  Grazing transits present a special problem for statistical studies of
exoplanets. Even though grazing planetary orbits are rare (due to geometric
selection effects), for many low to moderate signal-to-noise cases, a
significant fraction of the posterior distribution is nonetheless consistent
with a grazing geometry. A failure to accurately model grazing transits can
therefore lead to biased inferences even for cases where the planet is not
actually on a grazing trajectory. With recent advances in stellar
characterization, the limiting factor for many scientific applications is now
the quality of available transit fits themselves, and so the time is ripe to
revisit the transit fitting problem. In this paper, we model exoplanet transits
using a novel application of umbrella sampling and a geometry-dependent
parameter basis that minimizes covariances between transit parameters. Our
technique splits the transit fitting problem into independent Monte Carlo
sampling runs for the grazing, non-grazing, and transition regions of the
parameter space, which we then recombine into a single joint posterior
probability distribution using a robust weighting scheme. Our method can be
trivially parallelized and so requires no increase in the wall clock time
needed for computations. Most importantly, our method produces accurate
estimates of exoplanet properties for both grazing and non-grazing orbits,
yielding more robust results than standard methods for many common star-planet
configurations.
