On Optimal Estimating Functions in the Presence of Nuisance Parameters

Abstract

When there is only one interesting parameter 1 and one nuisance parameter 2 , Godambe and Thompson (1974) showed that the optimal estimating function for 1 essentially is a linear function of the 1 -score, the square of the 2 -score, and the derivative of 2 -score with respect to 2 . Mukhopadhyay (2000b) generalized this result to m nuisance parameters. Mukhopadhyay (2000, 2002a,b) obtained lower bounds to the variance of regular estimating functions in the presence of nuisance parameters. Taking cues from these results we propose a method of finding optimal estimating function for 1 by taking the multiple regression equation on 1 score and Bhattacharyya's (1946) scores with respect to 2 . The result is extended to the case of m nuisance parameters.