Nonparametric Construction of Multivariate Kernels

We propose a nonparametric method for constructing multivariate kernels tuned to the configuration of the sample, for density estimation in R-d, d moderate. The motivation behind the approach is to break down the construction of the kernel into two parts: determining its overall shape and then its global concentration. We consider a framework that is essentially nonparametric, as opposed to the usual bandwidth matrix parameterization. The shape of the kernel to be employed is determined by applying the backprojection operator, the dual of the Radon transform, to a collection of one-dimensional kernels, each optimally tuned to the concentration of the corresponding one-dimensional projections of the data. Once an overall shape is determined, the global concentration is controlled by a simple sealing. It is seen that the kernel estimators thus developed are easy and extremely fast to compute, and perform at least as well in practice as parametric kernels with cross-validated or otherwise tuned covariance structure. Connections with integral geometry are discussed, and the approach is illustrated under a wide range of scenarios in two and three dimensions, via an R package developed for its implementation.