A (nonlinear) measurement error model (MEM) consists of three parts: (1) a regression model relating an observable regressor variable z and an unobservable regressor variable ξ (the variables are independent and generally vector valued) to a response variable y, which is considered here to be observable without measurement errors; (2) a measurement model relating the unobservable ξ to an observable surrogate variable x; and (3) a distributional model for ξ.
Parts of MEM
The regression model can be described by a conditional distribution of y given (z, ξ) and given an unknown parameter vector θ. As usual this distribution is represented by a probability density function f(y | z, ξ; θ) with respect to some underlying measure on the Borel σ-field of R. We restrict our attention to distributions that belong to the exponential family, i.e., we assume f to be of the form