Statistical uncertainty estimation of higher-order cumulants with finite efficiency and its application in heavy-ion collisions

We derive the general analytical expressions of the statistical uncertainties of cumulants up to fourth order with considering an efficiency correction, which are approved by a toy Monte Carlo model analysis. An application to the study of particle multiplicity fluctuations in heavy-ion collisions is investigated. In this derivation, a mathematical proof is given that the validity of an averaged efficiency correction and the fluctuations induced by the non-uniformity of efficiency can be eliminated. The estimation of statistical uncertainties using the analytical formulae is found to be $10^6$ times faster than the commonly used bootstrap method. This advantage could appear in front of complexity with massive data analysis in many fields.