标题： 复合Lp分位数回归，近似分位数回归和Oracle模型选择理论 显示英文标题

标题： Composite Lp-quantile regression, near quantile regression and the oracle model selection theory

摘要： 在本文中，我们考虑高维Lp-分位数回归，它仅需要误差的低阶矩，并且也是上述方法和Lp回归的自然推广。 Lp-分位数回归的损失函数避免了绝对损失函数的不可微性以及平方损失函数要求误差方差有限的困难，因此保证了Lp-分位数回归的良好性质。 具体来说，我们首先开发了一种称为复合Lp-分位数回归（CLpQR）的新方法。 我们基于CLpQR研究了Oracle模型选择理论（称为CLpQR-oracle估计量），并证明在某些情况下，当误差方差为无限时，CLpQR-oracle的表现优于基于复合分位数回归的CQR-oracle。 此外，CLpQR具有高效率，有时比CQR和最小二乘回归更高效。 其次，我们提出另一种新的回归方法，即近似分位数回归，并证明当p趋近于1且样本量趋于无穷大时估计量的渐近正态性。 作为其应用，提供了一种平滑分位数目标函数的新思路以及分位数回归渐近协方差矩阵的新估计方法。 第三，我们通过结合循环坐标下降和增广邻近梯度算法，开发了一个统一的高效算法来拟合高维Lp-分位数回归。 值得注意的是，该算法在拟合分位数回归时，成为常用线性规划和内点算法的一个有利替代方案。 显示英文摘要

摘要： In this paper, we consider high-dimensional Lp-quantile regression which only requires a low order moment of the error and is also a natural generalization of the above methods and Lp-regression as well. The loss function of Lp-quantile regression circumvents the non-differentiability of the absolute loss function and the difficulty of the squares loss function requiring the finiteness of error's variance and thus promises excellent properties of Lp-quantile regression. Specifically, we first develop a new method called composite Lp-quantile regression(CLpQR). We study the oracle model selection theory based on CLpQR (call the estimator CLpQR-oracle) and show in some cases of p CLpQR-oracle behaves better than CQR-oracle (based on composite quantile regression) when error's variance is infinite. Moreover, CLpQR has high efficiency and can be sometimes arbitrarily more efficient than both CQR and the least squares regression. Second, we propose another new regression method,i.e. near quantile regression and prove the asymptotic normality of the estimator when p converges to 1 and the sample size infinity simultaneously. As its applications, a new thought of smoothing quantile objective functions and a new estimation are provided for the asymptotic covariance matrix of quantile regression. Third, we develop a unified efficient algorithm for fitting high-dimensional Lp-quantile regression by combining the cyclic coordinate descent and an augmented proximal gradient algorithm. Remarkably, the algorithm turns out to be a favourable alternative of the commonly used liner programming and interior point algorithm when fitting quantile regression.