Journal
STATISTICAL SCIENCE
Volume 33, Issue 4, Pages 568-594Publisher
INST MATHEMATICAL STATISTICS-IMS
DOI: 10.1214/18-STS665
Keywords
Adaptive risk bounds; bootstrap; Chernoff's distribution; convex regression; isotonic regression; likelihood ratio test; monotone function; order preserving function estimation; projection on a closed convex set; tangent cone
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Funding
- NSF CAREER [DMS-16-54589]
- NSF [DMS-17-12822, AST-16-14743]
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We consider the problem of nonparametric regression under shape constraints. The main examples include isotonic regression (with respect to any partial order), unimodal/convex regression, additive shape-restricted regression and constrained single index model. We review some of the theoretical properties of the least squares estimator (LSE) in these problems, emphasizing on the adaptive nature of the LSE. In particular, we study the behavior of the risk of the LSE, and its pointwise limiting distribution theory, with special emphasis to isotonic regression. We survey various methods for constructing pointwise confidence intervals around these shape-restricted functions. We also briefly discuss the computation of the LSE and indicate some open research problems and future directions.
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