We propose both a penalized quantile regression and an independence screening procedure to identify important covariates and to exclude unimportant ones for a general class of ultrahigh dimensional single-index models, in which the conditional distribution of the response depends on the covariates via a single-index structure. We observe that the linear quantile regression yields a consistent estimator of the direction of the index parameter in the single-index model. Such an observation dramatically reduces computational complexity in selecting important covariates in the single-index model. We establish an oracle property for the penalized quantile regression estimator when the covariate dimension increases at an exponential rate of the sample size. From a practical perspective, however, when the covariate dimension is extremely large, the penalized quantile regression may suffer from at least two drawbacks: computational expediency and algorithmic stability. To address these issues, we propose an independence screening procedure which is robust to model misspecification, and has reliable performance when the distribution of the response variable is heavily tailed or response realizations contain extreme values. The new independence screening procedure offers a useful complement to the penalized quantile regression since it helps to reduce the covariate dimension from ultrahigh dimensionality to a moderate scale. Based on the reduced model, the penalized linear quantile regression further refines selection of important covariates at different quantile levels. We examine the finite sample performance of the newly proposed procedure by Monte Carlo simulations and demonstrate the proposed methodology by an empirical analysis of a real data set.

中文翻译：

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单指标模型的正则化分位数回归和稳健特征筛选

我们提出了惩罚分位数回归和独立筛选程序来识别重要的协变量并排除一般类别的超高维单指数模型的不重要的协变量，其中响应的条件分布通过单指数取决于协变量我们观察到线性分位数回归在单指数模型中产生了指数参数方向的一致估计量。这样的观察极大地降低了单指数模型中选择重要协变量的计算复杂度。我们建立了一个预言属性。对于当协变量维数以样本量的指数速率增加时，惩罚分位数回归估计量。然而，从实际的角度来看，当协变量维数非常大时，惩罚分位数回归可能存在至少两个缺陷：计算便利性和算法稳定性。为了解决这些问题，我们提出了一种独立筛选程序，该程序对模型指定错误具有鲁棒性，并且在响应变量的分布严重拖尾时具有可靠的性能新的独立性筛选程序为惩罚分位数回归提供了有用的补充，因为它有助于将协变量维度从超高维减少到中等规模。基于简化模型，惩罚线性分位数回归进一步细化了不同分位数的重要协变量的选择水平。我们通过蒙特卡罗模拟检查新提出的程序的有限样本性能，并通过对真实数据集的经验分析来证明所提出的方法。