Prof. Dr. Alexander Kukush,
    
Kiev National Taras Shevchenko University, Ukraine

Quasi-Likelihood is More Efficient than Corrected Score in a Nonlinear
Measurement Error Model

Abstract:

A regression model is considered, where the response variable has a
density belonging to an exponential family and the covariate is
measured with Gaussian errors. The measurement error variance is
supposed to be known. The covariate is normally distributed with known
mean and variance. We compare two consistent estimators with respect
to their relative asymptotic efficiency. The quasi score (QS)
estimator uses the distribution of the regressor, while the corrected
score (CS) estimator does not make use of this distribution and is
therefore more robust. However, if the regression distribution is
known, QS is asymptotically more efficient than CS. In some cases it
is, in fact, even strictly more efficient, in the sense that the
difference of the asymptotic covariance matrices of CS and QS is
positive definite. The comparison is made by introducing a third
estimator, the efficiency of which turns out to be intermediate
between QS and CS. The results are applicable, e.g., to polynomial
model [3], log-linear Poisson model, and log-linear Gamma model.  Our
proof for the Poisson model is simpler than in [1].   The maximum
likelihood estimator is discussed as well. This estimator is much more
difficult to compute, and its asymptotic properties are not clear
since the regularity conditions of the log-likelihood function in
non-linear models are doubtful.   The results are joint with
Prof. Dr. H. Schneeweiss (Munich) and Dr.  S. Shklyar (Kiev) [2, 3].

References
    
     1. S. Shklyar and H. Schneeweiss, A comparison of asymptotic
     covariance matrices of three consistent estimators in the Poisson
     regression model with measurement errors. /Journal of
     Multivariate Analysis/, 2005, *94*, N2, 250-270.
    
     2. A. Kukush, H. Schneeweiss, and S. Shklyar, Quasi Score is more
     efficient than Corrected Score in a general nonlinear measurement
     error model, Discussion Paper 451, SFB 386.
     Ludwig-Maximilians-University of Munich, 2005.
    
     3. S. Shklyar, H. Schneeweiss, and A. Kukush, Quasi Score is more
     efficient than Corrected Score in a polynomial measurement error
     model. Discussion Paper 445, SFB 386.
     Ludwig-Maximilians-University of Munich, 2005. To appear in
     Metrika.