Multivariate Smoothing via the Fourier Integral Theorem and Fourier Kernel
This work offers improved estimation techniques for multivariate functions, which could benefit researchers and practitioners in various fields requiring robust statistical modeling.
This paper introduces Monte Carlo estimators for multivariate functions, including densities and regression functions, derived from the Fourier integral theorem. The proposed estimators achieve superior rates of convergence compared to standard kernel-based methods in many cases.
Starting with the Fourier integral theorem, we present natural Monte Carlo estimators of multivariate functions including densities, mixing densities, transition densities, regression functions, and the search for modes of multivariate density functions (modal regression). Rates of convergence are established and, in many cases, provide superior rates to current standard estimators such as those based on kernels, including kernel density estimators and kernel regression functions. Numerical illustrations are presented.