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We are interested in the nonparametric estimation of the probability densityof price returns, using the kernel approach. The output of the method heavilyrelies on the selection of a bandwidth parameter. Many selection methods havebeen proposed in the statistical literature. We put forward an alternativeselection method based on a criterion coming from information theory and fromthe physics of complex systems: the bandwidth to be selected maximizes a newmeasure of complexity, with the aim of avoiding both overfitting andunderfitting. We review existing methods of bandwidth selection and show thatthey lead to contradictory conclusions regarding the complexity of theprobability distribution of price returns. This has also some strikingconsequences in the evaluation of the relevance of the efficient markethypothesis. We apply these methods to real financial data, focusing on theBitcoin.

Complexity measure, kernel density estimation, bandwidth selection, and the efficient market hypothesis