A Non-Parametric Box-Cox Approach to Robustifying High-Dimensional Linear Hypothesis Testing

TL;DR

This paper develops a non-parametric Box-Cox framework for robust high-dimensional linear hypothesis testing, addressing model mis-specification and proposing efficient estimation and testing procedures with theoretical guarantees.

Contribution

It introduces a flexible non-parametric Box-Cox model for high-dimensional regression, along with novel estimation algorithms and hypothesis tests with known asymptotic distributions.

Findings

Proposed tests follow generalized chi-squared distributions under null and alternative hypotheses.

Simulation studies demonstrate the robustness and effectiveness of the new testing procedures.

Application to supermarket data reveals discrepancies with standard methods, emphasizing robustness.

Abstract

The mainstream theory of hypothesis testing in high-dimensional regression typically assumes the underlying true model is a low-dimensional linear regression model, yet the Box-Cox transformation is a regression technique commonly used to mitigate anomalies like non-additivity and heteroscedasticity. This paper introduces a more flexible framework, the non-parametric Box-Cox model with unspecified transformation, to address model mis-specification in high-dimensional linear hypothesis testing while preserving the interpretation of regression coefficients. Model estimation and computation in high dimensions poses challenges beyond traditional sparse penalization methods. We propose the constrained partial penalized composite probit regression method for sparse estimation and investigate its statistical properties. Additionally, we present a computationally efficient algorithm using augmented Lagrangian and coordinate majorization descent for solving regularization problems with folded concave penalization and linear constraints. For testing linear hypotheses, we propose the partial penalized composite likelihood ratio test, score test and Wald test, and show that their limiting distributions under null and local alternatives follow generalized chi-squared distributions with the same degrees of freedom and noncentral parameter. Extensive simulation studies are conducted to examine the finite sample performance of the proposed tests. Our analysis of supermarket data illustrates potential discrepancies between our testing procedures and standard high-dimensional methods, highlighting the importance of our robustified approach.