A general analytical solution to the problem of Malmquist bias due to lognormal distance errors

We present an analytical solution to the problem of statistically correcting for distance bias in a galaxy data set. Given an unknown intrinsic radial distribution of galaxies and their estimated distances with lognormal errors, we compute the optimal correction to the galaxy's estimated distances. These "Malmquist" corrections are calculated by utilizing the information contained in the distribution of estimated distances in the data set. This method precludes the need for assumptions concerning the real distribution of galactic distances, such as homogeneity, through the use of Bayesian statistical techniques involving nonuniform prior probability distributions. With regard to real data sets and distance estimator relations, the applicability of such a procedure depends intimately on the interplay between selection functions and the variables which define the distance estimator relation. Therefore, applications to real data must rely on data sets which have been selected using criteria independent of the scatter in the distance estimator relation. As no current data set meets this criteria, we confirm our result using Monte Carlo simulations. We further investigate the systematic errors introduced into galactic distances and the galactic velocity field by this and other Malmquist correction methods around an overdensity modeled after that of the Great Attractor region. We find that the inclusion of a zero-point constant in the Hubble flow fit to the galactic velocity field acts as a good indicator of inadequate Malmquist correction. We then apply these results to an elliptical galaxy set in the direction of the Great Attractor and show that much of the signal may be an artifact due to Malmquist correction.