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

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.

Figure 1

$yx$-scale plot of the left tail of $M$ independent random numbers identically distributed according to $H_4\left(A\right)$ for sample sizes $M={10}^4$ (top), ${10}^5$ (middle), and ${10}^6$ (bottom). Shaded areas around curves correspond to the $68.3\%$ and $95.4\%$ confidence intervals obtained from the bootstrap. The short-dashed lines indicate the analytical value of $y\left(0\right)$, and the long-dashed lines are the first-quantile lines, $y=\frac{1/2}{M}{x}^{-3}$, marking the region where the largest value of $A$ falls at each sample size.

Figure 2

$yx$-scale plot of the left tail of ${10}^6$ independent random numbers identically distributed according to $H_4\left(A\right)$ using the correct leading-order tail exponent ${\mu }^{\prime }=\mu =4$ and incorrect values ${\mu }^{\prime }=3$ and 5, which respectively go to zero and diverge as $x\to 0$. Shaded areas around curves correspond to the $68.3\%$ and $95.4\%$ confidence intervals obtained from the bootstrap. The short-dashed line indicates the analytical value of $y\left(0\right)$.

Figure 3

Probability distribution of the values of $w_1{y}_1/\xi$, Eq. (40), for various values of $\gamma$, using $\mu =4$ and $M={10}^6$.

Figure 4

Application of the tail-regression procedure to a sample of ${10}^6$ random variables distributed according to $P_A\left(A\right)=\frac{1}{2}{H}_{3.1}\left(A\right)+\frac{1}{2}{H}_{4.1}\left(A\right)$. The top panel shows the estimated probability distribution and the lower panels show $yx$ plots for each tail. Fits are shown as thick lines in the three panels and the analytical asymptotic form of the tail is shown as a dashed line in each of the bottom panels. $68.3\%$ and $95.4\%$ confidence intervals obtained from the bootstrap are shown as shaded areas.

Figure 5

Tail-regression estimator of the zeroth, first, and second moment of $P_A\left(A\right)=\frac{1}{2}{H}_{3.1}\left(A\right)+\frac{1}{2}{H}_{4.1}\left(A\right)$ and ${\chi }^2$ as a function of expansion order obtained using a sample of ${10}^6$ random numbers and a threshold of $-ln{q}_{\mathrm{R}}=2.25$. The “optimal” $n_{\mathrm{R}}=3$ is that at which all of these functions reach their respective plateaus.

Figure 6

Uncertainty on the tail-regression estimator of ${\sigma }^2$ obtained from ${10}^6$ random variables distributed according to $P_A\left(A\right)=\frac{1}{2}{H}_{3.1}\left(A\right)+\frac{1}{2}{H}_{4.1}\left(A\right)$ as a function of threshold $-ln{q}_{\mathrm{R}}$. Each curve corresponds to a different expansion order $n_{\mathrm{R}}$, as labeled. Combinations of $-ln{q}_{\mathrm{R}}$ and $n_{\mathrm{R}}$ that fail to satisfy our correctness criteria are not shown, and narrow fails are shown as crossed-out points. The “optimal” choice of threshold and expansion order, marked with a circle, is that which minimizes the uncertainty on $\mathcal{V}$, corresponding to $-ln{q}_{\mathrm{R}}=2.25$ and $n_{\mathrm{R}}=3$ in this case.

Figure 7

Convergence of (a) the estimators of $\langle A\rangle$ and (b) their uncertainties as a function of sample size $M$ for model distribution $P_A\left(A\right)=\frac{1}{2}{H}_{3.1}\left(A\right)+\frac{1}{2}{H}_{4.1}\left(A\right)$. The exact value $\langle A\rangle =0$ is marked with a dashed line in (a), and dashed lines proportional to $M^{-1/2}$ passing through the last point of each of the tail-regression estimator curves are shown in (b) as guides to the eye. $68.3\%$ confidence intervals are shown as shaded areas.

Figure 8

Convergence of (a) the estimators of ${\sigma }^2$ and (b) their uncertainties as a function of sample size $M$ for model distribution $P_A\left(A\right)=\frac{1}{2}{H}_{3.1}\left(A\right)+\frac{1}{2}{H}_{4.1}\left(A\right)$. The exact value ${\sigma }^2=4.6586$ is marked with a dashed line in (a), and dashed lines proportional to $M^{-1/2}$ passing through the last point of each of the tail-regression estimator curves are shown in (b) as guides to the eye. $68.3\%$ confidence intervals are shown as shaded areas.

