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:部分线性添加模型将线性模型笼统地归纳成线性模型,因为它们模拟反应变数和共变数之间的关系,假定一些共变数与反应有线性线性反应的线性关系,但其他各变数通过未知的单向光滑功能进入。线性组成部分的剩余部分或共变数的外部效应,在部分线性模型的情况下,即只涉及一个非参数部分时,即模型中只涉及一个非参数部分时,说明线性模型的有害影响。在处理添加成分时,在出现非典型数据时提供可靠的估计数据的问题具有实际重要性,促使需要强有力的程序。因此,我们建议对部分线性添加模型进行一套稳健的估算,将美元与稳健的线性线性回归估计值结合起来。我们根据温和的假设,取得一致的结果、趋同率和线性正常度,对线性组成部分进行一项蒙特卡洛研究,将稳健提案的绩效与不同模型和污染计划下的典型对应方进行比较。数字实验显示了拟议方法对有限样品的优点。我们还说明了拟议采用的实际数据集方法的效用。