Limit Distribution Theory for Multiple Isotonic Regression

• Activity Number: 103 - Semiparametric Inference with High-Dimensional and Complex Data
• Type: Invited
• Date/Time: Monday, August 3, 2020 : 1:00 PM to 2:50 PM
• Sponsor: ENAR
• Author(s): Qiyang Han and Cun-Hui Zhang*
• Companies: Rutgers University and Rutgers University

Abstract

We study limit distributions for the tuning-free max-min block estimators in multiple isotonic regression under both fixed lattice design and random design settings. We show that at a fixed interior point in the design space, the estimation error of the max-min block estimator converges in distribution to a non-Gaussian limit at certain rate depending on the number of vanishing derivatives and certain effective dimension and sample size that drive the asymptotic theory. The convergence rate is optimal in a local asymptotic minimax sense. There are two interesting features in our local theory. First, the max-min block estimator automatically adapts to the full spectrum of local smoothness levels and the intrinsic dimension of the isotonic regression function at the optimal rate. Second, the optimally adaptive local rates are in general not the same in fixed lattice and random designs.