This paper presents a comprehensive study of nonparametric estimation techniques on the circle using Fej\'er polynomials, which are analogues of Bernstein polynomials for periodic functions. Building upon Fej\'er's uniform approximation theorem, the paper introduces circular density and distribution function estimators based on Fej\'er kernels. It establishes their theoretical properties, including uniform strong consistency and asymptotic expansions. The proposed methods are extended to account for measurement errors by incorporating classical and Berkson error models, adjusting the Fej\'er estimator to mitigate their effects. Simulation studies analyze the finite-sample performance of these estimators under various scenarios, including mixtures of circular distributions and measurement error models. An application to rainfall data demonstrates the practical application of the proposed estimators, demonstrating their robustness and effectiveness in the presence of rounding-induced Berkson errors.

翻译：本文系统研究了利用费耶尔多项式在圆周上进行非参数估计的技术，该多项式是周期函数情形下伯恩斯坦多项式的类比。基于费耶尔一致逼近定理，本文提出了基于费耶尔核的圆周密度函数与分布函数估计量，并建立了其理论性质，包括一致强相合性与渐近展开。通过引入经典误差模型与伯克森误差模型，本文进一步扩展了所提方法以处理测量误差问题，并调整费耶尔估计量以减弱其影响。模拟研究分析了这些估计量在不同情境下的有限样本性能，包括圆周分布混合模型与测量误差模型。通过对降雨数据的实际应用，展示了所提估计量在存在舍入导致的伯克森误差时的稳健性与有效性。