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A robust proposal of estimation for the sufficient dimension reduction problem
TEST ( IF 1.3 ) Pub Date : 2021-01-03 , DOI: 10.1007/s11749-020-00745-9
Andrea Bergesio , María Eugenia Szretter Noste , Víctor J. Yohai

In nonparametric regression contexts, when the number of covariables is large, we face the curse of dimensionality. One way to deal with this problem when the sample is not large enough is using a reduced number of linear combinations of the explanatory variables that contain most of the information about the response variable. This leads to the so-called sufficient reduction problem. The purpose of this paper is to obtain robust estimators of a sufficient dimension reduction, that is, estimators which are not very much affected by the presence of a small fraction of outliers in the data. One way to derive a sufficient dimension reduction is by means of the principal fitted components (PFC) model. We obtain robust estimations for the parameters of this model and the corresponding sufficient dimension reduction based on a \(\tau \)-scale (\(\tau \)-estimators). Strong consistency of these estimators under weak assumptions of the underlying distribution is proven. The \(\tau \)-estimators for the PFC model are computed using an iterative algorithm. A Monte Carlo study compares the performance of \(\tau \)-estimators and maximum likelihood estimators. The results show clear advantages for \(\tau \)-estimators in the presence of outlier contamination and only small loss of efficiency when outliers are absent. A proposal to select the dimension of the reduction space based on cross-validation is given. These estimators are implemented in R language through functions contained in the package tauPFC. As the PFC model is a special case of multivariate reduced-rank regression, our proposal can be applied directly to this model as well.



中文翻译:

关于减少维数足够大的问题的可靠建议

在非参数回归的情况下,当协变量的数量很大时,我们将面临维度的诅咒。当样本不够大时解决此问题的一种方法是使用减少数量的解释变量线性组合,这些组合包含有关响应变量的大多数信息。这导致所谓的充分减少问题。本文的目的是获得足够减少维度的鲁棒估计量,即数据中存在少量异常值的影响不会很大。一种获得足够尺寸缩减的方法是借助主要装配组件(PFC)模型。我们基于该模型获得了该模型参数的可靠估计以及相应的充分降维\(\ tau \)- scale(\(\ tau \)- estimators)。这些估计量在基本分布的弱假设下具有强一致性。使用迭代算法计算PFC模型的\(\ tau \)估计量。蒙特卡洛研究比较了\(\ tau \)估计器和最大似然估计器的性能。结果表明,在存在离群值污染的情况下,\(\ tau \)估计量具有明显的优势,而在不存在离群值时,效率损失很小。提出了基于交叉验证选择缩减空间尺寸的建议。这些估算器通过tauPFC软件包中包含的功能以R语言实现。。由于PFC模型是多元降秩回归的特例,因此我们的建议也可以直接应用于该模型。

更新日期:2021-01-03
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