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August 2021 Confidence intervals for multiple isotonic regression and other monotone models
Hang Deng, Qiyang Han, Cun-Hui Zhang
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Ann. Statist. 49(4): 2021-2052 (August 2021). DOI: 10.1214/20-AOS2025


We consider the problem of constructing pointwise confidence intervals in the multiple isotonic regression model. Recently, Han and Zhang (2020) obtained a pointwise limit distribution theory for the so-called block max–min and min–max estimators (Fokianos, Leucht and Neumann (2020); Deng and Zhang (2020)) in this model, but inference remains a difficult problem due to the nuisance parameter in the limit distribution that involves multiple unknown partial derivatives of the true regression function.

In this paper, we show that this difficult nuisance parameter can be effectively eliminated by taking advantage of information beyond point estimates in the block max–min and min–max estimators. Formally, let uˆ(x0) (resp. vˆ(x0)) be the maximizing lower-left (resp. minimizing upper-right) vertex in the block max–min (resp. min–max) estimator, and fˆn be the average of the block max–min and min–max estimators. If all (first-order) partial derivatives of f0 are nonvanishing at x0, then the following pivotal limit distribution theory holds:


Here nuˆ,vˆ(x0) is the number of design points in the block [uˆ(x0),vˆ(x0)], σ is the standard deviation of the errors, and L1d is a universal limit distribution free of nuisance parameters. This immediately yields confidence intervals for f0(x0) with asymptotically exact confidence level and oracle length. Notably, the construction of the confidence intervals, even new in the univariate setting, requires no more efforts than performing an isotonic regression once using the block max–min and min–max estimators, and can be easily adapted to other common monotone models including, for example, (i) monotone density estimation, (ii) interval censoring model with current status data, (iii) counting process model with panel count data, and (iv) generalized linear models. Extensive simulations are carried out to support our theory.

Funding Statement

The research of H. Deng was supported in part by NSF DMS-1454817. The research of Q. Han was supported in part by NSF DMS-1916221. The research of C.-H. Zhang was supported in part by NSF DMS-1513378, DMS-1721495, IIS-1741390 and CCF-1934924.


The authors would like to thank three referees for their helpful comments and suggestions that improved the quality of the paper.


Download Citation

Hang Deng. Qiyang Han. Cun-Hui Zhang. "Confidence intervals for multiple isotonic regression and other monotone models." Ann. Statist. 49 (4) 2021 - 2052, August 2021.


Received: 1 January 2020; Revised: 1 July 2020; Published: August 2021
First available in Project Euclid: 29 September 2021

Digital Object Identifier: 10.1214/20-AOS2025

Primary: 60E15
Secondary: 62G05

Keywords: Confidence interval , Gaussian process , Limit distribution theory , multiple isotonic regression , shape constraints

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 4 • August 2021
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