The Geometry of Estimating Functions in the Presence of Nuisance Parameters 1

Abstract

The paper studies the geometry of estimating functions in the presence of nuisance parameters. The basic technique involves an idea of orthogonal projection first introduced by Small and McLeish (1989, 1991, 1992, 1994) in this context. The three main topics are : (A) globally optimal estimating functons; (B) locally optimal estimating functions; (C) conditionally optimal estimating functions. A general result is derived in each case. As special cases, we extend and unify some of the results already available in the literature. In particular, as special cases of our result on globally optimal estimating functions, we find the results of Godambe and Thompson (1974) and Godambe (1976) with nuisance parameters. We provide also a geometric interpretation of conditional and marginal inference of Bhapkar (1989, 1991) and Lloyd (1987) . As application of our result on locally optimal estimating functions, Godambe's (1985) result on optimal estimating functions for stochastic processes is extended to nuisance parameters. Finally, our general result on conditionally optimal estimating function helps to generalize the findings of Godambe and Thompson (1989) to situations which admit the presence of nuisance parameters.