Malenko A. Comparison of estimators in deneral nonlinear errors-in-variables regression model

The thesis is devoted to the study of new Quasi Score estimator in a general mean-variance model and errors-in-variables model, in which regressor density depends on unknown parameter as well. We consider the class of all consistent and asymptotically normal estimators constructed from linear-in-response estimating functions and propose a new structural Quasi Score like estimator (QL). It is proved that the new QL estimator has the least asymptotic covariance matrix (ACM) among all the estimators from class. Moreover one can easily compute the rank of the difference between ACM of QL and the ACM of another estimator in concrete model. We also study the cases in which it is possible to pre-estimate a part of parameters, using only regressor data, without solving a system of nonlinear equations. Theorems about comparison of ACMs are applied to polynomial, Poisson, Gamma, and logistic errors-in-variables models. We show the conditions under which QL estimator is strictly more efficient than the functional Corrected Score (CS) one. The difference between ACMs of QL and CS is also studied under small error variance. We consider also a generalization of mean-variance model to the case of vector response. In this case we elaborate a generalization of QL method and prove similar theorems about the comparison. The obtained result is applied to polynomial model with unknown responce variance and to Gamma model. We show the advantage of the new QL method over usual ''scalar'' one. In a scalar zero inflated Poisson model we managed to construct the QL estimator by means of additional component, indicator of event y=0. We show strict asymptotic advantage of this QL estimator over CS one. We also propose a new goodness-of-fit test which is based on estimating function and its partial derivatives. We study its local power in a polynomial errors-in-variables model.