On the choice of the smoothing parameter for the BHEP goodness-of-fit test

Abstract: The BHEP test for assessing univariate and multivariate normality introduced by Epps and Pulley (Biometrika 70 (1983) 723) and Baringhaus and Henze (Metrika 35 (1988) 339) has shown to be a relevant test procedure, recommended in some recent comparative studies. It is well known that the finite sample behaviour of the BHEP goodness-of-fit test strongly depends on the choice of a smoothing parameter h. In this paper we give a theoretical and finite sample based description of the role played by the smoothing parameter in the detection of departures from the null hypothesis of normality. Additionally, we develop a Monte Carlo study in order to propose an easy to use rule for the choice of h that produces a test with the omnibus property. In the important multivariate case, and contrary to the usual choice of h, the BHEP test with the smoothing parameter proposed in this paper presents a comparative good performance against a wide range of alternative distributions. In practice, if no relevant information about the tail of the alternatives is available, the use of this new bandwidth is strongly recommended. Otherwise, new choices of h which are suitable for short tailed and long tailed alternative distributions are also proposed.