  We derive two new forms of the K\'arm\'an-Howarth-Monin equation for decaying
compressible Hall magnetohydrodynamic (MHD) turbulence. We test them on results
of a weakly-compressible, two-dimensional, moderate-Reynolds-number Hall MHD
simulation and compare them with an isotropic spectral transfer (ST) equation.
The KHM and ST equations are automatically satisfied during the whole
simulation owing to the periodic boundary conditions and have complementary
cumulative behavior. They are used here to analyze the onset of turbulence and
its properties when it is fully developed. These approaches give equivalent
results characterizing: the decay of the kinetic + magnetic energy at large
scales, the MHD and Hall cross-scale energy transfer/cascade, the pressure
dilatation, and the dissipation. The Hall cascade appears when the MHD one
brings the energy close to the ion inertial range and is related to the
formation of reconnecting current sheets. At later times, the pressure-dilation
energy-exchange rate oscillates around zero with no net effect on the
cross-scale energy transfer when averaged over a period of its oscillations. A
reduced one-dimensional analysis suggests that all three methods may be useful
to estimate the energy cascade rate from in situ observations.
