使用细胞二叉树进行密度估计及其在单调密度上的应用
Density estimation using cellular binary trees and an application to monotone densities
AI总结：
本文提出基于细胞二叉树的直方图密度估计，复杂度一估计具有普遍一致性，复杂度二估计能以最优误差O(n^{-1/3})估计单调密度。
摘要：
考虑定义在[0,1]上的密度函数f，需从由f抽取的独立同分布样本X_1,...,X_n中估计f。本文研究基于二叉树的直方图估计，其采用区间递归分割。若分割区间的决策仅是该区间内数据点数量的（可能随机化的）函数，则称该估计为复杂度一。我们展示了一个复杂度一的普遍一致估计。若分割决策是k个等长子区间基数的函数，则称该估计为复杂度k。我们提出一个复杂度二的估计，它能以最优期望全变差误差O(n^{-1/3})估计[0,1]上的任意有界单调密度。
英文摘要：
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 $O(n^{-1/3})$.
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