  Several types of extra-galactic high-energy transients have been discovered,
which include high-luminosity and low-luminosity long-duration gamma-ray bursts
(GRBs), short-duration GRBs, supernova shock breakouts (SBOs), and tidal
disruption events (TDEs) without or with an associated relativistic jet. In
this paper, we apply a unified method to systematically study the
redshift-dependent event rate densities and the global luminosity functions
(ignoring redshift evolution) of these transients. We introduce some empirical
formulae for the redshift-dependent event rate densities for different types of
transients, and derive the local specific event rate density, which also
represents its global luminosity function. Long GRBs have a large enough sample
to reveal features in the global luminosity function, which is best
characterized as a triple power law. All the other transients are consistent
with having a single power law luminosity function. The total event rate
density depends on the minimum luminosity, and we obtain the following values
in units of ${\rm Gpc^{-3}~yr^{-1}}$: $0.8^{+0.1}_{-0.1}$ for high-luminosity
long GRBs above $ 10^{50}~{\rm erg~s^{-1}}$, $164^{+98}_{-65}$ for
low-luminosity long GRBs above $5\times 10^{46}~{\rm erg~s^{-1}}$,
$1.3^{+0.4}_{-0.3}$, $1.2^{+0.4}_{-0.3}$, and $3.3^{+1.0}_{-0.8}$ above $
10^{50}~ {\rm erg~s^{-1}}$ for short GRBs with three different merger delay
models (Gaussian, log-normal, and power law), $1.9^{+2.4}_{-1.2}\times 10^4$
above $ 10^{44}~{\rm erg~s^{-1}}$ for SBOs, $ 4.8^{+3.2}_{-2.1}\times10^2 $ for
normal TDEs above $10^{44}~ {\rm erg~s^{-1}}$, and $0.03^{+0.04}_{-0.02}$ above
$ 10^{48}~ {\rm erg~s^{-1}}$ for TDE jets as discovered by Swift. Intriguingly,
the global luminosity functions of different kinds of transients, which cover
over 12 orders of magnitude, are consistent with a single power law with an
index of -1.6.
