Effect of limited statistics on higher order cumulants measurement in heavy-ion collision experiments

TL;DR

This study investigates how limited data samples affect the measurement of higher order cumulants in heavy-ion collisions, highlighting the importance of statistical methods and minimum event requirements for accurate results.

Contribution

It introduces a comprehensive simulation analysis of statistical effects on cumulant measurements and evaluates error estimation methods, guiding experimental data analysis.

Findings

Bootstrap method is most robust for error estimation.

Higher order cumulants are sensitive to sample size and can become negative.

Minimum event statistics are identified for reliable cumulant determination.

Abstract

We have studied the effect of limited statistics of data on measurement of the different order of cumulants of net-proton distribution assuming that the proton and antiproton distributions follow Possionian and Binomial distributions with initial parameters determined from experimental results for two top center of mass energies ($\sqrt{s_{\mathrm{NN}}}=200$ and $62.4$ GeV) in most central ($0-5$%) Au$+$Au collisions at Relativistic Heavy Ion Collider (RHIC). In this simulation, we observe that the central values for higher order cumulants have a strong dependence on event sample size and due to statistical randomness the central values of higher order cumulants could become negative. We also present a study on the determination of the statistical error on cumulants using delta theorem, bootstrap and sub-group methods and verified their suitability by employing a Monte Carlo procedure. Based on our study we find that the bootstrap method provides a robust way for statistical error estimation on high order cumulants. We also present the exclusion limits on the minimum event statistics needed for determination of cumulants if the signal strength (phase transition or critical point) is at a level of $5$% and $10$% above the statistical level. This study will help the experiments to arrive at the minimum required event statistics and choice of proper method for statistical error estimation for high order cumulant measurements.