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 corresponds 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 to 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, when appropriately conditioned on the estimates of the original parameters, are often startlingly more effective than their conventional counterparts. This is apparently due to the local ancillarity of the original parameters for the tilting parameters.