In this paper, we study the spherical cardioid distribution, a higher-dimensional and higher-order generalization of the circular cardioid distribution. This distribution is rotationally symmetric and generates unimodal, multimodal, axial, and girdle-like densities. We show several characteristics of the spherical cardioid that make it highly tractable: simple density evaluation, closedness under convolution, explicit expressions for vectorized moments, and efficient simulation. The moments of the spherical cardioid up to a given order coincide with those of the uniform distribution on the sphere, highlighting its closeness to the latter. We derive estimators by the method of moments and maximum likelihood, their asymptotic distributions, and their asymptotic relative efficiencies. We give the machinery for a bootstrap goodness-of-fit test based on the projected-ecdf approach, including the projected distribution and closed-form expressions for test statistics. An application to modeling the orbits of long-period comets shows the usefulness of the spherical cardioid distribution in real data analyses.

翻译：本文研究了球面心形分布，这是圆形心形分布在高维和高阶上的推广。该分布具有旋转对称性，可生成单峰、多峰、轴向及带状密度函数。我们展示了球面心形分布的若干特性使其具有高度易处理性：密度计算简单、卷积封闭性、向量化矩的显式表达式以及高效模拟。球面心形分布直至给定阶数的矩与球面上均匀分布的矩相一致，这突显了其与后者的近似性。我们通过矩估计法和最大似然法推导了估计量及其渐近分布和渐近相对效率。基于投影经验分布函数方法，我们构建了自助法拟合优度检验的完整框架，包括投影分布和检验统计量的闭式表达式。通过对长周期彗星轨道建模的应用，展示了球面心形分布在真实数据分析中的实用性。