文章基本信息

标题：On convex least squares estimation when the truth is linear
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作者：Yining Chen ; Jon A. Wellner
期刊名称：Electronic Journal of Statistics
印刷版ISSN：1935-7524
出版年度：2016
卷号：10
期号：1
页码：171-209
DOI：10.1214/15-EJS1098
语种：English
出版社：Institute of Mathematical Statistics
摘要：We prove that the convex least squares estimator (LSE) attains a n-1/2 pointwise rate of convergence in any region where the truth is linear. In addition, the asymptotic distribution can be characterized by a modified invelope process. Analogous results hold when one uses the derivative of the convex LSE to perform derivative estimation. These asymptotic results facilitate a new consistent testing procedure on the linearity against a convex alternative. Moreover, we show that the convex LSE adapts to the optimal rate at the boundary points of the region where the truth is linear, up to a log-log factor. These conclusions are valid in the context of both density estimation and regression function estimation.
关键词：Adaptive estimation;convexity;density estima tion;least squares;regression function estimation;shape constraint.