Exact risk improvement of bandwidth selectors for kernel density estimation with directional data

The work of the author has been supported by FPU grant AP2010–0957 from the Spanish Ministry of Education. Support of Project MTM2008–03010, from the Spanish Ministry of Science and Innovation, Project 10MDS207015PR from Direccion Xeral de I+D, Xunta de Galicia and IAP network StUDyS, from Belgian Science Policy, are acknowledged

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