A Robust Monotonic Single-Index Model for Skewed and Heavy-Tailed Data: A Deep Neural Network Approach Applied to Periodontal Studies

TL;DR

This paper introduces a robust deep neural network-based single-index model tailored for skewed and heavy-tailed data, with applications in periodontal disease diagnosis, combining flexibility, robustness, and interpretability.

Contribution

It develops a novel monotonic neural network framework with a two-piece Student-t distribution for robust mode estimation in skewed, heavy-tailed data, with theoretical guarantees.

Findings

Demonstrates robustness to outliers in simulations

Achieves accurate modeling of skewed heavy-tailed data

Provides interpretable clinical insights

Abstract

Periodontal pocket depth is a widely used biomarker for diagnosing risk of periodontal disease. However, pocket depth typically exhibits skewness and heavy-tailedness, and its relationship with clinical risk factors is often nonlinear. Motivated by periodontal studies, this paper develops a robust single-index modal regression framework for analyzing skewed and heavy-tailed data. Our method has the following novel features: (1) a flexible two-piece scale Student-$t$ error distribution that generalizes both normal and two-piece scale normal distributions; (2) a deep neural network with guaranteed monotonicity constraints to estimate the unknown single-index function; and (3) theoretical guarantees, including model identifiability and a universal approximation theorem. Our single-index model combines the flexibility of neural networks and the two-piece scale Student-$t$ distribution, delivering robust mode-based estimation that is resistant to outliers, while retaining clinical interpretability through parametric index coefficients. We demonstrate the performance of our method through simulation studies and an application to periodontal disease data from the HealthPartners Institute of Minnesota. The proposed methodology is implemented in the \textsf{R} package \href{https://doi.org/10.32614/CRAN.package.DNNSIM}{\textsc{DNNSIM}}.