Testing the equality of two density functions based on independent samples is a classical problem in statistics. While the focus is often on global equality, it is also of interest to conduct local comparisons of density functions. Typically a type of Wald statistic is employed, where the local difference in densities is standardized by an estimate of the asymptotic standard error of that difference. We study the null distribution of this test statistic. The literature has suggested that this will be asymptotically standard normal, but we show that this is by no means always the case. In particular, when using bandwidth matrices of optimal order (for estimation), we prove that the asymptotic mean of this null distribution is nonzero when either the sample sizes differ, or when the Hessian matrices of the densities differ at the point where the densities are equal. In numerical studies we find the erroneous use of the standard normal null distribution in such cases can severely corrupt the test size. We show that these problems can be managed effectively by using common bandwidths when the Hessian matrices are equal, and applying adjusted undersmoothing bandwidth matrices when they are not.
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