On the least-squares fitting of correlated data: A priori vs a posteriori weighting

One of the methods in common use for analyzing large data sets is a two-step procedure, in which subsets of the full data are first least-squares fitted to a preliminary set of parameters, and the latter are subsequently merged to yield the final parameters. The second step of this procedure is properly a correlated least-squares fit and requires the variance-covariance matrices from the first step to construct the weight matrix for the merge. There is, however, an ambiguity concerning the manner in which the first-step variance-covariance matrices are assessed, which leads to different statistical properties for the quantities determined in the merge. The issue is one of a priori vs a posteriori assessment of weights, which is an application of what was originally called internal vs external consistency by Birge [Phys. Rev. 40, 207-227 (1932)] and Deming (''Statistical Adjustment of Data.'' Dover, New York, 1964). In the present work the simplest case of a merge fit-that of an average as obtained from a global fit vs a two-step fit of partitioned data-is used to illustrate that only in the case of a priori weighting do the results have the usually expected and desired statistical properties: normal distributions for residuals, t distributions for parameters assessed a posteriori, and chi(2) distributions for variances. (C) 1996 Academic Press, Inc.