  We present a mathematical analysis of the statistical parallax method. The
method yields physical insight into the maximum-likelihood determinations of
the luminosity and velocity distribution and enables us to conduct a vigorous
Monte Carlo investigation into various systematic effects. We apply our
analytic formalism to the RR Lyrae sample of Layden et al. The velocity
distribution of RR Lyrae stars is highly non-Gaussian, with kurtoses K_\pi =
2.04, K_\theta = 3.22 and K_z = 4.28 in the three principal directions, but
this has almost no effect on either the best fit or the uncertainty of the
luminosity determination. Indeed, our principal result is that the statistical
parallax method is extremely robust in the face of all systematic effects that
we considered. Our analysis, applied to the Layden et al. RR Lyrae sample,
strictly confirms the majority of their results. The mean RR Lyrae absolute
magnitude is M_V = 0.75 +/- 0.13 at the mean metallicity of the sample <[Fe/H]>
= -1.61, compared to M_V = 0.71 +/- 0.12 obtained by Layden et al. Most of the
difference is due to Malmquist bias which was not considered in previous
studies. We also analyze a semi-independent non-kinematically selected sample
of stars with metallicities at the [Fe/H] <= 1.5 taken from Layden et al. and
Beers & Sommer-Larsen and obtain M_V = 0.79 +/- 0.12 at <[Fe/H]> = -1.79.
Additionally, this analysis yields measurements of the radial bulk motion (4
+/- 10 km/s) and vertical bulk motion (0 +/- 6 km/s) of the halo relative to
the Local Standard of Rest.
