Systematics of RR Lyrae Statistical Parallax. I. Mathematics

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_π = 2.04, K_θ = 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 incorporated into previous studies.