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

2022/03/15 by Devroye, Luc, Hamdan, Jad
#FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2203.08006

Abstract

Consider a density f on [0,1] that must be estimated from an i.i.d. sample X₁,...,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).

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