This paper studies inference in the high-dimensional linear regression model with outliers. Sparsity constraints are imposed on the vector of coefficients of the covariates. The number of outliers can grow with the sample size while their proportion goes to 0. We propose a two-step procedure for inference on the coefficients of a fixed subset of regressors. The first step is a based on several square-root lasso l1-norm penalized estimators, while the second step is the ordinary least squares estimator applied to a well chosen regression. We establish asymptotic normality of the two-step estimator. The proposed procedure is efficient in the sense that it attains the semiparametric efficiency bound when applied to the model without outliers under homoscedasticity. This approach is also computationally advantageous, it amounts to solving a finite number of convex optimization programs.
翻译:本文对高维线性回归模型与外部线性回归模型的推论进行了研究。 对共变系数的矢量施加了差异性限制。 外向值的数量随着样本大小的增加而增加, 而其比例则达到0。 我们建议了一种两步程序,用于推论一个固定的递减子子子子的系数。 第一步基于若干受惩罚的平底 lasso l1- 诺姆测算器, 而第二步则是用于一个选择良好的回归的普通最小方位估计器。 我们建立了两步估测器的无症状常度。 拟议的程序是有效的, 因为它在对模型应用时实现了半对准效率的约束, 而没有在同质性下应用外差。 这个方法在计算上也是有利的, 它相当于解决数量有限的锥形优化程序。