Consistency of Nonparametric Density Estimators in CAT(0) Orthant Space

TL;DR

This paper proves the consistency of nonparametric density estimators, specifically kernel and log-concave maximum likelihood estimators, in CAT(0) orthant spaces, including phylogenetic tree space, extending their theoretical understanding.

Contribution

It establishes the first theoretical consistency results for these estimators in CAT(0) orthant spaces, broadening their applicability in evolutionary biology and related fields.

Findings

Proved consistency of kernel density estimators in CAT(0) orthant spaces.

Established consistency of log-concave maximum likelihood estimators in this setting.

Extended log-concave approximation techniques to CAT(0) orthant spaces.

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.