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.

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In statistical physics, Boltzmann-Shannon entropy provides good understanding for the equilibrium states of a number of phenomena. In statistics, the entropy corresponds to the maximum likelihood method, in which Kullback-Leibler divergence connects Boltzmann-Shannon entropy and the expected log-likelihood function. The maximum likelihood estimation has been supported for the optimal performance, which is known to be easily broken down in the presence of a small degree of model uncertainty. To deal with this problem, a new statistical method, closely related to Tsallis entropy, is proposed and shown to be robust for outliers, and we discuss a local learning property associated with the method.

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We discuss a one-parameter family of generalized cross entropy between two distributions with the power index, called the projective power entropy. The cross entropy is essentially reduced to the Tsallis entropy if two distributions are taken to be equal. Statistical and probabilistic properties associated with the projective power entropy are extensively investigated including a characterization problem of which conditions uniquely determine the projective power entropy up to the power index. A close relation of the entropy with the Lebesgue space L p and the dual L q is explored, in which the escort distribution associates with an interesting property. When we consider maximum Tsallis entropy distributions under the constraints of the mean vector and variance matrix, the model becomes a multivariate q-Gaussian model with elliptical contours, including a Gaussian and t-distribution model. We discuss the statistical estimation by minimization of the empirical loss associated with the projective power entropy. It is shown that the minimum loss estimator for the mean vector and variance matrix under the maximum entropy model are the sample mean vector and the sample variance matrix. The escort distribution of the maximum entropy distribution plays the key role for the derivation.