Tests for detecting probability mass points

The objective of this paper is developing test statistics to detect the presence of mass points when data are possibly generated by a mixture of a continuous and a discrete distribution. To serve our purpose we propose two versions of the probability mass point (PMP) test. We derive the limiting distributions of the PMP test statistics under the null and alternative hypothesis by exploiting the asymptotic difference between two kernel density estimators with different bandwidths. Specifically, the proposed PMP test statistic is shown to converge to either the standard normal distribution or a linear transformation of a positive Poisson distribution at a non-mass point depending on bandwidths choice, while it diverges to infinity at a mass point. Numerical experiments are conducted to demonstrate the validity of our proposed tests. Korean taxpayers’ bunching behavior is considered as an empirical application.