Title: What is asymptotically testable and what is not
Authors: Bas Kleijn - University of Amsterdam (Netherlands) [presenting]
Abstract: Given a statistical model for i.i.d. data, certain hypotheses can be tested consistently, while others cannot. If one thinks of consistent tests only in terms of converging sequences of test statistics, some immediate, simple conclusions can be drawn. But classical counterexamples demonstrate that the matter is more involved. We address the problem of what characterizes the asymptotic testability of hypotheses for uniform, pointwise and Bayesian tests. Posteriors distinguish measurable hypotheses (prior-almost-surely), but frequentist tests require more. Application of the Le Cam-Schwartz theorem (write U for the associated uniformity) leads to two equivalences: hypotheses are testable with uniform power if and only if they are separated by a uniformity U. Hypotheses are testable in a pointwise sense, if and only if the testing problem can be represented (continuously with respect to U) in a separable metric space. The above is illustrated with a large number of examples.