Wu Xiaofei, Liang Rongmei, Yang Hu
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331 People's Republic of China.
Stat Pap (Berl). 2022;63(1):53-95. doi: 10.1007/s00362-021-01229-0. Epub 2021 Mar 31.
Recently, many literatures have proved that prior information and structure in many application fields can be formulated as constraints on regression coefficients. Following these work, we propose a penalized LAD estimation with some linear constraints in this paper. Different from constrained lasso, our estimation performs well when heavy-tailed errors or outliers are found in the response. In theory, we show that the proposed estimation enjoys the Oracle property with adjusted normal variance when the dimension of the estimated coefficients is fixed. And when is much greater than the sample size , the error bound of proposed estimation is sharper than . It is worth noting the result is true for a wide range of noise distribution, even for the Cauchy distribution. In algorithm, we not only consider an typical linear programming to solve proposed estimation in fixed dimension , but also present an nested alternating direction method of multipliers (ADMM) in high dimension. Simulation and application to real data also confirm that proposed estimation is an effective alternative when constrained lasso is unreliable.
最近，许多文献证明，在许多应用领域中，先验信息和结构可以被表述为对回归系数的约束。基于这些工作，我们在本文中提出了一种带有一些线性约束的惩罚最小绝对偏差（LAD）估计。与约束套索估计不同，当响应中存在重尾误差或异常值时，我们的估计表现良好。在理论上，我们表明当估计系数的维度固定时，所提出的估计具有调整后的正态方差的神谕性质。并且当维度远大于样本量时，所提出估计的误差界比[未提及的某个估计的误差界]更精确。值得注意的是，该结果对于广泛的噪声分布都是成立的，甚至对于柯西分布也是如此。在算法方面，我们不仅考虑一种典型的线性规划来求解固定维度下的所提出估计，还在高维度下提出了一种嵌套交替方向乘子法（ADMM）。对实际数据的模拟和应用也证实，当约束套索估计不可靠时，所提出的估计是一种有效的替代方法。
