Potential Functions and Conservative Estimating Functions

Abstract

A quasiscore function, as defined by Wedderburn and by McCullagh, frequently fails to have a symmetric derivative matrix. Such a score function cannot be the gradient of any potential function on the parameter space; that is, there is no "quasilikelihood." Without a likelihood function it is difficult to distinguish good roots from bad roots or to set satisfactory confidence limits. From a different perspective, a potential function seems to be essential in order to give the theory an approximate Bayesian interpretation. The purpose of this paper is to satisfy these needs by developing a method of projecting the true score function onto a class of conservative estimating functions. By construction, a potential function for the projected score exists having many properties of a log-likelihood function.