  What role should assumptions play in inference? We present a small
theoretical case study of a simple, clean case, namely the nonparametric
comparison of two continuous distributions using (essentially) information
about quartiles, that is, the central information displayed in a pair of
boxplots. In particular, we contrast a suggestion of John Tukey--that the
validity of inferences should not depend on assumptions, but assumptions have a
role in efficiency--with a competing suggestion that is an aspect of Hansen's
generalized method of moments--that methods should achieve maximum asymptotic
efficiency with fewer assumptions. In our case study, the practical performance
of these two suggestions is strikingly different. An aspect of this comparison
concerns the unification or separation of the tasks of estimation assuming a
model and testing the fit of that model. We also look at a method (MERT) that
aims not at best performance, but rather at achieving reasonable performance
across a set of plausible models.
