A kernel density estimator for data on the polysphere $\mathbb{S}^{d_1}\times\cdots\times\mathbb{S}^{d_r}$, with $r,d_1,\ldots,d_r\geq 1$, is presented in this paper. We derive the main asymptotic properties of the estimator, including mean square error, normality, and optimal bandwidths. We address the kernel theory of the estimator beyond the von Mises-Fisher kernel, introducing new kernels that are more efficient and investigating normalizing constants, moments, and sampling methods thereof. Plug-in and cross-validated bandwidth selectors are also obtained. As a spin-off of the kernel density estimator, we propose a nonparametric $k$-sample test based on the Jensen-Shannon divergence. Numerical experiments illuminate the asymptotic theory of the kernel density estimator and demonstrate the superior performance of the $k$-sample test with respect to parametric alternatives in certain scenarios. Our smoothing methodology is applied to the analysis of the morphology of a sample of hippocampi of infants embedded on the high-dimensional polysphere $(\mathbb{S}^2)^{168}$ via skeletal representations ($s$-reps).

翻译：本文提出了一种针对多球面 $\mathbb{S}^{d_1}\times\cdots\times\mathbb{S}^{d_r}$（其中 $r,d_1,\ldots,d_r\geq 1$）上数据的核密度估计器。我们推导了该估计器的主要渐近性质，包括均方误差、正态性和最优带宽。我们探讨了超越 von Mises-Fisher 核的估计器核理论，引入了更高效的新核，并研究了其归一化常数、矩和抽样方法。同时，我们还获得了插件法和交叉验证带宽选择器。作为核密度估计器的一个副产品，我们提出了一种基于 Jensen-Shannon 散度的非参数 $k$ 样本检验。数值实验阐明了核密度估计器的渐近理论，并证明了 $k$ 样本检验在某些场景下相对于参数化替代方案的优越性能。我们的平滑方法被应用于分析一组通过骨架表示（$s$-reps）嵌入高维多球面 $(\mathbb{S}^2)^{168}$ 上的婴儿海马体样本的形态学。