## Annals of Statistics

### On estimation of isotonic piecewise constant signals

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 48, Number 2 (2020), 629-654.

Dates
Revised: November 2018
First available in Project Euclid: 26 May 2020

https://projecteuclid.org/euclid.aos/1590480028

Digital Object Identifier
doi:10.1214/18-AOS1792

Mathematical Reviews number (MathSciNet)
MR4102670

Subjects
Primary: 62G08: Nonparametric regression 62C20: Minimax procedures

#### Citation

Gao, Chao; Han, Fang; Zhang, Cun-Hui. On estimation of isotonic piecewise constant signals. Ann. Statist. 48 (2020), no. 2, 629--654. doi:10.1214/18-AOS1792. https://projecteuclid.org/euclid.aos/1590480028

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#### Supplemental materials

• Supplement to “On estimation of isotonic piecewise constant signals”. This supplement contains proofs of remaining results in Section 4, as well as some auxiliary lemmas.