Open Access
April 2020 On estimation of isotonic piecewise constant signals
Chao Gao, Fang Han, Cun-Hui Zhang
Ann. Statist. 48(2): 629-654 (April 2020). DOI: 10.1214/18-AOS1792


Consider a sequence of real data points $X_{1},\ldots ,X_{n}$ with underlying means $\theta ^{*}_{1},\dots ,\theta ^{*}_{n}$. This paper starts from studying the setting that $\theta ^{*}_{i}$ is both piecewise constant and monotone as a function of the index $i$. For this, we establish the exact minimax rate of estimating such monotone functions, and thus give a nontrivial answer to an open problem in the shape-constrained analysis literature. The minimax rate under the loss of the sum of squared errors involves an interesting iterated logarithmic dependence on the dimension, a phenomenon that is revealed through characterizing the interplay between the isotonic shape constraint and model selection complexity. We then develop a penalized least-squares procedure for estimating the vector $\theta ^{*}=(\theta^{*}_{1},\dots ,\theta ^{*}_{n})^{\mathsf{T}}$. This estimator is shown to achieve the derived minimax rate adaptively. For the proposed estimator, we further allow the model to be misspecified and derive oracle inequalities with the optimal rates, and show there exists a computationally efficient algorithm to compute the exact solution.


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Chao Gao. Fang Han. Cun-Hui Zhang. "On estimation of isotonic piecewise constant signals." Ann. Statist. 48 (2) 629 - 654, April 2020.


Received: 1 July 2018; Revised: 1 November 2018; Published: April 2020
First available in Project Euclid: 26 May 2020

zbMATH: 07241563
MathSciNet: MR4102670
Digital Object Identifier: 10.1214/18-AOS1792

Primary: 62C20 , 62G08

Keywords: adaptive estimation , Isotonic piecewise constant function , iterated logarithmic dependence , Oracle inequalities , reduced isotonic regression

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 2 • April 2020
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