In this paper, we deal with nonparametric regression for circular data, meaning that observations are represented by points lying on the unit circle. We propose a kernel estimation procedure with data-driven selection of the bandwidth parameter. For this purpose, we use a warping strategy combined with a Goldenshluger-Lepski type estimator. To study optimality of our methodology, we consider the minimax setting and prove, by establishing upper and lower bounds, that our procedure is nearly optimal on anisotropic Holder classes of functions for pointwise estimation. The obtained rates also reveal the specific nature of regression for circular responses. Finally, a numerical study is conducted, illustrating the good performances of our approach.

翻译：本文研究了圆形数据的非参数回归问题，其中观测值表示为位于单位圆上的点。我们提出了一种基于数据驱动带宽参数选择的核估计方法。为此，我们采用扭曲策略结合Goldenshluger-Lepski型估计量。为评估我们方法的最优性，我们考虑极小化极大框架，并通过建立上界和下界证明，我们的方法在各向异性Hölder函数类上的逐点估计几乎达到最优。所获得的收敛速率还揭示了圆形响应回归的特定性质。最后，通过数值研究验证了该方法的良好性能。