Estimation of the variance matrix in bivariate classical measurement error models

The presence of measurement errors is a ubiquitously faced problem and plenty of work has been done to overcome this when a single covariate is mismeasured under a variety of conditions. However, in practice, it is possible that more than one covariate is measured with error. When measurements are taken by the same device, the errors of these measurements are likely correlated. In this paper, we present a novel approach to estimate the covariance matrix of classical additive errors in the absence of validation data or auxiliary variables when two covariates are subject to measurement error. Our method assumes these errors to be following a bivariate normal distribution. We show that the variance matrix is identifiable under certain conditions on the support of the error-free variables and propose an estimation method based on an expansion of Bernstein polynomials. To investigate the performance of the proposed estimation method, the asymptotic properties of the estimator are examined and a diverse set of simulation studies is conducted. The estimated matrix is then used by the simulation-extrapolation (SIMEX) algorithm to reduce the bias caused by measurement error in logistic regression models. Finally, the method is demonstrated using data from the Framingham Heart Study.