Proofs of non-optimality of the standard least-squares method for track reconstructions

Abstract

It is a standard criterium in statistics to define an optimal estimator the one with the minimum variance. Thus, the optimality is proved with inequality among variances of competing estimators. The inequalities, demonstrated here, disfavor the standard least squares estimators. Inequalities among estimators are connected to names of Cramer, Rao and Frechet. The standard demonstrations of these inequalities require very special analytical properties of the probability functions, globally indicated as regular models. These limiting conditions are too restrictive to handle realistic problems in track fitting. A previous extension to heteroscedastic models of the Cramer-Rao-Frechet inequalities was performed with Gaussian distributions. These demonstrations proved beyond any possible doubts the superiority of the heteroscedastic models compared to the standard least squares method. However, the Gaussian distributions are typical members of the required regular models. Instead, the realistic probability distributions, encountered in tracker detectors, are very different from Gaussian distributions. Therefore, to have well grounded set of inequalities, the limitations to regular models must be overtaken. The aim of this paper is to demonstrate the inequalities for least squares estimators for irregular models of probabilities, explicitly excluded by the Cramer-Rao-Frechet demonstrations. Estimators for straight and parabolic tracks will be considered. The final part deals with the form of the distributions of simplified heteroscedastic track models reconstructed with optimal estimators and the standard (non-optimal) estimators. A comparison among the distributions of these different estimators shows the large loss in resolution of the standard least-squares estimators.

arXiv commentIt completes the results of INSTRUMENTS 2020 4,2-- 13 pages, 2 figures

References in corpus (5)

• The performance of the CMS muon detector in proton-proton collisions at sqrt(s) = 7 TeV at the LHC
• Performance of the reconstruction and identification of high-momentum muons in proton-proton collisions at $\sqrt{s}=$ 13 TeV
• Improvements of Track Fitting with Well Tuned Probability Distributions for Silicon Strip Detectors
• The Cramer-Rao inequality to go beyond the $\sqrt{\mathbf{N}}$-limit of the standard least-squares method in track fitting
• Optimizing momentum resolution with a new fitting method for silicon-strip detectors

Cited by in corpus (1)

• Probability Distributions of Positioning Errors for Some Forms of Center-of-Gravity Algorithms