Functional quadratic regression models postulate a polynomial relationship between a scalar response rather than a linear one. As in functional linear regression, vertical and specially high-leverage outliers may affect the classical estimators. For that reason, the proposal of robust procedures providing reliable estimators in such situations is an important issue. Taking into account that the functional polynomial model is equivalent to a regression model that is a polynomial of the same order in the functional principal component scores of the predictor processes, our proposal combines robust estimators of the principal directions with robust regression estimators based on a bounded loss function and a preliminary residual scale estimator. Fisher-consistency of the proposed method is derived under mild assumptions. The results of a numerical study show, for finite samples, the benefits of the robust proposal over the one based on sample principal directions and least squares. The usefulness of the proposed approach is also illustrated through the analysis of a real data set which also reveals that when the potential outliers are removed the classical and robust methods behave very similarly.

翻译：功能二次回归模型假定一个星标反应而不是线性反应之间的多元度关系。 正如功能线性回归一样,垂直和特别高杠杆离子可能会影响古典估计值。 因此,在这种情况下,提出可靠估算器的可靠程序提案是一个重要问题。考虑到功能性多面回归模型相当于一个在预测程序功能主要部分分数中同一顺序的多重回归模型,我们的提案将主要方向的稳健估测器与基于约束性损失函数的稳健回归度估测器和初步剩余比例估测器结合起来。拟议方法的渔业一致性是在温和假设下推算出来的。对于有限的样本,数字研究结果显示,稳健建议对基于样本主方向和最小方形的模型的好处。通过分析真实数据也说明了拟议方法的有用性,该数据还表明,当潜在外端被清除时,古典和稳健方法也表现非常相似。