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Biometrika 2004 91(4):929-941; doi:10.1093/biomet/91.4.929
© 2004 by Biometrika Trust
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A paradox concerning nuisance parameters and projected estimating functions

Masayuki Henmi1 and Shinto Eguchi2

1 Department of Statistics, University of Warwick, Coventry CV 4 7AL, U.K. m.henmi{at}warwick.ac.uk, 2 Institute of Statistical Mathematics, The Graduate University of Advanced Studies, Minami-azabu, Tokyo 106-8569, Japan eguchi{at}ism.ac.jp

This paper is concerned with a paradox associated with parameter estimation in the presence of nuisance parameters. In a statistical model with unknown nuisance parameters, the efficiency of an estimator of a parameter usually increases when the nuisance parameters are known. However the opposite phenomenon can sometimes occur. In this paper, we elucidate the occurrence of this paradox by examining estimating functions. In particular, we focus on the projected estimating function, which is defined by the projection of the score function on to a given estimating function. A sufficient condition for the paradox to occur is the orthogonality of the two components of the projected estimating functions corresponding to parameters of interest and nuisance parameters. In addition, a numerical assessment is conducted in the context of a simple model to investigate the improvement of the asymptotic efficiency of estimators.

Key Words: Orthogonality of estimating functions; Propensity score; Semiparametric model.

Received March 2003. Revised February 2004.


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