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Statistical properties of the estimator using covariance matrix

Alekhin Sergey

TL;DR

The paper assesses how to estimate parameters when point-to-point correlations from systematic errors are present, comparing the simple χ^2 estimator (SCE) with the covariance matrix estimator (CME) in a Bayesian treatment of systematics. It shows CME is consistent under realistic conditions, yields smaller dispersion than SCE for correlated data, and has negligible bias when the covariance is updated during the fit. It provides analytical formulas for the CME inversion to enable fast, accurate calculations for large data sets and discusses experimental planning via integrated ρ and φ quantities to optimize measurement regions. The work supports more reliable global fits in high-precision particle physics analyses by quantifying and mitigating systematic correlations.

Abstract

The statistical properties of estimator using covariance matrix for the account of point-to-point correlations due to systematic errors are analyzed. It is shown that the covariance matrix estimator (CME) is consistent for the realistic cases (when systematic errors on the fitted parameters are not extremely large comparing with the statistical ones) and its dispersion is always smaller, than the dispersion of the simplified $χ^2$ estimator applied to the correlated data. The CME bias is negligible for the realistic cases if the covariance matrix is calculated during the fit iteratively using the parameter estimator itself. Analytical formula for the covariance matrix inversion allows to perform fast and precise calculations even for very large data sets. All this allows for efficient use of the CME in the global fits.

Statistical properties of the estimator using covariance matrix

TL;DR

The paper assesses how to estimate parameters when point-to-point correlations from systematic errors are present, comparing the simple χ^2 estimator (SCE) with the covariance matrix estimator (CME) in a Bayesian treatment of systematics. It shows CME is consistent under realistic conditions, yields smaller dispersion than SCE for correlated data, and has negligible bias when the covariance is updated during the fit. It provides analytical formulas for the CME inversion to enable fast, accurate calculations for large data sets and discusses experimental planning via integrated ρ and φ quantities to optimize measurement regions. The work supports more reliable global fits in high-precision particle physics analyses by quantifying and mitigating systematic correlations.

Abstract

The statistical properties of estimator using covariance matrix for the account of point-to-point correlations due to systematic errors are analyzed. It is shown that the covariance matrix estimator (CME) is consistent for the realistic cases (when systematic errors on the fitted parameters are not extremely large comparing with the statistical ones) and its dispersion is always smaller, than the dispersion of the simplified estimator applied to the correlated data. The CME bias is negligible for the realistic cases if the covariance matrix is calculated during the fit iteratively using the parameter estimator itself. Analytical formula for the covariance matrix inversion allows to perform fast and precise calculations even for very large data sets. All this allows for efficient use of the CME in the global fits.

Paper Structure

This paper contains 4 sections, 61 equations, 2 figures.

Table of Contents

  1. THE SIMPLEST $\chi^2$ ESTIMATOR
  2. THE COVARIANCE MATRIX ESTIMATOR
  3. PLANNING OF THE COUNTING EXPERIMENTS
  4. CONCLUSION

Figures (2)

  • Figure 1: The standard deviations of SCE (circles) and CME (squares) for $\hat{U}$ at different scales of systematic errors $\kappa$. The lines correspond to the calculation performed with the two-dimensional generalization of Eqns. (9,16).
  • Figure 2: Bounds of gluon distribution obtained from the LO QCD fit to BCDMS data with different estimators (the SCE: a; the CME: b). Full lines correspond to the total experimental errors, dashed ones -- to the statistical only.