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 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 parameters. Conditions for unimodality are explored. Further properties of the proposed model are obtained, many by applying the theory of the Möbius transformation. Properties of a three-parameter symmetric submodel are also investigated; these include maximum likelihood estimation, its asymptotics and a reparametrisation that proves useful quite generally. A three-parameter asymmetric subfamily, which proves adequate in each of the examples in the paper, is also discussed with emphasis on its mean direction and circular skewness. The proposed family and subfamilies are used to model an asymmetrically distributed dataset and are then adopted as the angular error distribution of a circular-circular regression model, and an application given thereof. It is in this regression context that the Möbius transformation particularly comes into its own. Comparisons with other families of circular distributions are made.

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The nuclear receptor-type transcription factor retinoic acid receptor-related orphan receptor alpha (RORalpha) is a multifunctional molecule involved in tissue development and cellular function, such as inflammation, metabolism, and differentiation; however, the role of RORalpha during adipocyte differentiation has not yet been fully understood. Here we show that RORalpha inhibits the transcriptional activity of CCAAT/enhancer-binding protein beta (C/EBPbeta) without affecting its expression, thereby blocking the induction of both PPARgamma and C/EBPalpha, resulting in the suppression of C/EBPbeta-dependent adipogenesis. RORalpha interacted with C/EBPbeta so as to repress both the C/EBPbeta-p300 association and the C/EBPbeta-dependent recruitment of p300 to chromatin. In addition to the inhibitory effect on C/EBPbeta function, RORalpha also prevents the expression of the lipid droplet coating protein gene perilipin by peroxisome proliferators-activated receptor gamma (PPARgamma), acting through the specific mechanism of its promoter. We identified a suppressive ROR-responsive element overlapping the PPAR-responsive element in the perilipin promoter and verified that RORalpha competitively antagonizes the binding of PPARgamma. RORalpha inhibits PPARgamma-dependent adipogenesis along with the repression of perilipin induction. These findings suggest that RORalpha is a novel negative regulator of adipocyte differentiation that acts through dual mechanisms.