  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.
