The Taylor expansion of thermodynamic observables at a finite baryon chemical potential $\mu_B$ is an oft-used method to circumvent the well-known sign problem of Lattice QCD. A reliable Taylor estimate demands sufficiently high-ordered calculations in chemical potential $\mu$ for a proper estimate of its radius of convergence. Owing to the associated difficulty and limitations of precision in calculating these high-order Taylor coefficients, it becomes essential to look for various alternative resummation schemes which can work around this computational hurdle. Recently, a way to resum exponentially, the contributions of the first $N$ baryon charge density correlation functions $D_1,\dots,D_N$ to the Taylor series to all orders in $\mu_B$ was proposed in Phys. Rev. Lett. 128, 2, 022001 (2022). Since the correlation functions $D_n$ are calculated stochastically using estimates from different random volume sources, the resummation formulation gets affected by biased estimates, which can become very drastic and can radically misdirect the calculations for large values of $N$, $\mu$ and also higher order $\mu$ derivatives of free energy.
In this work, we present a cumulant expansion procedure that allows to investigate and regulate these biased estimates at different orders in $\mu$. We find that the unbiased estimates in the cumulant expansion can truly capture the genuine higher-order stochastic fluctuations of the higher order correlation functions, which got suppressed by the exponential resummation formulation. Finally, we introduce an unbiased formalism of exponential resummation, which when expanded in a series, can exactly reproduce the Taylor series upto a desired order in $\mu$. This allows to regain the knowledge of reweighting factor and many other important properties of the partition function, which got entirely lost while implementing the cumulant expansion scheme.