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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).

Past calibrations of statistical distance scales for planetary nebulae have been problematic, especially with regard to 'short' vs. 'long' scales. Reconsidering the calibration process naturally involves examining the precision and especially the systematic errors of various distance methods. Here we present a different calibration strategy, new for planetaries, that is anchored by precise trigonometric parallaxes for sixteen central stars published by Harris et al. (2007) of USNO, with four improved by Benedict et al. using the Hubble Space Telescope. We show how an internally consistent system of distances might be constructed by testing other methods against those and each other. In such a way systematic errors can be minimized.Several of the older statistical scales have systematic errors that can account for the short-long dichotomy. In addition to scale-factor errors all show signs of radius dependence, i.e. the distance ratio [scale/true] is some function of nebular radius. These systematic errors were introduced by choices of data sets for calibration, by the methodologies used, and by assumptions made about nebular evolution. The statistical scale of Frew and collaborators (2008, 2014) is largely free of these errors, although there may be a radius dependence for the largest objects. One set of spectroscopic parallaxes was found to be consistent with the trigonometric ones while another set underestimates distance consistently by a factor of two, probably because of a calibration difference. 'Gravity' distances seem to be overestimated for nearby objects but may be underestimated for distant objects, i.e. distance-dependent. Angular expansion distances appear to be suitable for calibration after correction for astrophysical effects (e.g. Mellema 2004). We find extinction distances to be often unreliable individually though sometimes approximately correct overall (total sample).Comparison of the Hipparcos parallaxes (van Leeuwen 2007) for large planetaries with our 'best estimate' distances confirms that those parallaxes are overestimated by a factor 2.5, as suggested by Harris et al.'s result for PHL 932. There may be negative implications for Gaia parallaxes for these objects. We suggest a possible connection with the much smaller overestimation recently shown for the Hipparcos Pleiades parallaxes by Melis et al. (2014).

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