*J. Japan Statist. Soc.,* Vol. 34 (No. 1), pp. 75-86, 2004

## A paradoxical effect of nuisance parameters on efficiency of estimators

Masayuki Henmi

**Abstract. **
This paper is concerned with parameter estimation in the presence of nuisance parameters. Usually, an estimator with known nuisance parameters is better than that with unknown nuisance parameters in reference to the asymptotic variance. However, it has been noted that the opposite can occur in some situations. In this paper we elucidate when and how this phenomenon occurs using the orthogonal decomposition of estimating functions. Most of the examples of this phenomenon are found in the case of semiparametric models, but this phenomenon can also occur in parametric models. As an example, we consider the estimation of the dispersion parameter in a generalized linear model.

*Key words and phrases*:
Asymptotic variance, estimating function, nuisance parameter, optimality, orthogonal decomposition, semiparametric model.

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