Density estimation using cellular binary trees and an application to monotone densities

TL;DR

This work develops and analyzes binary-tree-based density estimators on the unit interval, restricted to dyadic partitions and a cellular computation model. It introduces a universally consistent complexity-one estimator (randomized splitting based on the interval count) and a complexity-two estimator tailored for monotone densities, achieving the minimax-rate bound up to constants with an $O(n^{-1/3})$ total-variation-type error that scales with the bound $B=f(0)$. The analysis leverages connections to Galton–Watson branching processes and an infinite deterministic tree to obtain finite-tree size, error bounds, and sublinear-time algorithms, including a sorted-data implementation with $O(n^{1/3}\log n)$ runtime. The results establish a principled framework for cellular, distributed density estimation and outline extensions to broader shape constraints and higher dimensions. The contributions bridge nonparametric density estimation, partition-based histograms, and branching-process techniques, highlighting practical estimability under minimal smoothness assumptions.

Abstract

Consider a density $f$ on $[0,1]$ that must be estimated from an i.i.d. sample $X_1,...,X_n$ drawn from $f$. In this note, we study binary-tree-based histogram estimates that use recursive splitting of intervals. If the decision to split an interval is a (possibly randomized) function of the number of data points in the interval only, then we speak of an estimate of complexity one. We exhibit a universally consistent estimate of complexity one. If the decision to split is a function of the cardinalities of k equal-length sub-intervals, then we speak of an estimate of complexity k. We propose an estimate of complexity two that can estimate any bounded monotone density on $[0,1]$ with optimal expected total variation error $O(n^{-1/3})$.

Figures (6)

• Figure 1: Dyadic splitting until each interval contains at most one point.
• Figure 2: An estimate $f_n$ using a finite, random tree $T_n$.
• Figure 3: A depiction of an infinite tree $\mathcal{T}_\infty$.
• Figure 4: Definitions used in the proof of Lemma \ref{['lem9']}.
• Figure 5: Switching a balanced node with its parent.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• Remark 1
• Corollary 3
• proof
• Theorem 4
• Remark 2
• proof
• Proposition 5
• Lemma 6