Partially linear additive models generalize linear ones since they model the relation between a response variable and covariates by assuming that some covariates have a linear relation with the response but each of the others enter through unknown univariate smooth functions. The harmful effect of outliers either in the residuals or in the covariates involved in the linear component has been described in the situation of partially linear models, that is, when only one nonparametric component is involved in the model. When dealing with additive components, the problem of providing reliable estimators when atypical data arise, is of practical importance motivating the need of robust procedures. Hence, we propose a family of robust estimators for partially linear additive models by combining $B-$splines with robust linear regression estimators. We obtain consistency results, rates of convergence and asymptotic normality for the linear components, under mild assumptions. A Monte Carlo study is carried out to compare the performance of the robust proposal with its classical counterpart under different models and contamination schemes. The numerical experiments show the advantage of the proposed methodology for finite samples. We also illustrate the usefulness of the proposed approach on a real data set.

翻译：部分线性加性模型是对线性模型的推广，它通过假设某些协变量与响应变量呈线性关系，而其余每个协变量通过未知一元光滑函数进入模型，来刻画响应变量与协变量之间的关系。在仅包含一个非参数分量的部分线性模型场景中，残差或线性分量中涉及的协变量出现异常值的有害影响已被描述。当处理加性分量时，在异常数据出现时提供可靠估计的问题具有实际重要性，这促使需要稳健方法。因此，我们通过将 $B-$样条与稳健线性回归估计量相结合，提出了一类针对部分线性加性模型的稳健估计量。在温和假设下，我们获得了线性分量的一致性结果、收敛速度和渐近正态性。通过蒙特卡洛研究，比较了在不同模型和污染方案下稳健方法与其经典方法的性能。数值实验显示了所提方法在有限样本下的优势。我们还通过一个真实数据集说明了所提方法的实用性。