Inference on multiple quantiles in regression models by a rank-score approach

TL;DR

This work tackles the problem of conducting inference across multiple quantiles in regression while controlling the familywise error rate. It develops a multivariate rank-score test and embeds it in a closed-testing framework to achieve strong $FWER$ control, then extends the test with a weighting matrix to enhance power in selected directions. Theoretical results establish the null and alternative distributions (including noncentral chi-square behavior) and simulations show superior performance over standard Bonferroni-based corrections under heteroscedasticity. An R package $quasar$ is provided to facilitate practical application of the proposed method.

Abstract

This paper tackles the challenge of performing multiple quantile regressions across different quantile levels and the associated problem of controlling the familywise error rate, an issue that is generally overlooked in practice. We propose a multivariate extension of the rank-score test and embed it within a closed-testing procedure to efficiently account for multiple testing. Then we further generalize the multivariate test to enhance statistical power against alternatives in selected directions. Theoretical foundations and simulation studies demonstrate that our method effectively controls the familywise error rate while achieving higher power than traditional corrections, such as Bonferroni.

Figures (5)

• Figure 1: Tree of intersection hypotheses for three individual hypotheses. Adapted from goeman2011multiple.
• Figure 2: Type I error control for Wald-type and generalized rank-score tests with $B = I_5$ for intersection hypotheses. Average $p$-values; solid grey lines correspond to $\alpha$, and dotted lines represent the $95\%$ simulation confidence interval for the empirical type I error.
• Figure 3: Empirical power of the generalized rank-score tests for the $31$ intersection hypotheses, comparing the identity weighting ($B = I_5$) with the inverse-covariance weighting ($B = A^{-1}$).
• Figure 4: Empirical power for individual hypotheses. Solid black lines represent the closed testing procedure, while dashed and dotted grey lines represent the Holm-Bonferroni and Bonferroni corrections, respectively, all applied to generalized rank-score tests with $B = f \circ F^{-1}(\tau)$, $B=I_{3}$ and $B = \text{diag}\{\Delta\}^{-1}$.
• Figure 5: Empirical power for individual hypotheses. Solid black lines represent the closed testing procedure, while the dashed and dotted grey lines represent the Holm-Bonferroni and Bonferroni corrections, respectively, all applied to generalized rank-score tests with $B=I_{9}$.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof