A local modal estimation procedure is proposed for the regression function in a nonparametric regression model. A distinguishing characteristic of the proposed procedure is that it introduces an additional tuning parameter that is automatically selected using the observed data in order to achieve both robustness and efficiency of the resulting estimate. We demonstrate both theoretically and empirically that the resulting estimator is more efficient than the ordinary local polynomial regression (LPR) estimator in the presence of outliers or heavy-tail error distribution (such as t-distribution). Furthermore, we show that the proposed procedure is as asymptotically efficient as the LPR estimator when there are no outliers and the error distribution is a Gaussian distribution. We propose an expectation–maximisation-type algorithm for the proposed estimation procedure. A Monte Carlo simulation study is conducted to examine the finite sample performance of the proposed method. The simulation results confirm the theoretical findings. The proposed methodology is further illustrated via an analysis of a real data example.

中文翻译：

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局部模态回归

为非参数回归模型中的回归函数提出了局部模态估计程序。所提出的程序的一个显着特征是它引入了一个额外的调整参数，该参数使用观察到的数据自动选择，以实现结果估计的稳健性和效率。我们从理论和经验上证明，在存在异常值或重尾误差分布（例如 t 分布）的情况下，所得估计量比普通局部多项式回归 (LPR) 估计量更有效。此外，我们表明，当没有异常值且误差分布是高斯分布时，所提出的过程与 LPR 估计器一样渐近有效。我们为提议的估计程序提出了一种期望最大化型算法。进行蒙特卡罗模拟研究以检查所提出方法的有限样本性能。模拟结果证实了理论发现。通过对真实数据示例的分析进一步说明了所提出的方法。