Characterization theorems when variables are measured with error

Linear regression models are studied when variables of interest are observe d in the presence of measurement error. Techniques involving Fourier transf orms that lead to simple differential equations with unique solutions are u sed in the context of multiple regression. Necessary and sufficient conditi ons are proven for a random vector of measurement error of the independent variable to be multivariate normal. One characterization involves the Fishe r score of the observed vector. A second characterization involves the Hess ian matrix of the observed density. (C) 1999 Academic Press. AMS 1991 subje ct classifications: 62J05, 62H05.