One-step efficient GMM estimation has been developed in the recent papers of Back and Brown (1990), Imbens (1993), and Qin and Lawless (1994). These papers emphasized methods that correspond to using Owen's (1988) method of empirical likelihood to reweight the data so that the reweighted sample obeys all the moment restrictions at the parameter estimates. In this paper we consider an alternative KLIC motivated weighting and show how it and similar discrete reweightings define a class of unconstrained optimization problems which includes GMM as a special case. Such KLIC-motivated reweightings introduce M auxiliary "tilting" parameters, where M is the number of moments; parameter and overidentification hypotheses can be recast in terms of these tilting parameters. Such tests are often startlingly more effective than their conventional counterparts. These differences are not completely explained by differences in the leading terms of the asymptotic expansions of the test statistics.