A data-driven Fourier-mixture neural-network method for density estimation

TL;DR

This paper introduces a Fourier-trained neural network for density estimation that directly models probability densities from empirical characteristic functions, with theoretical error bounds and competitive numerical performance.

Contribution

It presents a novel Fourier-based neural network approach for density estimation, including theoretical analysis and extensions to multidimensional data.

Findings

Achieves competitive performance on Gaussian-mixture benchmarks.

Shows clear advantages on heavy-tailed distributions.

Demonstrates effective estimation of Australian equity return law.

Abstract

We propose a data-driven Fourier-trained neural-network method for estimating fixed-horizon probability densities from empirical characteristic-function (CF) information. The estimator is a positive Gaussian--Laplace mixture with closed-form CF, so training can be performed directly in Fourier space while preserving nonnegativity and unit mass. We consider two sampling settings. In the direct i.i.d. sampling setting, the method is trained against an empirical CF constructed from i.i.d. samples. In the resampling-based pseudo-sampling setting, it is trained against an empirical pseudo-CF constructed from dependent data by resampling. For the direct i.i.d. case, we derive an expected $L_2$ error bound that separates Fourier truncation, empirical training error, discretization, and CF sampling error. For the pseudo-sampling case, we obtain a conditional analogue with two additional pseudo-law discrepancy terms. We develop a multidimensional extension of the framework and analyze its computational complexity. Numerical experiments show competitive performance relative to Expectation--Maximization on Gaussian-mixture benchmarks, clear gains on heavy-tailed targets, $L_2$ error decay consistent with the theory in a well-specified setting, and effective estimation of one-year Australian equity return law from resampled dependent data.