Partially linear additive models generalize the linear models since they model the relation between a response variable and covariates by assuming that some covariates are supposed to have a linear relation with the response but each of the others enter with 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:部分线性添加模型概括线性模型,因为它们模拟反应变数和共变数之间的关系,假定有些共变体与反应有线性关系,但每个共变体都具有未知的单向光滑功能。线性组成部分的剩余部分或共变体的外差的有害影响在部分线性模型的情况中已有描述,即,在模型只涉及一个非对数组成部分的情况下,即,在模型中只涉及一个非对数组成部分时,在处理添加部分时,提供可靠的估算器的问题是实际需要稳健程序的问题。因此,我们建议对部分线性添加模型采用一套强势估计器,将美元-美元与强线性线性回归估计器结合起来。我们获得了一致性的结果、趋同率和线性正常度,在轻度假设下,对线性组成部分进行了一项蒙特卡洛研究,将稳健提案的性能与不同模型和污染计划下的典型对应方进行对比。数字实验显示了提议的定点样品方法的优势。我们还说明了拟议采用的实际数据集的效用。