Adaptive Test for High Dimensional Quantile Regression
Abstract
Testing high-dimensional quantile regression coefficients is crucial, as tail quantiles often reveal more than the mean in many practical applications. Nevertheless, the sparsity pattern of the alternative hypothesis is typically unknown in practice, posing a major challenge. To address this, we propose an adaptive test that remains powerful across both sparse and dense alternatives.We first establish the asymptotic independence between the max-type test statistic proposed by \citet{tang2022conditional} and the sum-type test statistic introduced by \citet{chen2024hypothesis}. Building on this result, we propose a Cauchy combination test that effectively integrates the strengths of both statistics and achieves robust performance across a wide range of sparsity levels. Simulation studies and real data applications demonstrate that our proposed procedure outperforms existing methods in terms of both size control and power.
Summary
The paper addresses the problem of testing high-dimensional quantile regression coefficients, a crucial task because tail quantiles often reveal more information than the mean in many applications. The authors note that the sparsity pattern of the alternative hypothesis is typically unknown, making it challenging to design a powerful test. To overcome this, they propose an adaptive test that maintains good power under both sparse and dense alternatives. Their approach involves establishing the asymptotic independence between the max-type test statistic proposed by Tang et al. (2022) and the sum-type test statistic introduced by Chen et al. (2024b). Building on this independence, they construct a Cauchy combination test, which effectively integrates the strengths of both statistics. Simulation studies and real-world data applications demonstrate that the proposed procedure outperforms existing methods in terms of both size control and power. This matters to the field because it provides a robust and adaptive tool for high-dimensional quantile regression testing, regardless of the underlying sparsity structure of the alternative hypothesis.
Key Insights
- •The core contribution is the adaptive test for high-dimensional quantile regression, constructed by combining a max-type test (sensitive to sparse alternatives) and a sum-type test (sensitive to dense alternatives).
- •The key theoretical result is the proof of asymptotic independence between the max-type and sum-type test statistics under the null hypothesis and specific alternative hypothesis (A1-A5). This justifies the combination of their p-values.
- •The authors use a Cauchy combination test to integrate the p-values from the max-type and sum-type tests. This method is known for its robustness under dependence.
- •Simulation results demonstrate the adaptive nature of the proposed test. The Cauchy Combination test (T_CC) consistently performs well across a range of sparsity levels, outperforming the individual max-type (T_MAX) and sum-type (T_SUM) tests in scenarios where the sparsity is mismatched to the test's strengths.
- •The real data application, using a wave energy farm dataset, shows that T_CC and T_MAX consistently reject the null hypothesis, indicating influential neighboring WECs. T_SUM's rejection rate is quantile-dependent, suggesting that aggregate coupling effects vary across the power output distribution.
- •A limitation is the assumption (A2) of multivariate normality for the covariates, which might not hold in all practical scenarios. Extending the framework to non-Gaussian settings is mentioned as a future research direction.
- •The paper builds upon previous work on combining sum-type and max-type tests in other high-dimensional testing problems, extending the idea to the quantile regression framework.
Practical Implications
- •The adaptive test can be used in various real-world applications where high-dimensional data is analyzed using quantile regression, and the sparsity pattern of the regression coefficients is unknown. Examples include finance, genomics, and environmental science.
- •Researchers and practitioners can use the proposed Cauchy combination test to improve the power and robustness of their hypothesis tests in high-dimensional quantile regression settings.
- •The findings suggest that relying solely on sparse-oriented or dense-oriented procedures may lead to substantial power loss, highlighting the importance of adaptive testing procedures.
- •The real data application provides insights into the performance of wave energy farms, demonstrating how quantile regression can be used to assess the reliability, resilience, and extreme-event behavior of these systems.
- •Future research directions include generalizing the framework to more flexible distributions for the covariates and exploring alternative test statistics that bridge the gap between max-type and sum-type approaches.