This paper focuses on the problem of testing the null hypothesis that the regression functions of several populations are equal under a general nonparametric homoscedastic regression model. It is well known that linear kernel regression estimators are sensitive to atypical responses. These distorted estimates will influence the test statistic constructed from them so the conclusions obtained when testing equality of several regression functions may also be affected. In recent years, the use of testing procedures based on empirical characteristic functions has shown good practical properties. For that reason, to provide more reliable inferences, we construct a test statistic that combines characteristic functions and residuals obtained from a robust smoother under the null hypothesis. The asymptotic distribution of the test statistic is studied under the null hypothesis and under root$-n$ contiguous alternatives. A Monte Carlo study is performed to compare the finite sample behaviour of the proposed test with the classical one obtained using local averages. The reported numerical experiments show the advantage of the proposed methodology over the one based on Nadaraya-Watson estimators for finite samples. An illustration to a real data set is also provided and enables to investigate the sensitivity of the $p-$value to the bandwidth selection.

翻译：本文聚焦于在一般非参数同方差回归模型下检验多个总体回归函数相等的原假设问题。众所周知，线性核回归估计量对异常响应值敏感。这些扭曲的估计量会影响基于它们构建的检验统计量，因此检验多个回归函数相等性时得出的结论也可能受到影响。近年来，基于经验特征函数的检验方法展现出良好的实际性能。为此，为提供更可靠的推断，我们构建了一个结合特征函数与在原假设下通过稳健平滑器获得的残差的检验统计量。研究了该检验统计量在原假设和根号$n$邻接备择假设下的渐近分布。通过蒙特卡洛研究，将所提出检验的有限样本表现与使用局部平均获得的经典检验进行了比较。数值实验结果表明，所提出方法在有限样本下优于基于Nadaraya-Watson估计量的方法。对真实数据集的示例分析也表明，该方法能够研究$p$值对带宽选择的敏感性。