Statistical bias in distance and peculiar velocity estimation. 1: The 'calibration' problem

This paper is the first in a series which treats comprehensively statistical biases that affect the analysis of galaxy distance indicator data. The discussion will be specialized to the case of the Tully-Fisher relation for spiral galaxies. Statistical biases arising in distance indicator studies fall into two general categories. Those in the first category, the topic of this paper, occur when the data analysis is conditioned upon assumptions about galaxy distances which are unrelated to the distance indicator information. The archetypical examples of such an approach are when a sample of galaxies is assigned a common distance based on assumed cluster membership, and when a sample is assumed to follow a well-defined redshift-distance relation. For reasons which are elaborated in the main body of the paper, the resultant statistical biases are described as comprising the "calibration problem. Statistical biases in the second category will be the topic of the second paper in this series. They occur when individual galaxy distances are inferred directly from a distance indicator, without regard to cluster membership or redshift information. This series will also emphasize the distinction between the statistical properties of the "forward" and "inverse" representations of distance indicator relations. In the case of Tully-Fisher, the former depicts the relation as a predictor of absolute magnitude, given a value of rotation velocity; the latter characterizes the relation as a predictor of rotation velocity given absolute magnitude. The use of the inverse method has been seen by some workers as a panacea for bias effects. A goal of this series is to show that in general the inverse method exhibits biases analogous to, though different in detail from, those exhibited by the forward relation. The form of calibration biases is determined by sample selection criteria. The main results of this paper are accurate expressions for the calibration biases which arise under several realistic types of sample selection, and a straightforward numerical technique to correct for such biases. The cases considered in detail are those in which sample selection is based upon photographic magnitude or diameter limits, while the Tully-Fisher photometry itself uses CCD or infrared magnitudes. Numerical simulations are presented to illustrate the bias effects and verify the efficacy of the correction procedure.