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

The BHEP (Baringhaus--Henze--Epps--Pulley) test for assessing univariate and multivariate normality has shown itself 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 $h$. 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 is given. Additionally, the results of a Monte Carlo study are reported in order to propose an easy-to-use rule for choosing $h$. In the important multivariate case, and contrary to the usual choice of $h$, the BHEP test with the proposed smoothing parameter presents a comparatively 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