Nonparametric testing the similarity of two unknown density functions: Local power and bootstrap analysis

In this paper, we compare two kernel-based tests for testing the closeness between two unknown density functions, one test has a smoothing parameter it that goes to zero as sample size tends to infinity, the other one has a fixed smoothing parameter. We show that the former test (with a shrinking h) can be more powerful than the later for "singular" local alternatives considered by Rosenblatt (1975). We also demonstrate that bootstrap procedure provide a better approximation than the asymptotic normal approximation. Monte Carlo simulations are reported to examine the finite sample performances of the nonparametric kernel-based tests based on both the asymptotic and bootstrap critical values.