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 (semiparametrically) weakly identified. I characterise the appropriate limit experiment in which to study local (asymptotic) optimality of tests in the non-regular case and generalise 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：本文考虑在半参数模型中进行假设检验，这些模型可能具有非正则性。我证明了在温和条件下，C($\alpha$) 型检验是局部正则的，包括在局部正则估计量不存在的情况，例如（半参数）弱识别模型。我刻画了用于研究非正则情形下检验的局部（渐近）最优性的恰当极限实验，并将经典的功效界推广到此种情形。我给出了所提出的 C($\alpha$) 型检验达到这些功效界的条件。文中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。