Estimation of semiparametric models when the criterion function is not smooth

We provide easy to verify suffcient conditions for the consistency and asymptotic normality of a class of semiparametric optimization estimators where the criterion function does not obey standard smoothness conditions and simultaneously depends on some preliminary nonparametric estimators. Our results extend existing theories like those of Pakes and Pollard (1989), Andrews (1994a), and Newey (1994). We apply our results to two examples: a 'hit rate' and a partially linear median regression with some endogenous regressors.