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

2026-05-18

AI Summary

This study addresses the challenges of non-negativity, normalization, and accuracy in estimating probability densities from empirical characteristic functions over a fixed time window. The authors propose a neural network approach trained directly in the Fourier domain, leveraging the closed-form characteristic function of a Gaussian–Laplace mixture model to enforce non-negativity and unit integral of the resulting density. The method is applicable to both independent and identically distributed data and dependent data resampling scenarios. As the first work to integrate neural networks with Fourier-domain training for density estimation, it derives a multidimensional L₂ error bound accounting for truncation, empirical, discretization, and sampling errors. Experiments demonstrate that the method matches the EM algorithm on Gaussian mixture benchmarks, substantially outperforms existing approaches on heavy-tailed distributions, exhibits L₂ error decay consistent with theoretical predictions, and successfully estimates the annual return distribution of the Australian stock market.

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.

Problem

• density estimation
• characteristic function
• Fourier space
• dependent data
• probability density

Innovation

• Fourier-space training
• characteristic function
• Gaussian-Laplace mixture
• density estimation
• pseudo-sampling