  For a class of precision CPT-invariance test measurements using antihydrogen,
a deficit in the data indicates the presence of the signal. The construction of
classical confidence intervals for the properties of the antiatoms from
measurements may pose a challenge due to the limited statistics experimentally
available. We use the Feldman-Cousins method to estimate model parameters for
such a low count rate measurement. First, we construct confidence intervals for
the Poisson process with a known background and an unknown signal deficit. Then
the generalized Monte Carlo version of the method is applied to the use case of
the hyperfine transition frequency measurement of the ground-state antihydrogen
atom, where the expected double-dip resonance line shape and the mean
background is assumed to be known. We find that confidence intervals of the
antihydrogen properties could be obtained already from low statistics data. We
also discuss how the method may be extended to allow estimation of additional
model parameters.
