  We derive analytical expressions for the bias of the Jarzynski free-energy
estimator from N nonequilibrium work measurements, for a generic work
distribution. To achieve this, we map the estimator onto the Random Energy
Model in a suitable scaling limit parametrized by (log N)/m, where m measures
the width of the lower tail of the work distribution, and then compute the
finite-N corrections to this limit with different approaches for different
regimes of (log N)/m. We show that these expressions describe accurately the
bias for a wide class of work distributions, and exploit them to build an
improved free-energy estimator from unidirectional work measurements. We apply
the method to optical tweezers unfolding/refolding experiments on DNA hairpins
of varying loop size and dissipation, displaying both near-Gaussian and
non-Gaussian work distributions.
