Minimax Risk Bounds for Piecewise Constant Models
Minimax Risk Bounds for Piecewise Constant Models
复制标题
分段常数模型的极小极大风险界限
DOI：
--
复制
发表时间：
2017
期刊：
影响因子：
--
通讯作者：
Cun
中科院分区：
文献类型：
--
作者：
Chao Gao;Fang Han;Cun
考虑数据点X1，. . .，Xn，其基本平均θ ∈ Rn是至多k个分片的分段常数。本文建立了θ θ_（？）的最小二乘估计（LSE）的非渐近风险界.主要结果有两个方面。首先，当没有额外的形状约束时，我们揭示了LSE风险的新相变：随着k从2增加到更高，比率从log logn变化到k log（en/k）。其次，当进一步假设θ_n为非减函数时，我们证明了在2 ≤ k_n ≤ n上，速率改进为k_（16 n/k_n）。这些界限是尖锐的意义上说，他们匹配的极大极小下界的研究问题（不牺牲任何对数因子）。它们补充了变点检测文献中的对应物，并填补了最近发现的与保序回归分段常数模型相关的一些显着空白。证明中开发的技术，这是建立在Levy的部分和和Doob的鞅理论，是独立的兴趣，并可能有潜在的应用研究的一些其他形状约束的回归问题。
Consider a sequence of data points X1, . . . , Xn whose underlying mean θ ∗ ∈ Rn is piecewise constant of at most k∗ pieces. This paper establishes sharp nonasymptotic risk bounds for the least squares estimator (LSE) on estimating θ∗. The main results are twofold. First, when there is no additional shape constraint assumed, we reveal a new phase transition for the risk of LSE: As k∗ increases from 2 to higher, the rate changes from log logn to k∗ log(en/k∗). Secondly, when θ∗ is further assumed to be nondecreasing, we show the rate is improved to be k∗ log log(16n/k∗) over 2 ≤ k∗ ≤ n. These bounds are sharp in the sense that they match the minimax lower bounds of the studied problems (without sacrificing any logarithmic factor). They complement their counterpart in the change-point detection literature and fill some notable gaps in recent discoveries relating isotonic regression to piecewise constant models. The techniques developed in the proofs, which are built on Levy’s partial sum and Doob’s martingale theory, are of independent interest and may have potential applications to the study of some other shape-constrained regression problems.
影响因子：
1.6
作者：
Amelunxen, Dennis;Lotz, Martin;Tropp, Joel A.
通讯作者：
Tropp, Joel A.
影响因子：
1.4
作者：
Hao N;Niu YS;Zhang H
通讯作者：
Zhang H