We show that the operations of mixing and wrapping linear distributionsaround a unit circle commute, and can produce a wide variety of circularmodels. In particular, we show that many wrapped circular models studied inthe literature can be obtained as scale mixtures of just the wrapped Gaussianand the wrapped exponential distributions, and inherit many properties fromthese two basic models. We also point out how this general approach canproduce flexible asymmetric circular models, the need for which has beennoted by many authors.

A generalization of the von Mises distribution, which is broad enough to cover unimodality as well as multimodality, symmetry as well as asymmetry of circular data, is discussed here. We study this distribution in some detail and discuss its many features, some inferential and computational aspects, and provide some important results including characterization properties for this distribution. © 2006 Elsevier B.V. All rights reserved.

Extreme value theory is part and parcel of any study of order statistics in one dimension. Our aim here is to consider such large sample theory for the maximum distance to the origin, and the related maximum "interpoint distance," in multidimensions. We show that for a family of spherically symmetric distributions, these statistics have a Gumbel-type limit, generalizing several existing results. We also discuss the other two types of limit laws and suggest some open problems. This work complements our earlier study on the minimum interpoint distance.

A very general measure of skewness based on the quantiles is introduced, which includes several well-known measures as special cases. Sample versions of our measure may be used as test statistics for testing the hypothesis of symmetry about an unknown value. We provide large sample theory for such a statistic and discuss the asymptotic relative efficiencies of this against some competing test statistics for symmetry. © 2012 Elsevier B.V..