Shklyar S. Measurement error models and their application in radioepidemiology

Consider linear regression with errors in variables. The true points lie in p-dimensional subspace of the p+q-dimensional Euclidean space . They are observed with measurement errors, which are supposed to be of zero mean and exactly known covariance matrix . Known consistency conditions of the Total Least Squares estimator are relaxed, and new consistency condition are found. Particularly, the Gallo condition, which restricts the rate of growth of the design matrix and thus discriminates against high-leverage points, is not needed anymore. We also consider the model where the true points lie on a conic section or on a couple of two lines. The true points are observed with measurement errors, which are i.i.d. bivariate normal variables with zero mean and scalar times identity covariance matrix. The Adjusted Least Squares estimator for unknown variance of the measurement errors and conditions of its consistency in the conic section filling model were available as a prior art. Here, the asymptotic normality of the estimator is proved, and the estimators of the asymptotic covariance matrix are constructed. The Adjusted Least Squares estimator is “projected” to be used in the two-line fitting problem, and the resulting estimator is compared with other estimators. We also consider nonlinear regression models. In the Berkson models of logistic and Poisson regressions the conditions for consistency of the Maximum Likelihood and, respectively, Quasi-Likelihood estimators are constructed. In inverse exponential regression model with classical measurement errors, the Sufficiency estimator and the Corrected Score estimator are constructed (by known methods) and compared to each other. In the autoregressive AR(1) model with measurement errors with known variance, we compare the Corrected Score estimator and the Maximum Likelihood estimator designed for the ARMA(1,1) model.