  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.
