Consistency of Nonparametric Density Estimators in CAT(0) Orthant Space
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| Format: | Preprint |
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2025
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| _version_ | 1866915566298595328 |
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| author | Takazawa, Yuki Sei, Tomonari |
| author_facet | Takazawa, Yuki Sei, Tomonari |
| contents | The inference of evolutionary histories is a central problem in evolutionary biology. The analysis of a sample of phylogenetic trees can be conducted in Billera-Holmes-Vogtmann tree space, which is a CAT(0) metric space of phylogenetic trees. The globally non-positively curved (CAT(0)) property of this space enables the extension of various statistical techniques. In the problem of nonparametric density estimation, two primary methods, kernel density estimation and log-concave maximum likelihood estimation, have been proposed, yet their theoretical properties remain largely unexplored.
In this paper, we address this gap by proving the consistency of these estimators in a more general setting$\unicode{x2014}$CAT(0) orthant spaces, which include BHV tree space. We extend log-concave approximation techniques to this setting and establish consistency via the continuity of the log-concave projection map. We also modify the kernel density estimator to correct boundary bias and establish uniform consistency using empirical process theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_18290 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Consistency of Nonparametric Density Estimators in CAT(0) Orthant Space Takazawa, Yuki Sei, Tomonari Statistics Theory Methodology The inference of evolutionary histories is a central problem in evolutionary biology. The analysis of a sample of phylogenetic trees can be conducted in Billera-Holmes-Vogtmann tree space, which is a CAT(0) metric space of phylogenetic trees. The globally non-positively curved (CAT(0)) property of this space enables the extension of various statistical techniques. In the problem of nonparametric density estimation, two primary methods, kernel density estimation and log-concave maximum likelihood estimation, have been proposed, yet their theoretical properties remain largely unexplored. In this paper, we address this gap by proving the consistency of these estimators in a more general setting$\unicode{x2014}$CAT(0) orthant spaces, which include BHV tree space. We extend log-concave approximation techniques to this setting and establish consistency via the continuity of the log-concave projection map. We also modify the kernel density estimator to correct boundary bias and establish uniform consistency using empirical process theory. |
| title | Consistency of Nonparametric Density Estimators in CAT(0) Orthant Space |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2510.18290 |