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We propose a five-parameter bivariate wrapped Cauchy distribution as a unimodal model for toroidal data. It is highly tractable, displays numerous desirable properties, including marginal and conditional distributions that are all wrapped Cauchy, and arises as an appealing submodel of a six-parameter distribution obtained by applying Möbius transformation to a pre-existing bivariate circular model. Method of moments and maximum likelihood estimation of its parameters are fast, and tests for independence and goodness-of-fit are available. An analysis involving dihedral angles of the proteinogenic amino acid Tyrosine illustrates the distribution's application. A Markov process for circular data is also explored. [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 discrete time Markov process which takes values on the unit circle. Some properties of the process, including the limiting behaviour and ergodicity, are investigated. Many computations associated with this process are shown to be greatly simplified if the variables and parameters of the model are represented in terms of complex numbers. A further discussion is given on some submodels, in particular on the stationary process. The proposed model is compared with some existing Markov processes for circular data. Some statistical issues of the model, such as statistical inference, model selection and diagnostic checks, are considered. Finally, an application of the model to wind direction data is provided. [ABSTRACT FROM AUTHOR]