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 reveals that when the potential outliers are removed the classical and robust methods behave very similarly.

翻译：函数型二次回归模型假设标量响应与预测变量之间存在多项式关系，而非线性关系。与函数型线性回归类似，垂直异常值以及高杠杆异常值可能影响经典估计量。因此，提出能在这种情况下提供可靠估计的稳健方法是一个重要课题。考虑到函数型多项式模型等价于一个回归模型，该模型在预测过程的函数主成分得分上呈现相同阶数的多项式形式，我们的方法结合了主方向的稳健估计与基于有界损失函数和初步残差尺度估计的稳健回归估计量。在温和假设下导出了所提方法的Fisher相合性。数值研究结果表明，对于有限样本，稳健方法优于基于样本主方向和最小二乘的方法。通过一个真实数据集的分析进一步说明了所提方法的实用性，该分析揭示当潜在异常值被移除时，经典方法与稳健方法的表现非常相似。