On the estimation of distances from trigonometric parallaxes

A general series is derived for the expectation of the distance computed from a measured trigonometric parallax, given the true distance, on the assumption that the measurement errors are normally distributed with known standard deviation around the true parallax. This expectation is in general different from the true distance; the difference can be either positive or negative. The variance of the computed distances is shown to be infinite, which makes the application of a correction for bias pointless. Two approaches to render the variance finite and thereby make possible application of such a correction are explored: (1) transformation of the observed parallax, and (2) a weighting that virtually eliminates distances for measured parallaxes near zero. The first approach turns out to allow estimation of distance with a relatively small bias, of the order of 10 per cent for a standard error equal to twice the true parallax. It also can be modified to yield an estimate of the absolute magnitude to within 0.1 – 0.2 mag for the same error. The method is tested for sets of synthetic data incorporating effects of observational selection, including an apparent magnitude limit or selection according to proper motion. Typically the bias in these tests is a few per cent in distance or less than 0.1 in absolute magnitude.