Quantile regression is used to study effects of covariates on a particular quantile of the data distribution. Here we are interested in the question whether a covariate has any effect on the entire data distribution, i.e., on any of the quantiles. To this end, we treat all the quantiles simultaneously and consider global tests for the existence of the covariate effect in the presence of nuisance covariates. This global quantile regression can be used as the extension of linear regression or as the extension of distribution comparison in the sense of Kolmogorov-Smirnov test. The proposed method is based on pointwise coefficients, permutations and global envelope tests. The global envelope test serves as the multiple test adjustment procedure under the control of the family-wise error rate and provides the graphical interpretation which automatically shows the quantiles or the levels of categorical covariate responsible for the rejection. The Freedman-Lane permutation strategy showed liberality of the test for extreme quantiles, therefore we propose four alternatives that work well even for extreme quantiles and are suitable in different conditions. We present a simulation study to inspect the performance of these strategies, and we apply the chosen strategies to two data examples.

翻译：分位数回归用于研究协变量对数据分布特定分位数的影响。本文关注的问题是：协变量是否对整体数据分布（即所有分位数）存在影响。为此，我们将所有分位数同时纳入考量，并在存在干扰协变量的情况下，对协变量效应的存在性进行全局检验。这种全局分位数回归既可作为线性回归的扩展，也可作为Kolmogorov-Smirnov检验意义上分布比较的扩展。所提出的方法基于逐点系数、置换检验和全局包络检验。全局包络检验作为在控制族系误差率条件下的多重检验校正程序，能够提供图形化解释，自动显示导致拒绝原假设的分位数或分类协变量的水平。Freedman-Lane置换策略在极端分位数处表现出检验的自由性，因此我们提出了四种适用于不同条件的替代方案，即使在极端分位数情况下也能良好运作。我们通过模拟研究检验了这些策略的性能，并将选定的策略应用于两个数据实例。