## Comparing the efficiency of estimates in concrete errors-in-variables models under unknown nuisance parameters.(English)Zbl 1164.62021

Suppose that the relation between a response variable $$y$$ and a covariate (or regressor) $$x$$ is given by a pair of conditional mean and variance functions: $$E(y| x)=:m(x,\theta),\;V(y| x)=:v(x,\theta)$$. Here $$\theta$$ is an unknown $$d$$-dimensional parameter vector to be estimated. The parameter $$\theta$$ belongs to an open parameter set $$\Theta$$. The variable $$x$$ has a density $$\rho(x,\theta)$$ with respect to a $$\sigma$$-finite measure $$\nu$$ on a Borel $$\sigma$$-field on the real line. It is assumed that $$v(x,\theta)>0$$, for all $$x$$ and $$\theta$$, and that all the functions are sufficiently smooth. Such a model is called a mean-variance model.
The estimation of $$\theta$$ is based on an extended quasi score (QS) function. Of special interest is the case where the distribution of $$x$$ depends only on a subvector $$\alpha$$ of $$\theta$$, which may be considered as a nuisance parameter. A major application of this model is the classical measurement error model, where the corrected score (CS) estimator is an alternative to the QS estimator. Under unknown nuisance parameters the authors derive conditions under which the QS estimator is strictly more efficient than the CS estimator. Considered are the loglinear Poisson, gamma, and logit models.

### MSC:

 62H12 Estimation in multivariate analysis 62J05 Linear regression; mixed models 62J12 Generalized linear models (logistic models) 62F12 Asymptotic properties of parametric estimators 62J10 Analysis of variance and covariance (ANOVA)