Density estimation in RKHS with application to Korobov spaces in high dimensions

A kernel method for estimating a probability density function from an independent and identically distributed sample drawn from such density is presented. Our estimator is a linear combination of kernel functions, the coefficients of which are determined by a linear equation. An error analysis for the mean integrated squared error is established in a general reproducing kernel Hilbert space setting. The theory developed is then applied to estimate probability density functions belonging to weighted Korobov spaces, for which a dimension-independent convergence rate is established. Under a suitable smoothness assumption, our method attains a rate arbitrarily close to the optimal rate. Numerical results support our theory.