Global Inference for High Dimensional Quantile Regression with Heterogeneity

原文摘要

Quantile regression has received considerable attention in high dimensional methodology due to its robustness and ability to accommodate heteroskedasticity. Despite this attention, current inferential methods for high dimensional quantile regression have significant limitations---only testing significance, testing at a single quantile at a time, or assuming homoskedasticity. To generalize the inferences that can be drawn in high dimensional quantile regression, we introduce partial penalized Wald and rank score tests for testing linear hypotheses uniformly over a set of quantiles $\mathcal T$ under a heteroskedastic model. We propose globally adaptive local linear approximation (LLA) algorithms for folded concave penalized estimation and show that our global LLA algorithms recover ``testing oracles'' for the full and reduced models with high probability. We then show that our global LLA estimators achieve the $\sqrt{s/n}$ oracle estimation rate uniformly over $\mathcal T$, establish Bahadur representations, and derive asymptotic distributions for the test statistics under the null and local alternatives. We develop a dual formulation incorporating adaptive penalty weights for the rank scores and use the strong oracle property to establish their oracle equivalence. Using empirical process techniques, we derive reference distributions based on suprema of squared Bessel processes, extending partial penalized inference beyond pointwise chi-square calibration.

摘要来源：arXiv