  In this paper we consider the construction of optimal tests of equivalence
hypotheses. Specifically, assume X_1,..., X_n are i.i.d. with distribution
P_{\theta}, with \theta \in R^k. Let g(\theta) be some real-valued parameter of
interest. The null hypothesis asserts g(\theta)\notin (a,b) versus the
alternative g(\theta)\in (a,b). For example, such hypotheses occur in
bioequivalence studies where one may wish to show two drugs, a brand name and a
proposed generic version, have the same therapeutic effect. Little optimal
theory is available for such testing problems, and it is the purpose of this
paper to provide an asymptotic optimality theory. Thus, we provide asymptotic
upper bounds for what is achievable, as well as asymptotically uniformly most
powerful test constructions that attain the bounds. The asymptotic theory is
based on Le Cam's notion of asymptotically normal experiments. In order to
approximate a general problem by a limiting normal problem, a UMP equivalence
test is obtained for testing the mean of a multivariate normal mean.
