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

TL;DR

This paper introduces a new binary-tree-based histogram method for density estimation on [0,1], capable of adaptively estimating monotone densities with optimal error rates using recursive interval splitting.

Contribution

It proposes a universally consistent binary-tree-based density estimator of complexity two that achieves optimal convergence rates for monotone densities.

Findings

Achieves $O(n^{-1/3})$ total variation error for monotone densities.

Introduces a complexity-two estimator based on recursive splitting.

Demonstrates universal consistency of the proposed method.

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 $O(n^{-1/3})$.