Undesirable effects of covariance matrix techniques for error analysis

TL;DR

This paper warns against using covariance matrix techniques in regression for certain data and functions, as it can amplify errors and produce misleading results, and offers a more robust alternative.

Contribution

It identifies the limitations of covariance matrix methods in error analysis and proposes a test and a more reliable fitting approach for correlated data.

Findings

Covariance matrix techniques can amplify systematic errors.

Incorrect errors for fit parameters can occur with these techniques.

A more robust fitting method is proposed.

Abstract

Regression with $\chi^2$ constructed from the covariance matrix should not be used for some combinations of covariance matrices and fitting functions. Using the technique for unsuitable combinations can amplify systematic errors. This amplification is uncontrolled, and can produce arbitrarily inaccurate results that might not be ruled out by a $\chi^2$ test. In addition, this technique can give incorrect (artificially small) errors for fit parameters. I give a test for this instability and a more robust (but computationally more intensive) method for fitting correlated data.