  We show that a regression of unsmoothed peculiar velocity measurements
against peculiar velocities predicted from a smoothed galaxy density field
leads to a biased estimate of the cosmological density parameter Omega, even
when galaxies trace the underlying mass distribution and galaxy positions and
velocities are known perfectly. The bias arises because the errors in the
predicted velocities are correlated with the predicted velocities themselves.
We investigate this bias using cosmological N-body simulations and analytic
arguments. In linear perturbation theory, for cold dark matter power spectra
and Gaussian or top hat smoothing filters, the bias in Omega is always
positive, and its magnitude increases with increasing smoothing scale. This
linear calculation reproduces the N-body results for Gaussian smoothing radii
R_s > 10 Mpc/h, while non-linear effects lower the bias on smaller smoothing
scales, and for R_s < 3 Mpc/h Omega is underestimated rather than
overestimated. The net bias in Omega for a given smoothing filter depends on
the underlying cosmological model. The effect on current estimates of Omega
from velocity-velocity comparisons is probably small relative to other
uncertainties, but taking full advantage of the statistical precision of future
peculiar velocity data sets will require either equal smoothing of the
predicted and measured velocity fields or careful accounting for the biases
discussed here.
