On Integrated $L^{1}$ Convergence Rate of an Isotonic Regression Estimator for Multivariate Observations

TL;DR

This paper introduces a modified isotonic regression estimator for multivariate data that automatically adapts to data shape, achieves a specific convergence rate, and does not require tuning, with validation through simulations and real data.

Contribution

It proposes a new isotonic regression estimator for multivariate observations that automatically determines smoothing without auxiliary tuning and establishes its convergence rate.

Findings

Estimator achieves optimal integrated L1 convergence rate.

Method captures data shape without parametric assumptions.

Validated through simulations and real data examples.

Abstract

We consider a general monotone regression estimation where we allow for independent and dependent regressors. We propose a modification of the classical isotonic least squares estimator and establish its rate of convergence for the integrated $L_1$-loss function. The methodology captures the shape of the data without assuming additivity or a parametric form for the regression function. Furthermore, the degree of smoothing is chosen automatically and no auxiliary tuning is required for the theoretical analysis. Some simulations and two real data illustrations complement the study of the proposed estimator.