Modelling Circular Data Using a Mixture of Von Mises and Uniform Distributions

The von Mises distribution is often useful for modelling circular data problems. We consider a model for which von Mises data is contaminated with a certain proportion of points uniformly distributed around the circle. Maximum likelihood estimation is used to produce parameter estimates for this mixture model. Computational issues involved with obtaining the maximum likelihood estimates for the mixture model are discussed. Both parametric and goodness-of-fit based test procedures are presented for selecting the appropriate model (uniform, von Mises, mixture) and determining its adequacy. Parametric tests presented in this project are based on the likelihood ratio test statistic and goodness-of-fit tests are based on Watson’s goodness-of-fit statistic for the circle. A parametric bootstrap is performed to obtain the approximate distribution of Watson’s statistic in situations where the true parameter values are unknown.