Kernel Density Estimation with Polyspherical Data and its Applications
本文提出了多球面数据上的核密度估计方法，推导了渐近性质并引入更高效的核函数，还基于此构造了非参数k样本检验，在婴儿海马体形态分析中展示了应用。
A kernel density estimator for data on the polysphere Sd1×⋯×Sdr, with r,d1,…,dr≥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 that is consistent against alternatives with non-homogeneous densities. 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 (S2)168 through skeletal representations (s-reps).