  Numerical inversion is a general detector calibration technique that is
independent of the underlying spectrum. This procedure is formalized and
important statistical properties are presented, using high energy jets at the
Large Hadron Collider as an example setting. In particular, numerical inversion
is inherently biased and common approximations to the calibrated jet energy
tend to over-estimate the resolution. Analytic approximations to the closure
and calibrated resolutions are demonstrated to effectively predict the full
forms under realistic conditions. Finally, extensions of numerical inversion
are presented which can reduce the inherent biases. These methods will be
increasingly important to consider with degraded resolution at low jet energies
due to a much higher instantaneous luminosity in the near future.
