  This paper studies optimal hypothesis testing for nonregular statistical
models with parameter-dependent support. We consider both one-sided and
two-sided hypothesis testing and develop asymptotically uniformly most powerful
tests based on the likelihood ratio process. The proposed one-sided test
involves randomization to achieve asymptotic size control, some tuning constant
to avoid discontinuities in the limiting likelihood ratio process, and a
user-specified alternative hypothetical value to achieve the asymptotic
optimality. Our two-sided test becomes asymptotically uniformly most powerful
without imposing further restrictions such as unbiasedness. Simulation results
illustrate desirable power properties of the proposed tests.
