Modified likelihood ratio test for homogeneity in a mixture of von Mises distributions

Directional data often arise in many sciences, including astronomy, biology, ecology, geology and medicine. One particular statistical problem of interest is whether the data are from a mixture of two von Mises distributions or one single von Mises distribution. Motivating examples include a DNA microarray experiment, where it is suggested that a proportion of circadian genes have systematically different phase/peak expressions in two different tissues. We study the use of the modified likelihood ratio test (MLRT) to this class of problems. The MLRT statistic is shown to have a simple χ12 null limiting distribution. The result is extended to mixture models with general parametric kernels. The simulation study gives additional insight into the finite-sample performance of the test. Two real data examples are used to illustrate the proposed method.