A two stage k-monotone b-spline regression estimator: Uniform lipschitz property and optimal convergence rate
Date
2018Journal
Electronic Journal of StatisticsPublisher
Institute of Mathematical StatisticsType
Article
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This paper considers k-monotone estimation and the related asymptotic performance analysis over a suitable H�lder class for general k. A novel two stage k-monotone B-spline estimator is proposed: in the first stage, an unconstrained estimator with optimal asymptotic performance is considered; in the second stage, a k-monotone B-spline estimator is constructed (roughly) by projecting the unconstrained estimator onto a cone of k-monotone splines. To study the asymptotic performance of the second stage estimator under the sup-norm and other risks, a critical uniform Lipschitz property for the k-monotone B-spline estimator is established under the ??-norm. This property uniformly bounds the Lipschitz constants associated with the mapping from a (weighted) first stage input vector to the B-spline coefficients of the second stage k-monotone estimator, independent of the sample size and the number of knots. This result is then exploited to analyze the second stage estimator performance and develop convergence rates under the sup-norm, pointwise, and Lp-norm (with p ? [1, ?)) risks. By employing recent results in k-monotone estimation minimax lower bound theory, we show that these convergence rates are optimal. � 2018, Institute of Mathematical Statistics. All rights reserved.Sponsors
?Supported in part by the NSF grants CMMI-1030804 and DMS-1042916.Keyword
Asymptotic analysisB-splines
Convergence rates
K-monotone estimation
Nonparametric regression
Shape constraints
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https://www.scopus.com/inward/record.uri?eid=2-s2.0-85047371886&doi=10.1214%2f18-EJS1426&partnerID=40&md5=10cfac70b4a89ea69ec70c125a246e00; http://hdl.handle.net/10713/8824ae974a485f413a2113503eed53cd6c53
10.1214/18-EJS1426