Methods of Error Estimation for Delay Power Spectra in $21\,\textrm{cm}$
  Cosmology
  Precise measurements of the 21 cm power spectrum are crucial for
understanding the physical processes of hydrogen reionization. Currently, this
probe is being pursued by low-frequency radio interferometer arrays. As these
experiments come closer to making a first detection of the signal, error
estimation will play an increasingly important role in setting robust
measurements. Using the delay power spectrum approach, we have produced a
critical examination of different ways that one can estimate error bars on the
power spectrum. We do this through a synthesis of analytic work, simulations of
toy models, and tests on small amounts of real data. We find that, although
computed independently, the different error bar methodologies are in good
agreement with each other in the noise-dominated regime of the power spectrum.
For our preferred methodology, the predicted probability distribution function
is consistent with the empirical noise power distributions from both simulated
and real data. This diagnosis is mainly in support of the forthcoming HERA
upper limit, and also is expected to be more generally applicable.
