DIVIDE AND CONQUER IN NONSTANDARD PROBLEMS AND THE SUPER-EFFICIENCY PHENOMENON
DIVIDE AND CONQUER IN NONSTANDARD PROBLEMS AND THE SUPER-EFFICIENCY PHENOMENON
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DOI：
10.1214/17-aos1633
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发表时间：
2019-04-01
影响因子：
4.5
通讯作者：
Sen, Bodhisattva
中科院分区：
文献类型：
--
作者：
Banerjee, Moulinath;Durot, Cecile;Sen, Bodhisattva
我们研究了分而治之原理在收敛速度通常慢于根n且极限分布为非高斯的非标准问题中的工作原理，并对涉及单调函数的非参数估计的各种重要且研究得很好的问题提供了详细的处理。我们发现，对于固定模型，通过对互斥子样本上的非标准估计求平均得到的混合估计在函数的逐点估计意义上优于基于整个样本的非标准单调性约束(全局)估计。我们还证明了，在适当的条件下，如果允许子样本的数量以适当的速度增加，则集合估计是渐近正态分布的，其方差是经验上可以估计的。此外，在单调回归的背景下，我们证明了在固定模型下这种效率的提高是有代价的--在一类模型上，集合估计器在一致意义上的性能(最大风险)随子样本数量的增加而恶化，从而导致某种形式的超效率现象。在这个过程中，我们发展了关于保序回归中偏差的阶数的分析结果，这是独立感兴趣的。
We study how the divide and conquer principle works in non-standard problems where rates of convergence are typically slower than root n and limit distributions are non-Gaussian, and provide a detailed treatment for a variety of important and well-studied problems involving nonparametric estimation of a monotone function. We find that for a fixed model, the pooled estimator, obtained by averaging nonstandard estimates across mutually exclusive subsamples, outperforms the nonstandard monotonicity-constrained (global) estimator based on the entire sample in the sense of pointwise estimation of the function. We also show that, under appropriate conditions, if the number of subsamples is allowed to increase at appropriate rates, the pooled estimator is asymptotically normally distributed with a variance that is empirically estimable from the subsample-level estimates. Further, in the context of monotone regression, we show that this gain in efficiency under a fixed model comes at a price-the pooled estimator's performance, in a uniform sense (maximal risk) over a class of models worsens as the number of subsamples increases, leading to a version of the super-efficiency phenomenon. In the process, we develop analytical results for the order of the bias in isotonic regression, which are of independent interest.