Quantiles and expectiles have been receiving much attention in many areas such as economics, ecology, and finance. By means of L^p optimization, both quantiles and expectiles can be embedded in a more general class of M-quantiles. Inspired by this point of view, we propose a generalized regression called L^p-quantile regression to study the whole conditional distribution of a response variable given predictors in a heterogeneous regression setting. In this article, we focus on the variable selection aspect of high-dimensional penalized L^p-quantile regression, which provides a flexible application and makes a complement to penalized quantile and expectile regressions. This generalized penalized L^p-quantile regression steers an advantageous middle course between ordinary penalized quantile and expectile regressions without sacrificing their virtues too much when 1 < p < 2, that is, offers versatility and flexibility with these ‘quantile-like’ and robustness properties. We develop the penalized L^p-quantile regression with scad and adaptive lasso penalties. With properly chosen tuning parameters, we show that the proposed estimators display oracle properties. Numerical studies and real data analysis demonstrate the competitive performance of the proposed penalized L^p-quantile regression when 1 < p < 2, and they combine the robustness properties of quantile regression with the efficiency of penalized expectile regression. These properties would be helpful for practitioners.

中文翻译：

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惩罚高维 M 分位数回归：从 L1 到 Lp 优化

分位数和期望值在经济学、生态学、金融学等诸多领域受到广泛关注。通过L ^p优化，分位数和期望值都可以嵌入到更一般的 M 分位数类别中。受这一观点的启发，我们提出了一种称为L ^{p 分}位数回归的广义回归，以研究在异质回归设置中给定预测变量的响应变量的整个条件分布。在本文中，我们专注于高维惩罚L ^{p 分}位数回归的变量选择方面，它提供了灵活的应用，并对惩罚分位数和期望回归进行了补充。这个广义惩罚L ^p当 1 < p  < 2 时，-分位数回归在普通惩罚分位数和期望回归之间占据有利的中间路线，而不会过多地牺牲它们的优点 ，也就是说，提供了具有这些“类分位数”和稳健性属性的多功能性和灵活性。我们开发了带有scad和自适应套索惩罚的惩罚L ^{p 分}位数回归。通过正确选择调整参数，我们表明建议的估计器显示了预言机属性。数值研究和真实数据分析证明了当 1 < p时提出的惩罚L ^{p 分}位数回归 的竞争性能 < 2，他们将分位数回归的稳健性与惩罚期望回归的效率相结合。这些属性将对从业者有所帮助。