DIVIDE AND CONQUER IN NONSTANDARD PROBLEMS AND THE SUPER-EFFICIENCY PHENOMENON

• 成果类型: Article
• 署名作者: Banerjee, Moulinath; Durot, Cecile; Sen, Bodhisattva
• 署名单位: University of Michigan System; University of Michigan; Columbia University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1633
• 发表日期: 2019
• 页码: 720-757
• 关键词: Monotone; regression

摘要

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.