  The formation and evolution of leading jets can be described by jet functions
which satisfy non-linear DGLAP-type evolution equations. Different than for
inclusive jets, the leading jet functions constitute normalized probability
densities for the leading jet to carry a longitudinal momentum fraction
relative to the initial fragmenting parton. We present a parton shower
algorithm which allows for the calculation of leading-jet cross sections where
logarithms of the jet radius and threshold logarithms are resummed to
next-to-leading logarithmic (NLL$'$) accuracy. By calculating the mean of the
leading jet distribution, we are able to quantify the average out-of-jet
radiation, the so-called jet energy loss. When an additional reference scale is
measured, we are able to determine the energy loss of leading jets at the cross
section level which is identical to parton energy loss at leading-logarithmic
accuracy. We identify several suitable cross sections for an extraction of the
jet energy loss and we present numerical results for leading subjets at the
LHC. In addition, we consider hemisphere and event-wide leading jets in
electron-positron annihilation similar to measurements performed at LEP.
Besides the average energy loss, we also consider its variance and other
statistical quantities such as the KL divergence which quantifies the
difference between quark and gluon jet energy loss. We expect that our results
will be particularly relevant for quantifying the energy loss of quark and
gluon jets that propagate through hot or cold nuclear matter.
