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

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.

arXiv comment17 pages, 17 figures; accepted in Phys. Rev. E

References in corpus (20)

• Continuum variational and diffusion quantum Monte Carlo calculations
• Jastrow correlation factor for atoms, molecules, and solids
• Inhomogeneous backflow transformations in quantum Monte Carlo calculations
• Full optimization of Jastrow-Slater wave functions with application to the first-row atoms and homonuclear diatomic molecules
• Pfaffian pairing wave functions in electronic structure quantum Monte Carlo
• Zero-variance principle for Monte Carlo algorithms
• Stable liquid Hydrogen at high pressure by a novel ab-initio molecular dynamics
• Zero-Variance Zero-Bias Principle for Observables in quantum Monte Carlo: Application to Forces
• Backflow Correlations for the Electron Gas and Metallic Hydrogen
• Quantum Monte Carlo study of the phase diagram of solid molecular hydrogen at extreme pressures
• Scheme for adding electron-nucleus cusps to Gaussian orbitals
• Quantum Monte Carlo study of the first-row atoms and ions
• Accurate, efficient and simple forces with Quantum Monte Carlo methods
• Shape and Energy Consistent Pseudopotentials for Correlated Electron systems
• Framework for constructing generic Jastrow correlation factors
• Strategies for improving the efficiency of quantum Monte Carlo calculations
• Heavy-tailed random error in quantum Monte Carlo
• Correlated Electron Pseudopotentials for 3d-Transition Metals
• Methods for calculating forces within quantum Monte Carlo simulations
• Total forces in the diffusion Monte Carlo method with nonlocal pseudopotentials

Cited by in corpus (6)

• Variational and Diffusion Quantum Monte Carlo Calculations with the CASINO Code
• Towards Quantum Monte Carlo Forces on Heavier Ions: Scaling Properties
• Calculation of the energies of the multideterminant states of the nitrogen vacancy center in diamond with quantum Monte Carlo
• Diffusion Monte Carlo evaluation of disiloxane linearization barrier
• Making the most of data: Quantum Monte Carlo Post-Analysis Revisited
• Fast Evaluation of Unbiased Atomic Forces in ab initio Variational Monte Carlo via the Lagrangian Technique