An Extension of the Cardioid Distributions on Circle

TL;DR

This paper introduces a new family of circular distributions extending Cardioid distributions, analyzing their properties, modes, symmetry, and Fourier series characteristics, with implications for statistical modeling on the circle.

Contribution

It presents a novel family of circular distributions generalizing Cardioid distributions, with detailed property analysis and Fourier series characterization.

Findings

Distribution can be unimodal or bimodal

Parameters are uniquely determined by the distribution

Distribution's Fourier series degree is at most 2

Abstract

A new family of distributions on the circle is introduced which are a generalization of the Cardioid distributions. The elementary properties such as mean, variance and the characteristic function are computed. The distribution is 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). 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.