  In a stochastic heat engine driven by a cyclic non-equilibrium protocol,
fluctuations in work and heat give rise to a fluctuating efficiency. Using
computer simulations and tools from large deviation theory, we have examined
these fluctuations in detail for a model two-state engine. We find in general
that the form of efficiency probability distributions is similar to those
described by Verley et al. [2014 Nat Comm, 5 4721], in particular featuring a
local minimum in the long-time limit. In contrast to the time-symmetric engine
protocols studied previously, however, this minimum need not occur at the value
characteristic of a reversible Carnot engine. Furthermore, while the local
minimum may reside at the global minimum of a large deviation rate function, it
does not generally correspond to the least likely efficiency measured over
finite time. We introduce a general approximation for the finite-time
efficiency distribution, $P(\eta)$, based on large deviation statistics of work
and heat, that remains very accurate even when $P(\eta)$ deviates significantly
from its large deviation form.
