A projection approach for multiple monotone regression

TL;DR

This paper introduces a projection-based method for estimating multivariate monotone regression functions, improving accuracy over initial estimators and providing practical algorithms and asymptotic analysis.

Contribution

It proposes a novel projection approach for multivariate monotone regression, extending shape-constrained inference to multiple predictors with computational algorithms.

Findings

Projection reduces estimation error compared to initial estimators

Asymptotic distribution of the projection estimator is derived

Method is illustrated with an environmental toxicology example

Abstract

Shape-constrained inference has wide applicability in bioassay, medicine, economics, risk assessment, and many other fields. Although there has been a large amount of work on monotone-constrained univariate curve estimation, multivariate shape-constrained problems are much more challenging, and fewer advances have been made in this direction. With a focus on monotone regression with multiple predictors, this current work proposes a projection approach to estimate a multiple monotone regression function. An initial unconstrained estimator -- such as a local polynomial estimator or spline estimator -- is first obtained, which is then projected onto the shape-constrained space. A shape-constrained estimate (with multiple predictors) is obtained by sequentially projecting an (adjusted) initial estimator along each univariate direction. Compared to the initial unconstrained estimator, the projection estimate results in a reduction of estimation error in terms of both $L^p$ ($p\geq 1$) distance and supremum distance. We also derive the asymptotic distribution of the projection estimate. Simple computational algorithms are available for implementing the projection in both the unidimensional and higher dimensional cases. Our work provides a simple recipe for practitioners to use in real applications, and is illustrated with a joint-action example from environmental toxicology.