{"ID":2852043,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.18290","arxiv_id":"2510.18290","title":"Consistency of Nonparametric Density Estimators in CAT(0) Orthant Space","abstract":"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.","short_abstract":"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 th...","url_abs":"https://arxiv.org/abs/2510.18290","url_pdf":"https://arxiv.org/pdf/2510.18290v1","authors":"[\"Yuki Takazawa\",\"Tomonari Sei\"]","published":"2025-10-21T04:42:42Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"stat.ME\"]","methods":"[]","has_code":false}