  We introduce the Pad\'e--Z$_2$ (PZ) stochastic estimator for calculating
determinants and determinant ratios. The estimator is applied to the
calculation of fermion determinants from the two ends of the Hybrid Monte Carlo
trajectories with pseudofermions. Our results on the $8^3 \times 12$ lattice
with Wilson action show that the statistical errors from the stochastic
estimator can be reduced by more than an order of magnitude by employing an
unbiased variational subtraction scheme which utilizes the off-diagonal
matrices from the hopping expansion. Having been able to reduce the error of
the determinant ratios to about 20 % with a relatively small number of noise
vectors, this may become a feasible algorithm for simulating dynamical fermions
in full QCD. We also discuss the application to the density of states in
Hamiltonian systems.
