  Ensuring a satisfactory statistical convergence of anharmonic thermodynamic
properties requires sampling of many atomic configurations, however the methods
to obtain those necessarily produce correlated samples, thereby reducing the
effective sample size and increasing the uncertainty compared to purely random
sampling. In previous works procedures have been implemented to accelerate the
computations by first performing simulations using an approximate Hamiltonian
which is computationally more efficient than the accurate one and then using
various methods to correct for the resulting error. Those rely on recalculating
the accurate energies of a random subset of configurations obtained using the
approximate Hamiltonian thereby maximizing the effective sample size. This
procedure can be particularly suitable for calculating thermodynamic properties
using density-functional theory in which case the accurate and approximate
Hamiltonians may be represented by parametrically suitably converged and
non-converged ones. Whereas it is qualitatively known that there needs to be a
sufficient overlap between the phase spaces of the approximate and the accurate
Hamiltonians, the quantitative limits of applicability and the relative
efficiencies of such methods is not well known. In this paper a statistical
analysis is performed first theoretically and then quantitatively by numerical
analysis. The sampling distributions of different free energy estimators are
obtained and the dependence of their bias and variance with respect to
convergence parameters, simulation times and reference potentials is estimated.
