According to the nonequilibrium work relations, path-ensembles generated by irreversible processes in which a system is driven out of equilibrium according to a predetermined protocol may be used to compute equilibrium free energy differences and expectation values. Estimation has previously been improved by considering data collected from the reverse process, which starts in equilibrium in the final thermodynamic state of the forward process and is driven according to the time-reversed protocol. Here, we develop a theoretically rigorous statistical estimator for nonequilibrium path-ensemble averages specialized for symmetric protocols, in which forward and reverse processes are identical. The estimator is tested with a number of model systems: a symmetric 1D potential, an asymmetric 1D potential, the unfolding of deca-alanine, separating a host-guest system, and translocating a potassium ion through a gramicidin A ion channel. When reconstructing free energies using data from symmetric protocols, the new estimator outperforms existing rigorous unidirectional and bidirectional estimators, converging more quickly and resulting in smaller error. However, in most cases, using the bidirectional estimator with data from a forward and reverse pair of asymmetric protocols outperforms the corresponding symmetric protocol and estimator with the same amount of simulation time. Hence, the new estimator is only recommended when the bidirectional estimator is not feasible or is expected to perform poorly. The symmetric estimator has similar performance to a unidirectional protocol of half the length and twice the number of trajectories.
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