One of the most basic questions in Statistics is "Are the centers of two distributions the same?". The question is answered formally through a hypothesis test with a null of "no difference", whether the analysis be classical or Bayesian. Traditional formulations of the problem rely on the belief that the models are perfectly specified. Robustness to violations of assumptions is typically studied under conditions that do not change the validity of the null hypothesis (e.g., symmetric contaminations). The actual deficiencies in model and data are likely to change a “true null” into a false statement.
The imperfections of model and data suggest the use of flexible nonparametric models for the two distributions and also suggest consideration of a family of measures of center (e.g., for realvalued data, those based upon Huber’s loss function). Each measure of the center generates a testing problem. The resulting family of null hypotheses constitutes a familial null hypothesis. Profile methods replace the original question with “Is there a measure of center in the family for which the centers of the two distributions are the same?”
