Measurement error models with a general class of error distribution

The ordinary maximum likelihood (ML) approach in classical regression models, fails when the independent variables are subject to error. The most noticeable and well known problem reported in the literature is the inconsistency of the ML estimators [1]. To solve this problem, a number of alternatives were proposed. The measurement error model (MEM) is the most fashionable of them, but it has some limitations. It is necessary to know some parameters to avoid inconsistencies resulting from unbounded likelihoods (functional version) and non-identifiability (structural version). For more details see, for example [2] and references therein. Typically [1], it is assumed that the errors are normally distributed, but the normal assumption is not always tenable. Actually, this is a strong assumption that cannot always be satisfied in practice. There are situations where the observed covariate is positive, so that its distribution may not be appropriately approximated by a normal distribution. The most important contributions of this article are to introduce a multiple regression model in which some covariates subject to error are not necessarily normally distributed and to propose a method for obtaining consistent estimators for all parameters of the model, based on the corrected score approach. The method is computationally simple and can be implemented with any statistical package. It extends the results in [3] that studied an MEM where the surrogate for the unobservable true covariate is the event count per unit time. They regarded the Poisson distribution for the surrogate variable, the rate of which is the unobserved true covariate. The authors justified the proposed model with a medical example.