Small nonparametric tolerance regions for directional data

We present a natural approach, based on minimum volume sets, for constructing nonparametric tolerance regions for directional data. The tolerance regions have desirable features like invariance and are asymptotically minimal under certain conditions. We establish the asymptotic correctness of our tolerance regions by using the theory of empirical processes and generalized quantiles. The results are obtained under minimal conditions. In case of circular data, the finite sample properties of the tolerance arcs are studied through simulations. The method is also applied to a real data example.