New Asymptotic Limit Theory and Inference for Monotone Regression

TL;DR

This paper advances the asymptotic theory and inference methods for monotone regression, providing new limit results and confidence intervals that are valid across diverse distributions.

Contribution

It extends the asymptotic limit theory for monotone regression to triangular arrays and develops uniformly valid confidence intervals.

Findings

Extended asymptotic limit theory to triangular arrays.

Developed confidence intervals valid over a broad class of distributions.

Enhanced understanding of the asymptotic behavior of isotonic estimators.

Abstract

Nonparametric regression problems with qualitative constraints such as monotonicity or convexity are ubiquitous in applications. For example, in predicting the yield of a factory in terms of the number of labor hours, the monotonicity of the conditional mean function is a natural constraint. One can estimate a monotone conditional mean function using nonparametric least squares estimation, which involves no tuning parameters. Several interesting properties of the isotonic LSE are known including its rate of convergence, adaptivity properties, and pointwise asymptotic distribution. However, we believe that the full richness of the asymptotic limit theory has not been explored in the literature which we do in this paper. Moreover, the inference problem is not fully settled. In this paper, we present some new results for monotone regression including an extension of existing results to triangular arrays, and provide asymptotically valid confidence intervals that are uniformly valid over a large class of distributions.