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由 Möbius 变换诱导的环面分布AI 解读

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…

与 Möbius 变换相关的圆周分布族及其应用AI 解读

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…