This paper considers the first order large sample properties of the GEL class of estimators for models specified by non-smooth indicators. The GEL class includes a number of estimators recently introduced as alternatives to the efficient GMM estimator which may suffer from substantial biases in finite samples. These include EL, ET and the CUE. This paper also establishes the validity of tests suggested in the smooth moment indicators case for over-identifying restrictions and specification. In particular, a number of these tests avoid the necessity of providing an estimator for the Jacobian matrix which may be problematic for the sample sizes typically encountered in practice.