We describe sophisticated new Bayesian analysis methods that make it possible to estimate quickly the masses and radii of rapidly rotating, oblate neutron stars by fitting oblate-star waveform models to energy-resolved observations of the waveforms of X-ray burst oscillations produced by such stars. We find that a 25% variation of the temperature of the hot spot in the north-south direction does not significantly bias estimates of the mass M and equatorial radius derived by fitting a model that assumes a uniform-temperature spot. Our results show that fits of oblate-star waveform models to waveform data can simultaneously determine M and with 1σ uncertainties 7% if (1) the star's rotation rate is 600 Hz; (2) the spot center and observer's sightline are both within of the star's rotational equator; and (3) counts are collected during burst oscillations that have a fractional rms amplitude 10%. A fractional amplitude of ~10% is realistic, and the accepted NICER and proposed LOFT and AXTAR space missions could collect counts from a single star by combining data from many X-ray bursts from the star. Uncertainties 7% are small enough to improve substantially our understanding of cold, ultradense matter.
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