  Cosmological parameter estimation techniques that robustly account for
systematic measurement uncertainties will be crucial for the next generation of
cosmological surveys. We present a new analysis method, superABC, for obtaining
cosmological constraints from Type Ia supernova (SN Ia) light curves using
Approximate Bayesian Computation (ABC) without any likelihood assumptions. The
ABC method works by using a forward model simulation of the data where
systematic uncertainties can be simulated and marginalized over. A key feature
of the method presented here is the use of two distinct metrics, the `Tripp'
and `Light Curve' metrics, which allow us to compare the simulated data to the
observed data set. The Tripp metric takes as input the parameters of models fit
to each light curve with the SALT-II method, whereas the Light Curve metric
uses the measured fluxes directly without model fitting. We apply the superABC
sampler to a simulated data set of $\sim$1000 SNe corresponding to the first
season of the Dark Energy Survey Supernova Program. Varying $\Omega_m, w_0,
\alpha$ and $\beta$ and a magnitude offset parameter, with no systematics we
obtain $\Delta(w_0) = w_0^{\rm true} - w_0^{\rm best \, fit} = -0.036\pm0.109$
(a $\sim11$% 1$\sigma$ uncertainty) using the Tripp metric and $\Delta(w_0) =
-0.055\pm0.068$ (a $\sim7$% 1$\sigma$ uncertainty) using the Light Curve
metric. Including 1% calibration uncertainties in four passbands, adding 4 more
parameters, we obtain $\Delta(w_0) = -0.062\pm0.132$ (a $\sim14$% 1$\sigma$
uncertainty) using the Tripp metric. Overall we find a $17$% increase in the
uncertainty on $w_0$ with systematics compared to without. We contrast this
with a MCMC approach where systematic effects are approximately included. We
find that the MCMC method slightly underestimates the impact of calibration
uncertainties for this simulated data set.
