Deep Transformation Model

TL;DR

This paper introduces a flexible, nonparametric deep transformation model that generalizes many existing models, offers robustness to heavy-tailed errors, and includes novel loss functions and variable selection methods, with proven theoretical and practical effectiveness.

Contribution

It proposes a new nonparametric transformation model with a novel loss function, a DReLU-based estimator, and variable selection techniques suitable for high-dimensional and censored data.

Findings

Model encompasses many popular models as special cases.

Estimator demonstrates strong theoretical properties.

Numerical studies confirm practical utility.

Abstract

There has been a significant recent surge in deep neural network (DNN) techniques. Most of the existing DNN techniques have restricted model formats/assumptions. To overcome their limitations, we propose the nonparametric transformation model, which encompasses many popular models as special cases and hence is less sensitive to model mis-specification. This model also has the potential of accommodating heavy-tailed errors, a robustness property not broadly shared. Accordingly, a new loss function, which fundamentally differs from the existing ones, is developed. For computational feasibility, we further develop a double rectified linear unit (DReLU)-based estimator. To accommodate the scenario with a diverging number of input variables and/or noises, we propose variable selection based on group penalization. We further expand the scope to coherently accommodate censored survival data. The estimation and variable selection properties are rigorously established. Extensive numerical studies, including simulations and data analyses, establish the satisfactory practical utility of the proposed methods.