Alexander Kukush,
Kyiv National Taras Shevchenko University
Efficient Estimation in Nonlinear Measurement Error Model under Unknown Nuisance Parameters
A mean-variance (MV) regression model is considered where the stochastic dependence between y and x is given via conditional mean and variance, which include unknown regression parameter. We assume that the explanatory variable x is normally distributed. The mean and variance of x are unknown nuisance parameters.
We construct the Quasi-Score (QS) estimator for all unknown parameters, which is a modification of the Quasi-Likelihood approach. The asymptotic efficiency of the QS estimator is shown within a broad class of estimators generated by linear-in-y unbiased scores [1].
In more detail we study a particular case of the MV model, namely a nonlinear errors-in-variables model. For the polynomial, Poisson, and Gamma model, we show that the QS estimators of nuisance parameters will be just empirical mean and empirical variance of the observed x’s. We prove that the
QS estimator of the regression parameter is strictly more asymptotically efficient than the Corrected
Score estimator [1]. The latter does not utilize the information about the form of the distribution of x and therefore more robust compared with QS. Similar problems are studied for the zero-inflated
Poisson model [2].
The efficiency of QS estimate for the polynomial model under nuisance parameters known was studied in [3].
The results are joint with Prof. H. Schneeweiss (Munich), Dr. Shalabh (Kanpur, India), and my Ph.D. student A. Malenko.
References
[1] A. Kukush, A. Malenko, and H. Schneeweiss, Optimality of the quasi-score estimator in a meanvariance model with applications to measurement error models, Discussion Paper 494, SFB 386.
Ludwig-Maximilians-University of Munich, 2006.
[2] A. Kukush, A. Malenko, H. Schneeweiss, and Shalabh, Optimality of Quasi-Score in the multivariate mean-variance model with application to the zero-inflated Poisson model with measurement errors, Discussion Paper 498, SFB 386. Ludwig-Maximilians-University of Munich, 2007.
[3] S. Shklyar, H. Schneeweiss, and A. Kukush, Quasi Score is more efficient than Corrected Score in a polynomial measurement error model. Metrika, 2007, 65, N3, 275-295.