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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2510.18068 |
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| _version_ | 1866914105027198976 |
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| author | Beran, Rudolf |
| author_facet | Beran, Rudolf |
| contents | Directional data consists of unit vectors in q-dimensions that can be described in polar or Cartesian coordinates. Axial data can be viewed as a pair of directions pointed in opposite directions or as a projection matrix of rank 1. Historically, their statistical analysis has largely been based on a few low-order exponential family models of distributions for random directions or axes. A lack of tractable algebraic forms for the normalizing constants has hindered the use of higher-order exponential families for less constrained modeling. Of interest are functionals of the unknown distribution of the directional/axial data, such as the directional/axial mean, dispersion, or distribution itself. This paper outlines nonparametric estimators and bootstrap confidence sets for such functionals. The procedures are based on the empirical distribution of the directional/axial sample expressed in Cartesian coordinates. Sketched as well are nonparametric comparisons among multiple mean directions or axes, estimation of trend in mean directions, and analysis of q-dimensional observations restricted to lie in a specified compact subset. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18068 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cartesian Statistics on Spheres Beran, Rudolf Methodology 62H11, 62G09 Directional data consists of unit vectors in q-dimensions that can be described in polar or Cartesian coordinates. Axial data can be viewed as a pair of directions pointed in opposite directions or as a projection matrix of rank 1. Historically, their statistical analysis has largely been based on a few low-order exponential family models of distributions for random directions or axes. A lack of tractable algebraic forms for the normalizing constants has hindered the use of higher-order exponential families for less constrained modeling. Of interest are functionals of the unknown distribution of the directional/axial data, such as the directional/axial mean, dispersion, or distribution itself. This paper outlines nonparametric estimators and bootstrap confidence sets for such functionals. The procedures are based on the empirical distribution of the directional/axial sample expressed in Cartesian coordinates. Sketched as well are nonparametric comparisons among multiple mean directions or axes, estimation of trend in mean directions, and analysis of q-dimensional observations restricted to lie in a specified compact subset. |
| title | Cartesian Statistics on Spheres |
| topic | Methodology 62H11, 62G09 |
| url | https://arxiv.org/abs/2510.18068 |