# A tail-regression estimator for heavy-tailed distributions of known tail indices and its application to continuum quantum Monte Carlo data

## TL;DR

This paper introduces a regression-based estimator for heavy-tailed distributions with known tail indices, enabling reliable confidence intervals in quantum Monte Carlo data analysis where traditional methods fail.

## Contribution

It presents a novel tail-regression estimator that leverages known asymptotic decay exponents to accurately estimate moments and confidence intervals for heavy-tailed distributions.

## Key findings

- Accurately estimates variance of local energy in electron gas
- Provides confidence intervals for Hellmann-Feynman forces in carbon dimer
- Reduces uncertainty by 45-60 times compared to standard error

## Abstract

Standard statistical analysis is unable to provide reliable confidence intervals on expectation values of probability distributions that do not satisfy the conditions of the central limit theorem. We present a regression-based estimator of an arbitrary moment of a probability distribution with power-law heavy tails that exploits knowledge of the exponents of its asymptotic decay to bypass this issue entirely. Our method is applied to synthetic data and to energy and atomic force data from variational and diffusion quantum Monte Carlo calculations, whose distributions have known asymptotic forms [J. R. Trail, Phys. Rev. E 77, 016703 (2008); A. Badinski et al., J. Phys.: Condens. Matter 22 074202 (2010)]. We obtain convergent, accurate confidence intervals on the variance of the local energy of an electron gas and on the Hellmann-Feynman force on an atom in the all-electron carbon dimer. In each of these cases the uncertainty on our estimator is 45% and 60 times smaller, respectively, than the nominal (ill-defined) standard error.