Comparing the efficiency of structural and functional methods in measurement error models

The paper is a survey of recent investigations by the authors and
others into the relative efficiencies of structural and functional
estimators of the regression parameters in a measurement error
model. While structural methods, in particular the quasi-score (QS)
method, take advantage of the knowledge of the regressor
distribution (if available), functional methods, in particular the
corrected score (CS) method, discards such knowledge and works even
if such knowledge is not available. Among other results, it has
been shown that QS is more efficient than CS as long as the
regressor distribution is completely known. However, if nuisance
parameters in the regressor distribution have to be estimated, this
is no more true in general. But by modifying the QS method, the
adverse effect of the nuisance parameters can be overcome. For
small measurement errors, the efficiencies of QS and CS become
almost indistinguishable, whether nuisance parameters are present
or not. QS is (asymptotically) biased if the regressor distribution
has been misspecified, while CS is always consistent and thus more
robust than QS.