Efficiency of estimating equations and the use of pivots

Methods are proposed for deriving unbiased estimating equations for the parameters of interest in a statistical model which includes further incidental or nuisance parameters. We start with pivot-like quantities which eliminate the dependence on the incidental parameters, and derive functions of these which give estimating equations which are most efficient in a certain sense. These equations involve the incidental parameters, which are then replaced by estimators. The methods overcome some of the difficulties encountered in likelihood and least squares estimation when there are many incidental parameters, and our theory reduces to likelihood and least squares estimation when there are none. Thus the optimality results apply to these methods too. Some examples are discussed.