  This paper considers hypothesis testing in semiparametric models which may be
non-regular. I show that C($\alpha$) style tests are locally regular under mild
conditions, including in cases where locally regular estimators do not exist,
such as models which are (semi-parametrically) weakly identified. I
characterise the appropriate limit experiment in which to study local
(asymptotic) optimality of tests in the non-regular case, permitting the
generalisation of classical power bounds to this case. I give conditions under
which these power bounds are attained by the proposed C($\alpha$) style tests.
The application of the theory to a single index model and an instrumental
variables model is worked out in detail.
