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.

翻译：本文研究可能非正则的半参数模型中的假设检验问题。作者证明，在温和条件下，C($\alpha$)型检验具有局部正则性，包括在局部正则估计量不存在的情形（如半参数弱识别模型）中仍成立。本文刻画了适用于非正则情形下检验局部（渐近）最优性研究的极限实验，从而将经典势界推广至该情形。作者给出了这些势界能被所提出的C($\alpha$)型检验达到的条件，并详细阐述了该理论在单指标模型和工具变量模型中的应用。