On Optimal Estimating Functions in the Presence of Nuisance Parameters

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