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A general method of constructing a circular model is to apply transformation to its argument. In this paper, we propose a new transformation in order to construct a large class of new skew-symmetric circular models. We consider a symmetric unimodal distribution as our base model and provide general results for the modality, skewness and shape properties of the resulting circular distributions. The new class enfolds two well-known and common classes of unimodal symmetric and asymmetric circular distributions. The skewness parameter in the new class greatly improves the flexibility of our model to cover symmetric, asymmetric, unimodal and multimodal data. Further, we consider Wrapped Cauchy distribution as a special case for our base distribution and introduce the Skew-Symmetric Wrapped Cauchy (SSWC) distribution, discuss its sub-models and estimate its parameters by the maximum likelihood method. Finally, an application of SSWC distribution and its inferential methods are illustrated using two real data examples. [ABSTRACT FROM AUTHOR]

This article presents a class of four-parameter distributions for circular data that are unimodal, possess simple characteristic and density functions and a tractable distribution function, can be interpretably parameterized directly in terms of their trigonometric moments, afford a very wide range of skewness and kurtosis, envelop numerous interesting submodels including the wrapped Cauchy and cardioid distributions, allow straightforward parameter estimation by both method of moments and maximum likelihood, and are closed under convolution. This class of distributions exhibits the widest range of attractive properties yet available while retaining unimodality. [ABSTRACT FROM PUBLISHER]

We propose a family of four-parameter distributions on the circle that contains the von Mises and wrapped Cauchy distributions as special cases. The family is derived by transforming the von Mises distribution via a Möbius transformation, which maps the unit circle onto itself. The densities in the family have a symmetric or asymmetric, unimodal or bimodal shape, depending on the values of the parameters. Conditions for unimodality are explored. Further properties of the proposed model are obtained, many by applying the theory of Möbius transformation. Properties of a three-parameter symmetric submodel are investigated as well; these include maximum likelihood estimation, its asymptotics, and a reparameterization that proves useful quite generally. A three-parameter asymmetric subfamily, which often proves to be an adequate model, is also discussed, with emphasis on its mean direction and circular skewness. The proposed family and subfamilies are used to model an asymmetrically distributed data set and are then adopted as the angular error distribution of a circular–circular regression model. Two applications of the latter are given. It is in this regression context that the Möbius transformation especially comes into its own. Comparisons with other families of circular distributions are made. Supplemental materials for this article are available online. [ABSTRACT FROM AUTHOR]

This paper considers the three-parameter family of symmetric unimodal distributions obtained by wrapping the location-scale extension of Student's t distribution onto the unit circle. The family contains the wrapped normal and wrapped Cauchy distributions as special cases, and can be used to closely approximate the von Mises distribution. In general, the density of the family can only be represented in terms of an infinite summation, but its trigonometric moments are relatively simple expressions involving modified Bessel functions. Point estimation of the parameters is considered, and likelihood-based methods are used to fit the family of distributions in an illustrative analysis of cross-bed measurements. The use of the family as a means of approximating the von Mises distribution is investigated in detail, and new efficient algorithms are proposed for the generation of approximate pseudo-random von Mises variates. [ABSTRACT FROM AUTHOR]