Some probabilistic models on the torus have been proposed in the literature. Mardia (1975) proposed a distribution on the torus as a maximum entropy distribution under the specified condition. Jammalamadaka and SenGupta (2001) proposed a distribution which maximizes the entropy given the mean direction, concentration and circular correlation. Both models are symmetric and fitted to the symmetrically distributed data. However, it is almost impossible to fit these models to the data which are distributed asymmetrically.

My goal is to provide a probabilistic model on the torus which fits the asymmetrically distributed data. I plan to investigate the properties, estimation and inference of the model. In addition to that, I am also interested in studying the models on the sphere and cylinder, which could be applied to the data on the sphere and cylinder, respectively.