Nonoptimally Weighted Least Squares

This research was supported in part by National Institutes of Health Grants AR20610 (Multipurpose Arthritis Center) and 2R01GM21215.12 (Adaptation of New Statistical Ideas for Medicine) awarded to Stanford University.Most of the ideas here originate with John Tukey, in published or unpublished work.The authors indicate that it was especially helpful (while they were trying to prove the result for the regression case) to learn from him, when they asked, that the proposition was true.