Quasi-interpolation for multivariate density estimation on bounded domain

The paper proposes a new nonparametric scheme for multivariate density estimation under the framework of quasi-interpolation, a classical function approximation scheme in approximation theory. Given samples of a random variable obeyed by an unknown density function with compact support, we first partition the support into several bins and compute frequency of samples falling into each bin. Then, by viewing these frequencies as (approximate) integral functionals of density function over corresponding bins, we construct a quasi-interpolation scheme for approximating the density function. Maximal mean squared errors of the scheme demonstrates that our scheme keeps the same optimal convergence rate as classical nonparametric density estimations. In addition, the scheme includes classical boundary kernel density estimation as a special case when the number of bins equals to the number of samples. Moreover, it can dynamically allocate different amounts of smoothing via selecting the bin widths and shape parameters of kernels with the prior knowledge. Numerical simulations provide evidence that the proposed nonparametric scheme is robust and is capable of producing high-performance estimation of density function.