Statistical Analysis from the Fourier Integral Theorem

TL;DR

This paper develops Monte Carlo-based, nonparametric estimators for multivariate and conditional distributions derived directly from the Fourier integral theorem, avoiding complex dependence modeling.

Contribution

It introduces explicit Monte Carlo estimators for multivariate distributions and conditionals that do not require covariance or dependence structure estimation.

Findings

Estimators are fully nonparametric and explicit.

No recursive or iterative algorithms needed.

Applicable to prediction, mixing distribution estimation, and multivariate data analysis.

Abstract

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.