论文标题
使用细胞二进制树的密度估计，并应用于单调密度
Density estimation using cellular binary trees and an application to monotone densities
论文作者
论文摘要
考虑必须从I.I.D估算的$ [0,1] $上的密度$ f $。 $ x_1，...，x_n $从$ f $中绘制。在本说明中，我们研究了基于二元基的直方图估计，这些直方图使用间隔的递归分裂。如果将间隔划分的决定仅是间隔中数据点数量的（可能是随机的）函数，那么我们谈到复杂性的估计值。我们表现出对复杂性的普遍估计。如果拆分的决定是K相等长度间隔的基础性的函数，那么我们谈到复杂性k的估计值。我们提出了一个复杂性二的估计，可以估算$ [0,1] $上的任何有限单调密度，最佳预期总变化错误$ O（n^{ - 1/3}）$。
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 $O(n^{-1/3})$.
