Density estimation via periodic scaled Korobov kernel method with exponential decay condition

Abstract

We propose the periodic scaled Korobov kernel (PSKK) method for nonparametric density estimation on $\mathbb{R}^d$. By first wrapping the target density into a periodic version through modulo operation and subsequently applying kernel ridge regression in scaled Korobov spaces, we extend the kernel approach proposed by Kazashi and Nobile (SIAM J. Numer. Anal., 2023) and eliminate its requirement for inherent periodicity of the density function. This key modification enables effective estimation of densities defined on unbounded domains. We establish rigorous mean integrated squared error (MISE) bounds, proving that for densities with smoothness of order $\alpha$ and exponential decay, our PSKK method achieves an $\mathcal{O}(M^{-1/(1+1/(2\alpha)+\epsilon)})$ MISE convergence rate with an arbitrarily small $\epsilon>0$. While matching the convergence rate of the previous kernel approach, our method applies to non-periodic distributions at the cost of stronger differentiability and exponential decay assumptions. Numerical experiments confirm the theoretical results and demonstrate a significant improvement over traditional kernel density estimation in large-sample regimes.