Tests against inequality constraints in semiparametric models

We consider the problem of testing that a subvector of a finite dimensional parameter is zero against that it satisfies some inequality constraints in a semiparametric model. To our knowledge, this problem has not been studied in the literature in a general context. Huang (Ann. Statist. 24 (1996) 540) developed maximum likelihood estimation in a class of semiparametric models, and Choi et al. (Ann. Statist. 24 (1996) 84) developed efficient tests against unrestricted alternatives in a large class of semiparametric models which includes the adaptable ones. In this paper, we introduce local tests as a general approach to developing tests against inequality constraints, and apply them in different semiparametric models. In particular, we apply our local tests to develop adaptive tests and efficient tests in the classes of semiparametric models studied by Huang (1996) and Choi et al. (1996), thus extending their results to inequality constrained framework.