This paper is concerned with a partially linear semiparametric regression model containing an unknown regression coefficient, an unknown nonparametric function, and an unobservable Gaussian distributed random error. We focus on the case of simultaneous variable selection and estimation with a divergent number of covariates under the assumption that the regression coefficient is sparse. We consider the applications of the least squares to semiparametric regression and particularly present an adaptive lasso penalized least squares (PLS) method to select the regression coefficient. We note that there are many algorithms for PLS in various applications, but they seem to be rarely used in semiparametric regression. This paper focuses on using a semismooth Newton augmented Lagrangian (SSNAL) algorithm to solve the dual of PLS which is the sum of a smooth strongly convex function and an indicator function. At each iteration, there must be a strongly semismooth nonlinear system, which can be solved by semismooth Newton by making full use of the penalized term. We show that the algorithm offers a significant computational advantage, and the semismooth Newton method admits fast local convergence rate. Numerical experiments on simulated and real data have demonstrated the effectiveness of the PLS method and the progressiveness of the SSNAL algorithm.

翻译：本文研究一个包含未知回归系数、未知非参数函数以及不可观测高斯分布随机误差的部分线性半参数回归模型。我们关注在回归系数具有稀疏性假设下，协变量数量发散时的同步变量选择与估计问题。我们探讨了最小二乘法在半参数回归中的应用，特别提出了一种自适应Lasso惩罚最小二乘（PLS）方法用于回归系数选择。我们注意到虽然存在多种适用于不同场景的PLS算法，但这些算法在半参数回归中应用甚少。本文重点采用半光滑牛顿增广拉格朗日（SSNAL）算法求解PLS的对偶问题，该对偶问题由光滑强凸函数与示性函数之和构成。每次迭代都需要求解一个强半光滑非线性方程组，该方程组可通过充分利用惩罚项特性，采用半光滑牛顿法有效求解。我们证明该算法具有显著的计算优势，且半光滑牛顿法具有快速的局部收敛速率。基于模拟数据与真实数据的数值实验验证了PLS方法的有效性及SSNAL算法的先进性。