We are interested in the nonparametric estimation of the probability density of price returns, using the kernel approach. The output of the method heavily relies on the selection of a bandwidth parameter. Many selection methods have been proposed in the statistical literature. We put forward an alternative selection method based on a criterion coming from information theory and from the physics of complex systems: the bandwidth to be selected maximizes a new measure of complexity, with the aim of avoiding both overfitting and underfitting. We review existing methods of bandwidth selection and show that they lead to contradictory conclusions regarding the complexity of the probability distribution of price returns. This has also some striking consequences in the evaluation of the relevance of the efficient market hypothesis. We apply these methods to real financial data, focusing on the Bitcoin.

翻译：我们关注于利用核方法对价格收益的概率密度进行非参数估计。该方法的输出结果高度依赖于带宽参数的选择。统计学文献中提出了多种带宽选择方法。我们提出了一种基于信息论与复杂系统物理学标准的新型替代选择方法：所选的带宽能够最大化一种新的复杂性度量，旨在避免过拟合与欠拟合。我们对现有带宽选择方法进行了综述，并揭示这些方法在价格收益概率分布复杂性方面得出了相互矛盾的结论。这一现象对有效市场假说相关性的评估也产生了显著影响。我们以比特币为重点研究对象，将这些方法应用于实际金融数据。