  We describe the X-ray pulse profile models we use, and how we use them, to
analyze Neutron Star Interior Composition Explorer (NICER) observations of
rotation-powered millisecond pulsars to obtain information about the
mass-radius relation of neutron stars and the equation of state of the dense
matter in their cores. Here we detail our modeling of the observed profile of
PSR J0030+0451 that we analyzed in Miller et al. (2019) and Riley et al. (2019)
and describe a cross-verification of computations of the pulse profiles of a
star with R/M 3, in case stars this compact need to be considered in future
analyses. We also present our early cross-verification efforts of the parameter
estimation procedures used by Miller et al. (2019) and Riley et al. (2019) by
analyzing two distinct synthetic data sets. Both codes yielded credible regions
in the mass-radius plane that are statistically consistent with one another and
both gave posterior distributions for model parameter values consistent with
the values that were used to generate the data. We also summarize the
additional tests of the parameter estimation procedure of Miller et al. (2019)
that used synthetic pulse profiles and the NICER pulse profile of PSR
J0030+0451. We then illustrate how the precision of mass and radius estimates
depends on the pulsar's spin rate and the size of its hot spot by analyzing
four different synthetic pulse profiles. Finally, we assess possible sources of
systematic error in these estimates made using this technique, some of which
may warrant further investigation.
