributions with a certain symmetric structure, the distributions related to the two-sample Wishart problem and the distributions of preference rankings.

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“…Suppose 8 E 8 1 is a coordinatization of the points in the orbits. That is, any pre-shape T can be completely specified by the pair (a, 8), where a is the shape of the configuration and 8 is its orientation. In more technical language, we can say that S3 is a fiber bundle with fibers congruent to S1 and base space S2(1/2).…”

Section: Landmarks From the Spherical Normalmentioning

confidence: 99%

“…ThenApplying the factorization of formula (6.12), integrating over the variables w,8, and Xl, and ignoring the boundary effects of configurations X that lie within a distance of WI from 8A, we see that the expectation becomesWe shall continue to ignore boundary effects in subsequent formulas. Now suppose that m > n particles are uniformly and independently scattered throughout A.…”

mentioning

confidence: 99%