  Using the theoretical models of the QSO formation, we can reproduce optical
QSO luminosity functions (LFs) at high redshifts (z > 2.2). Two different
models can reproduce LFs successfully, though the lifetime of QSOs, t_Q, and
the relation between the black hole mass and the host halo mass are different
each other; t_Q = 10^6yr, in one model, t_Q > 10^7yr, in other models. Here, we
propose a method to break this degeneracy.
  We calculate the mass function of supermassive black holes (SMBHs) at z=2.5,
and compare the result with the current mass function obtained by Salucci et
al.(1999). In the shorter lifetime model, the mass function at z=2.5 exceeds
that of z=0.0 by one order of magnitude, then it should be ruled out. We
conclude that the lifetime is at least t_Q > 10^7yr.
  Next, we examine the difference of the formation epoch of SMBHs existing at
z=3.0 for each model under the model assumptions. We simply discuss the
difference of formation epoch as another possible model-discriminator.
