A Circular Distribution Constructed from the Product of Cardioid-Type Densities with (Hyper-) Toroidal Extension

Abstract

This study defines the cardioid-type product distribution which is a possibly asymmetric and multimodal distribution on the circle. A parsimonious distribution as a special case, which is denoted by CTP 2, includes sine-skewed cardioid and cosine-perturbed cardioid distributions. Although the cardioid-type product distribution is a special case of the nonnegative trigonometric sums distribution, the nonnegativity and the role of the parameters are clear. The CTP 2 has a simple normalizing constant and is sufficiently flexible with respect to fitting purposes. A cardioid-type product distribution on the hypertorus is defined by utilizing an approach analogous to the construction of CTP 2. The properties of a toroidal distribution, denoted by TCTP 2, are investigated in detail. Both the marginal and conditional distributions of the TCTP 2 belong to the CTP 2 family. Joint trigonometric moments, correlation coefficients, random number generation, estimation, test for independence and discretization were also studied. Thunder and circular genome data are used as illustrative examples.