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Title: Nonparametric density estimation on the polysphere Authors:  Andrea Meilan-Vila - Universidad Carlos III de Madrid (Spain) [presenting]
Eduardo Garcia-Portugues - Universidad Carlos III de Madrid (Spain)
Abstract: Polyspherical data refer to observations on $S^d_1\times\ldots\times S^d_r$, $d_1,\ldots,d_r\ge 1$, where $S^d$ denotes the hypersphere of dimension $d\ge 1$. The poly sphere comprises the circle ($r=d_1=1$), sphere ($r=1$, $d_1=2$), and torus $(d_1=...=d_r=1)$, as particular cases. The goal is to propose and study a kernel density estimator for this type of data. Using a tailored tangent-normal decomposition, the main asymptotic properties of the estimator, such as bias, variance, pointwise normality, and optimal bandwidth, are obtained. Some guidelines, based on cross-validation and plug-in methods, to select the asymptotically optimal bandwidth parameter are also provided in practice. Moreover, the kernel efficiency with respect to a certain ``Epanechnikov'' kernel is studied. An application of the methodology to the hippocampus via s-reps on the polysphere $(S^2)^168$ is discussed.