A test of singularity for distribution functions

Many non- and semi- parametric estimators have asymptotic properties that have been established under conditions that exclude the possibility of singular parts in the distribution. It is thus important to be able to test for absence of singularities. Methods of testing that focus on specific singularities do exist, but there are few generally applicable approaches. A general test based on kernel density estimation was proposed by Frigyesi and Hössjer (1998), but this statistic can diverge for some absolutely continuous distributions. Here we use a result in Zinde-Walsh (2008) to characterize distributions with varying degrees of smoothness, via functionals that reveal the behavior of the bias of the kernel density estimator. The statistics proposed here have well defined asymptotic distributions that are asymptotically pivotal in some class of distributions (e.g. for continuous density) and diverge for distributions in an alternative class, at a rate that can be explicitly evaluated and controlled.