Until now the problem of estimating circular densities when data are observed with errors has been mainly treated by Fourier series methods. We propose kernel-based estimators exhibiting simple construction and easy implementation. Specifically, we consider three different approaches: the first one is based on the equivalence between kernel estimators using data corrupted with different levels of error. This proposal appears to be totally unexplored, despite its potential for application also in the Euclidean setting. The second approach relies on estimators whose weight functions are circular deconvolution kernels. Due to the periodicity of the involved densities, it requires ad hoc mathematical tools. Finally, the third one is based on the idea of correcting extra bias of kernel estimators which use contaminated data and is essentially an adaptation of the standard theory to the circular case. For all the proposed estimators, we derive asymptotic properties, provide some simulation results, and also discuss some possible generalizations and extensions. Real data case studies are also included.

中文翻译：

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观察到有错误的圆形数据的密度估计

到目前为止，当观测到有误差的数据时估计圆形密度的问题主要通过傅里叶级数方法来处理。我们提出了基于内核的估计器，它们具有简单的构造和易于实现的特点。具体来说，我们考虑了三种不同的方法：第一种方法是基于内核估计器之间的等效性，使用的数据被不同程度的错误破坏。尽管它在欧几里得环境中也有应用的潜力，但这个提议似乎完全没有被探索过。第二种方法依赖于权重函数是循环反卷积核的估计器。由于所涉及密度的周期性，它需要特殊的数学工具。最后，第三个是基于修正的思路使用受污染数据的内核估计器的额外偏差，本质上是标准理论对循环情况的适应。对于所有提出的估计量，我们推导了渐近性质，提供了一些模拟结果，并讨论了一些可能的推广和扩展。真实数据案例研究也包括在内。