Sample size effects in multivariate fitting of correlated data

A common problem in analysis of experiments or in lattice QCD simulations is fitting a parameterized model to the average over a number of samples of correlated data values. If the number of samples is not infinite, estimates of the variance of the parameters (“error bars”) and of the goodness of fit are affected. We illustrate these problems with numerical simulations, and calculate approximate corrections to the variance of the parameters for estimates made in the standard way from derivatives of the parameters’ probability distribution as well as from jackknife and bootstrap estimates.

💡 Research Summary

The paper addresses a pervasive issue in experimental analysis and lattice QCD simulations: fitting a parametrized model to the mean of a set of correlated multivariate data when the number of samples is finite. In such cases the sample covariance matrix, estimated from the data, deviates from the true covariance, leading to systematic biases in both the estimated parameter uncertainties (error bars) and goodness‑of‑fit statistics such as χ².

The authors begin by assuming a Gaussian model and analytically expand the inverse covariance matrix estimated from a finite sample. They show that while the maximum‑likelihood estimates of the parameters remain unbiased, the curvature of the log‑likelihood (the Fisher information) acquires a correction term of order 1/N, where N is the number of samples. This correction reduces the apparent variance of the parameters when standard error estimates are obtained from the second derivatives of the log‑likelihood.

Through extensive Monte‑Carlo simulations the authors demonstrate that the conventional error‑propagation formula underestimates the true variance, especially for small N. They derive an explicit 1/N correction to the variance and verify that applying this correction brings the estimated uncertainties into agreement with the empirical distribution of the fitted parameters.

The study then turns to resampling techniques. For the jackknife, which recomputes the fit after omitting each observation in turn, the authors find a similar 1/N bias in the covariance of the jackknife replicates. For the bootstrap, which draws many resampled data sets with replacement, the bias arises because each bootstrap sample does not faithfully reproduce the original covariance structure when N is limited. In both cases the authors provide analytic correction terms and show, via simulation, that the corrected jackknife and bootstrap error estimates match the true parameter variance.

The key conclusions are: (1) finite‑sample effects introduce a systematic O(1/N) underestimation of parameter variances and distort χ² values; (2) standard analytic error estimates, jackknife, and bootstrap all suffer from this bias; (3) simple correction formulas derived in the paper effectively remove the bias and are straightforward to implement.

Practically, the results are highly relevant for lattice QCD analyses where multiple correlated observables are fitted simultaneously, as well as for any field dealing with high‑dimensional correlated data (e.g., astrophysics, statistical physics, bioinformatics). By incorporating the proposed corrections, researchers can obtain reliable error bars and goodness‑of‑fit assessments without needing prohibitively large sample sizes, thereby improving the robustness of scientific conclusions drawn from correlated multivariate fits.