Quantile regression (QR) is becoming increasingly popular due to its relevance in many scientific investigations. There is a great amount of work about linear and nonlinear QR models. Specifically, nonparametric estimation of the conditional quantiles received particular attention, due to its model flexibility. However, nonparametric QR techniques are limited in the number of covariates. Dimension reduction offers a solution to this problem by considering low-dimensional smoothing without specifying any parametric or nonparametric regression relation. The existing dimension reduction techniques focus on the entire conditional distribution. We, on the other hand, turn our attention to dimension reduction techniques for conditional quantiles and introduce a new method for reducing the dimension of the predictor \(\mathbf {X}\). The novelty of this paper is threefold. We start by considering a single index quantile regression model, which assumes that the conditional quantile depends on \(\mathbf {X}\) through a single linear combination of the predictors, then extend to a multi-index quantile regression model, and finally, generalize the proposed methodology to any statistical functional of the conditional distribution. The performance of the methodology is demonstrated through simulation examples and real data applications. Our results suggest that this method has a good finite sample performance and often outperforms the existing methods.

中文翻译：

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中心分位数子空间

由于分位数回归（QR）在许多科学研究中的重要性，因此越来越受欢迎。关于线性和非线性QR模型，有大量工作要做。具体来说，由于条件模型的灵活性，对条件分位数的非参数估计引起了特别的关注。但是，非参数QR技术的协变量数量有限。通过考虑低维平滑而不指定任何参数或非参数回归关系，降维为该问题提供了解决方案。现有的降维技术专注于整个条件分布。另一方面，我们将注意力转移到条件分位数的降维技术上，并介绍了一种减少预测变量维的新方法\（\ mathbf {X} \）。本文的新颖性是三方面的。我们首先考虑一个单指数分位数回归模型，该模型假设条件分位数通过预测变量的单个线性组合取决于\（\ mathbf {X} \），然后扩展到一个多指数分位数回归模型，最后，将提出的方法推广到条件分布的任何统计功能。通过仿真示例和实际数据应用证明了该方法的性能。我们的结果表明，该方法具有良好的有限样本性能，并且通常优于现有方法。