Statistical Analysis from the Fourier Integral Theorem
This provides a fully nonparametric method for statistical analysis of multivariate data, addressing challenges in modeling dependencies without iterative algorithms.
The paper tackles the problem of estimating multivariate and conditional distribution functions without needing to estimate covariance or dependence structures, by deriving explicit Monte Carlo estimators from the Fourier integral theorem, enabling applications like prediction for Markov processes and covariate-dependent mixing distributions.
Taking the Fourier integral theorem as our starting point, in this paper we focus on natural Monte Carlo and fully nonparametric estimators of multivariate distributions and conditional distribution functions. We do this without the need for any estimated covariance matrix or dependence structure between variables. These aspects arise immediately from the integral theorem. Being able to model multivariate data sets using conditional distribution functions we can study a number of problems, such as prediction for Markov processes, estimation of mixing distribution functions which depend on covariates, and general multivariate data. Estimators are explicit Monte Carlo based and require no recursive or iterative algorithms.