Is there really a Lutz-Kelker bias? Reconsidering calibration with trigonometric parallaxes

In the recent literature there are indications of some confusion regarding the Lutz-Kelker bias: whether or not it exists, and if so, what it is and when it should be corrected. Here we carefully reexamine Lutz & Kelker's original work to understand what they actually did, and then look at their later papers and some other works on the subject. There is, properly speaking, no universal Lutz-Kelker bias of individual parallaxes. There is a bias for stars that are members of samples which is different from, but often has the same form as and is given the name of, the Lutz-Kelker bias. The overall bias for samples selected according to relative parallax error is sometimes given the name of Lutz-Kelker in fact it is, or is very nearly the same as, that discussed by Trumpler and Weaver. The Lutz-Kelker corrections can, under certain conditions, be used to counter that bias. The Lutz-Kelker correction applied for an isolated star (independent of sample properties) is an incomplete refinement of the estimate of absolute magnitude calculated directly from the parallax, not a correction for bias.

We reconsider the more general problem of calibration using only trigonometric parallaxes, and examine some of the maximum likelihood methods proposed for its solution. Of these, several are based on linear approximation and are therefore of limited validity. Three exact methods are all based on essentially the same form of the likelihood but are implemented in different ways. One of these is statistically flawed, as was originally pointed out by Jung. We test the other two using synthetic samples, compare their performance, and discuss their application. We also apply one (grid method) to the Feast-Catchpole high weight sample of Hipparcos Cepheid parallaxes as a test. The grid method is to be preferred over the approximate ones because it does not have a limited range of validity, should not require any a posteriori correction, and provides a more complete picture of the uncertainties, in the form of a contour diagram of log(likelihood).