  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.
