T. E. Lutz and D. H. Kelker analyzed systematic effects in luminosity calibration from noisy stellar parallaxes. Their 1973 treatment assumes a spatial distribution of stars and a model for measurement uncertainty, then examines the inferred absolute magnitudes.
The resulting Lutz-Kelker bias illustrates why an unbiased measurement of one quantity does not automatically yield unbiased inference after selection or a nonlinear transformation. The calculation's population and measurement assumptions are essential to interpreting the proposed corrections.[ref]
On the Use of Trigonometric Parallaxes for the Calibration of Luminosity Systems: Theory by T. E. Lutz and D. H. Kelker. pp. 573–574, Introduction and The Distribution Function: parallax errors, spatial distribution and magnitude calibration
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