🤖 AI Summary
This paper addresses optimal hypothesis testing under parametrically dependent support sets—i.e., nonregular models. For both one-sided and two-sided testing problems, we propose asymptotically uniformly most powerful (AUMP) tests based on the likelihood ratio process. We introduce, for the first time, a randomized one-sided test achieving exact asymptotic level control; for two-sided testing, the proposed test remains AUMP under standard unbiasedness constraints. A key innovation is the incorporation of a tuning constant that eliminates discontinuities in the limiting likelihood ratio process. Rigorous theoretical analysis establishes the asymptotic optimality of both tests. Monte Carlo simulations demonstrate that the proposed procedures attain substantially higher finite-sample power than existing methods, particularly in small- to moderate-sized samples.
📝 Abstract
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.