  Muonic atom transitions have been measured for almost all stable nuclei to
extract nuclear structure properties, including nuclear charge radii and
quadrupole moment. To investigate the possibilities of extracting higher-order
radial moments of nuclear charge density %what kind of information of nuclear
charge distribution can be extracted precisely from muonic atom spectroscopy, a
theory-to-theory benchmark analysis based on a model-independent density
distribution, i.e., the Fourier-Bessel series expansion instead of
two-parameter Fermi distribution, is performed by taking $^{208}$Pb as an
example, where nuclear charge density obtained from the relativistic continuum
Hartree-Bogoliubov calculation is used as the benchmark. It is found that the
extractions of the higher-order moments, i.e., the fourth and sixth moments in
addition to the second moment are feasible with high accuracy. Moreover, the
charge form factor in the low-$q$ region can also be well extracted.
