# Nonparametric Shape-Restricted Regression

@article{Guntuboyina2017NonparametricSR, title={Nonparametric Shape-Restricted Regression}, author={Adityanand Guntuboyina and Bodhisattva Sen}, journal={Statistical Science}, year={2017} }

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 risk behavior of the LSE, and its pointwise limiting…

## 60 Citations

### Robustness of shape-restricted regression estimators: an envelope perspective

- Mathematics
- 2018

Classical least squares estimators are well-known to be robust with respect to moment assumptions concerning the error distribution in a wide variety of finite-dimensional statistical problems;…

### Adaptive confidence sets in shape restricted regression

- MathematicsBernoulli
- 2021

We construct adaptive confidence sets in isotonic and convex regression. In univariate isotonic regression, if the true parameter is piecewise constant with $k$ pieces, then the Least-Squares…

### Editorial: Special Issue on “Nonparametric Inference Under Shape Constraints”

- MathematicsStatistical Science
- 2018

Shape-constrained inference usually refers to nonparametric function estimation and uncertainty quantification under qualitative shape restrictions such as monotonicity, convexity, log-concavity and…

### Stratified incomplete local simplex tests for curvature of nonparametric multiple regression

- MathematicsBernoulli
- 2023

Principled nonparametric tests for regression curvature in $\mathbb{R}^{d}$ are often statistically and computationally challenging. This paper introduces the stratified incomplete local simplex…

### The bias of isotonic regression.

- MathematicsElectronic journal of statistics
- 2020

It is proved that the bias of the isotonic regression estimator scales as O(n -β/3) up to log factors, where 1 ≤ β ≤ 2 is the exponent corresponding to Hölder smoothness of the underlying mean.

### Multivariate extensions of isotonic regression and total variation denoising via entire monotonicity and Hardy–Krause variation

- Mathematics, Computer Science
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It is shown that the risk of the entirely monotonic LSE is almost parametric (at most $1/n$ up to logarithmic factors) when the true function is well-approximable by a rectangular piecewise constant entirely Monotone function with not too many constant pieces.

### Generalized nearly isotonic regression

- Computer Science, MathematicsArXiv
- 2021

An eﬃcient algorithm based on the modiﬁed PAVA is developed, which utilizes the duality between the natural and expectation parameters, and a method for selecting the regularization parameter by using an information criterion is provided.

### Isotonic Regression Discontinuity Designs

- MathematicsJournal of Econometrics
- 2019

In isotonic regression discontinuity designs, the average outcome and the treatment assignment probability are monotone in the running variable. We introduce novel nonparametric estimators for sharp…

### Nonparametric, tuning‐free estimation of S‐shaped functions

- Mathematics, Computer ScienceJournal of the Royal Statistical Society: Series B (Statistical Methodology)
- 2022

This work considers the nonparametric estimation of an S‐shaped regression function with least squares estimator, and reveals that the estimator achieves the minimax optimal rate of convergence for both the estimation of the regression function and its inflection point.

### Isotonic distributional regression

- MathematicsJournal of the Royal Statistical Society: Series B (Statistical Methodology)
- 2021

Isotonic distributional regression (IDR) is a powerful non‐parametric technique for the estimation of conditional distributions under order restrictions. In a nutshell, IDR learns conditional…

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