Characterization Theorems when Variables Are Measured with Error

Linear regression models are studied when variables of interest are observed in the presence of measurement error. Techniques involving Fourier transforms that lead to simple differential equations with unique solutions are used in the context of multiple regression. Necessary and sufficient conditions are proven for a random vector of measurement error of the independent variable to be multivariate normal. One characterization involves the Fisher score of the observed vector. A second characterization involves the Hessian matrix of the observed density.