On robust learning in the canonical change point problem under heavy tailed errors in finite and growing dimensions
On robust learning in the canonical change point problem under heavy tailed errors in finite and growing dimensions
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DOI：
10.1214/21-ejs1927
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发表时间：
2021-05
影响因子：
1.1
通讯作者：
Debarghya Mukherjee;M. Banerjee;Y. Ritov
中科院分区：
文献类型：
--
作者：
Debarghya Mukherjee;M. Banerjee;Y. Ritov
本文给出了一些关于正则变点估计问题的新发现。第一部分研究了利用基于1（绝对偏差）和2（最小二乘）准则的鲁棒Huber估计函数对简单残桩模型实线上变化点的估计。虽然‘ 2准则已被广泛研究，但它的鲁棒对应物，特别是’ 1最小化问题尚未得到研究。导出了Huber估计函数下估计变点的极限分布，并与' 2准则下的极限分布进行了比较。理论和实证研究表明，在大尾误差下使用Huber估计函数（特别是1准则）更有利，因为与2准则相比，它在通常水平上导致更小的渐近置信区间。我们还比较了并行设置中的“1”和“2”方法，其中有m个独立的单变化点问题，目标是控制估计的变化点与真实值的最大偏差，并严格确定“1”估计准则提供了优于“2”的收敛速度，并且这种相对优势是由误差分布尾部的重量驱动的。最后，我们导出了变平面估计问题的最小最大最优率，并证明了Huber估计达到了最优率，而' 2格式对重尾误差产生了率次最优估计。在推导结果的过程中，我们建立了关于复合二项式过程和复合泊松过程的极小值的若干性质，这些性质具有独立的意义。
This paper presents a number of new findings about the canonical change point estimation problem. The first part studies the estimation of a change point on the real line in a simple stump model using the robust Huber estimating function which interpolates between the `1 (absolute deviation) and `2 (least squares) based criteria. While the `2 criterion has been studied extensively, its robust counterparts and in particular, the `1 minimization problem have not. We derive the limit distribution of the estimated change point under the Huber estimating function and compare it to that under the `2 criterion. Theoretical and empirical studies indicate that it is more profitable to use the Huber estimating function (and in particular, the `1 criterion) under heavy tailed errors as it leads to smaller asymptotic confidence intervals at the usual levels compared to the `2 criterion. We also compare the `1 and `2 approaches in a parallel setting, where one has m independent single change point problems and the goal is to control the maximal deviation of the estimated change points from the true values, and establish rigorously that the `1 estimation criterion provides a superior rate of convergence to the `2, and that this relative advantage is driven by the heaviness of the tail of the error distribution. Finally, we derive minimax optimal rates for the change plane estimation problem in growing dimensions and demonstrate that Huber estimation attains the optimal rate while the `2 scheme produces a rate sub-optimal estimator for heavy tailed errors. In the process of deriving our results, we establish a number of properties about the minimizers of compound Binomial and compound Poisson processes which are of independent interest.