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Register a new userHierarchical orbital decompositions and extended decomposable distributions
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Added by
Akimichi Takemura
Publication date
2006
fields
Mathematical Statistics
and research's language is
English
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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 (2006) 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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In this paper, we explore a connection between binary hierarchical models, their marginal polytopes and codeword polytopes, the convex hulls of linear codes. The class of linear codes that are realizable by hierarchical models is determined. We classify all full dimensional polytopes with the property that their vertices form a linear code and give an algorithm that determines them.
Elliptically contoured distributions can be considered to be the distributions for which the contours of the density functions are proportional ellipsoids. We generalize elliptically contoured densities to ``star-shaped distributions with concentric star-shaped contours and show that many results in the former case continue to hold in the more general case. We develop a general theory in the framework of abstract group invariance so that the results can be applied to other cases as well, especially those involving random matrices.
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We introduce new shape-constrained classes of distribution functions on R, the bi-$s^*$-concave classes. In parallel to results of Dumbgen, Kolesnyk, and Wilke (2017) for what they called the class of bi-log-concave distribution functions, we show that every $s$-concave density $f$ has a bi-$s^*$-concave distribution function $F$ for $s^*leq s/(s+1)$. Confidence bands building on existing nonparametric bands, but accounting for the shape constraint of bi-$s^*$-concavity, are also considered. The new bands extend those developed by Dumbgen et al. (2017) for the constraint of bi-log-concavity. We also make connections between bi-$s^*$-concavity and finiteness of the CsorgH{o} - Revesz constant of $F$ which plays an important role in the theory of quantile processes.
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