Kernel Methods for Unimodality Test

Abstract

Finding the number of modes is of great interest in density estimation. Well known nonparametric unimodality tests are including the dip test, excess mass test, and Silverman's test. The dip and excess mass statistic are based on the empirical distribution and supremum distance, while Silverman's test depends on the bandwidth of kernel density estimator. A main issue of these tests is conservatism and often calibration methods are used to address this issue. We propose kernel methods of unimodality based on the dip and excess mass statistics to address the aforementioned issue. We proposed to use the total variation distance to identify the closest unimodal distribution to kernel distribution and construct the kernel dip test based on the unimodal distribution from calculating test statistics. Our numerical studies show that the proposed tests outperform. We also introduce a kernel excess mass statistics. Under the strong unimodal condition, the limiting distribution of the kernel excess mass statistic is the same as that of the empirical excess mass statistic. However the numerical studies indicate that the calibration of kernel excess mass test has a greater power and better level accuracy than the calibration of empirical excess mass test. We apply the proposed method to astronomy data, physical properties of minor planets in the solar system.