Figure 9

Convergence of (a) the estimators of $\langle A\rangle$ and (b) their uncertainties as a function of sample size $M$ for model distribution $P_A\left(A\right)=\frac{1}{2}{H}_{2.1}\left(A\right)+\frac{1}{2}{H}_{3.1}\left(A\right)$. The exact value $\langle A\rangle =0$ is marked with a dashed line in (a) and dashed lines proportional to $M^{-1/2}$ passing through the last point of each of the tail-regression estimator curves are shown in (b) as guides to the eye. $68.3\%$ confidence intervals are shown as shaded areas.

Figure 10

Convergence of (a) the constrained tail-regression estimator of $\langle A\rangle$ and (b) its uncertainty as a function of sample size $M$ for model distribution $P_A\left(A\right)=\frac{1}{2}{H}_{1.1}\left(A\right)+\frac{1}{2}{H}_{2.1}\left(A\right)$. The exact value $\langle A\rangle =0$ is marked with a dashed line in (a) and a dashed line proportional to $M^{-1/2}$ passing through the last point of the tail-regression estimator curve is shown in (b) as a guide to the eye. $68.3\%$ confidence intervals are shown as shaded areas.

Figure 11

Application of the tail-regression procedure to a sample of ${10}^6$ VMC local energies for the 54-electron gas at $r_{\mathrm{s}}=1$ using the SJ wave function. The top panel shows the estimated probability distribution and the lower panels show $yx$ plots of both tails. Fits are shown as thick lines in the three panels. $68.3\%$ and $95.4\%$ confidence intervals obtained from the bootstrap are shown as shaded areas.

Figure 12

Convergence of (a) the estimators of ${\sigma }^2$ and (b) their uncertainties as a function of sample size $M$ for the VMC local energy of a 54-electron gas at $r_{\mathrm{s}}=1$ using the SJ wave function. Our best estimate of the value of the variance of the local energy for this system, ${\sigma }^2\approx 0.00057545\left(24\right)$ a.u., is marked with a dashed line in (a) and dashed lines proportional to $M^{-1/2}$ passing through the last point of each of the tail-regression estimator curves are shown in (b) as guides to the eye. $68.3\%$ confidence intervals are shown as shaded areas.

Figure 13

$yx$ plots of the left (left) and right (right) tails of the VMC local Pulay force on a carbon atom along the molecular axis of the ${\mathrm{C}}_2$ molecule at $r_{\mathrm{CC}}=2$ a.u. at sample sizes $M={10}^6$ (top), ${10}^7$ (middle), and ${10}^8$ (bottom). Fits are shown as thick lines in the three panels. $68.3\%$ and $95.4\%$ confidence intervals obtained from the bootstrap are shown as shaded areas.

Figure 14

Convergence of (a) the estimators of $\langle A\rangle$ and (b) their uncertainties as a function of sample size $M$ for the VMC local Pulay force on a carbon atom along the molecular axis of the ${\mathrm{C}}_2$ molecule at $r_{\mathrm{CC}}=2$ a.u. The best value of the Pulay force of 1.1643(35) a.u. is marked with a dashed line in (a) and a dashed line proportional to $M^{-1/2}$ passing through the last point of the tail-regression estimator curve is shown in (b) as a guide to the eye. $68.3\%$ confidence intervals are shown as shaded areas.

Figure 15

Application of the tail-regression procedure to a sample of ${10}^8$ VMC local Hellmann-Feynman forces on a carbon atom along the molecular axis of the ${\mathrm{C}}_2$ molecule at $r_{\mathrm{CC}}=2$ a.u. The top panel shows the estimated probability distribution and the lower panels show $yx$ plots of both tails. Fits are shown as thick lines in the three panels. $68.3\%$ and $95.4\%$ confidence intervals obtained from the bootstrap are shown as shaded areas.

Figure 16

Convergence of (a) the estimators of $\langle A\rangle$ and (b) their uncertainties as a function of sample size $M$ for the VMC local Hellmann-Feynman force on a carbon atom along the molecular axis of the ${\mathrm{C}}_2$ molecule at $r_{\mathrm{CC}}=2$ a.u. The best value of the Hellmann-Feynman force of $-0.5758\left(29\right)$ a.u. is marked with a dashed line in (a) and dashed lines proportional to $M^{-1/2}$ passing through the last point of each of the tail-regression estimator curves are shown in (b) as guides to the eye. $68.3\%$ confidence intervals are shown as shaded areas.

Figure 17

Convergence of (a) the estimators of $\langle A\rangle$ and (b) their uncertainties as a function of sample size $M$ for the mixed DMC local Hellmann-Feynman force on a carbon atom along the molecular axis of the ${\mathrm{C}}_2$ molecule at $r_{\mathrm{CC}}=2$ a.u. The best value of the Hellmann-Feynman force of 0.306(58) a.u. is marked with a dashed line in (a) and dashed lines proportional to $M^{-1/2}$ passing through the last point of each of the tail-regression estimator curves are shown in (b) as guides to the eye. $68.3\%$ confidence intervals are shown as shaded areas.