Mathematics of Statistical Parallax and the Local Distance Scale

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. By far the most worrying systematic effect is that the likelihood estimator used in all previous studies explicitly assumes that the velocity distribution is Gaussian (kurtosis K=3), while the actual RR Lyraes are highly non-Gaussian, with kurtoses K_\pi=2.04, K_\theta=3.22 and K_z=4.28 in the three principal directions. In particular, the radial distribution is almost box-like (K=1.8). Nevertheless, we find analytically and numerically that the assumption of Gaussianity has almost no effect on either the best fit or the uncertainty of the luminosity determination. In fact, we find the statistical parallax method to be extremely robust in the face of all systematic effects that we considered. The mean RR Lyrae absolute magnitude is M_V=0.76 +/- 0.12 at the mean metallicity of the sample =-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. Unless the RR Lyraes in the Large Magellanic Cloud (LMC) differ from those in the Galaxy in some unknown way, its distance modulus is \mu_LMC=18.25 +/-0.13 where, we believe, the statistical error should be taken at face value