Amirkabir University of TechnologyAUT Journal of Mathematics and Computing2120210201An extension of the Cardioid distributions on circle4552420910.22060/ajmc.2020.18285.1029ENErfanSalavatiDepartment of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran0000-0001-8214-5518Journal Article20200418A new family of distributions on the circle is introduced which is a generalization of the Cardioid distributions. The elementary properties such as mean, variance, and the characteristic function are computed. The distribution is shown to be either unimodal or bimodal. The modes are computed. The symmetry of the distribution is characterized. The parameters are shown to be canonic (i.e. uniquely determined by the distribution). This implies that the estimation problem is welldefined. We also show that this new family is a subset of distributions whose Fourier series has degree at most 2 and study the implications of this property. Finally, we study the maximum likelihood estimation for this family.https://ajmc.aut.ac.ir/article_4209_384f506c165ebe2a3d5dda21192cfab2.pdf