A minimax approach to consistency and efficiency for estimating equations

The consistency of estimating equations has been studied, in the main, along the lines of Cramér's classical argument, which only asserts the existence of consistent solutions. The statement similar to that of Doob and Wald, which identifies the consistent solutions, has not yet been established. The obstacle is that the solutions of estimating equations cannot in general be defined as the maximum of likelihood functions. In this paper we demonstrate that the consistent solutions can be identified as the minimax of a function R, whose properties resemble those of a log likelihood ratio, but which exists in a much wider context. Furthermore, since we do not need R to be differentiable, the minimax is consistent even when the estimating equation does not exist. In this respect, the minimax is a new estimator. We first convey the idea by focusing on the quasi-likelihood estimate, and then indicate its full generality by providing a set of sufficient conditions for consistency and studying a number of important cases. Efficiency will also be verified.