Confidence Intervals for Multiple Isotonic Regression and Other Monotone Models

• Activity Number: 336 - Nonparametric Test in Unusual Data Structure or of Independence
• Type: Contributed
• Date/Time: Wednesday, August 5, 2020 : 10:00 AM to 2:00 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Hang Deng* and Qiyang Han and Cun-Hui Zhang
• Companies: Rutgers University and Rutgers University and Rutgers University
• Keywords: limit distribution theory; confidence interval; multiple isotonic regression; Gaussian process; shape constraints

Abstract

We consider the problem of constructing point-wise confidence intervals in the multiple isotonic regression model. Recently, Han and Zhang (2019) obtained a point-wise limit distribution theory for the block max-min and min-max estimators in this model, but inference remains a difficult problem due to the nuisance parameter in the limit distribution.

In this paper, we show that this difficulty can be effectively eliminated by taking advantage of information beyond point estimates in the block max-min and min-max estimators, primarily their block sizes, so that a pivotal limit distribution theory can be derived. This immediately yields confidence intervals for $f_0(x_0)$ with asymptotically exact confidence level and optimal length, where $f_0$ is the true mean. Notably, the construction of the confidence intervals, even new in the univariate setting, requires no more efforts than performing an isotonic regression for once using the block max-min and min-max estimators, and can be easily adapted to other common monotone models. Extensive simulation results demonstrate the accuracy of the coverage probability of the proposed confidence intervals, giving strong support to our theory.