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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.