Abstract
SUMMARY 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 para meters 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.
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