  We study the accuracy and precision for estimating the fraction of observed
levels $\varphi$ in quantum chaotic spectra through long-range correlations. We
focus on the main statistics where theoretical formulas for the fraction of
missing levels have been derived, the $\Delta_3$ of Dyson and Mehta and the
power spectrum of the $\delta_n$ statistic. We use Monte Carlo simulations of
the spectra from the diagonalization of Gaussian Orthogonal Ensemble matrices
with a definite number of levels randomly taken out to fit the formulas and
calculate the distribution of the estimators for different sizes of the
spectrum and values of $\varphi$. A proper averaging of the power spectrum of
the $\delta_n$ statistic needs to be performed for avoiding systematic errors
in the estimation. Once the proper averaging is made the estimation of the
fraction of observed levels has quite good accuracy for the two methods even
for the lowest dimensions we consider $d=100$. However, the precision is
generally better for the estimation using the power spectrum of the $\delta_n$
as compared to the estimation using the $\Delta_3$ statistic. This difference
is clearly bigger for larger dimensions. Our results show that a careful
analysis of the value of the fit in view of the ensemble distribution of the
estimations is mandatory for understanding its actual significance and give a
realistic error interval.
