Modelling Count Data with Heteroscedastic Measurement Error in the Covariates

This paper is concerned with the estimation of the regression coefficients for a count data model when one of the explanatory variables is subject to heteroscedastic measurement error. The observed values W are related to the true regressor X by the additive error model W=X+U. The errors U are assumed to be normally distributed with zero mean but heteroscedastic variances, which are known or can be estimated from repeated measurements. Inference is done by using quasi likelihood methods, where a model of the observed data is specified only through a mean and a variance function for the response Y given W and other correctly observed covariates. Although this approach weakens the assumption of a parametric regression model, there is still the need to determine the marginal distribution of the unobserved variable X, which is treated as a random variable. Provided appropriate functions for the mean and variance are stated, the regression parameters can be estimated consistently. We illustrate our methods through an analysis of lung cancer rates in Switzerland. One of the covariates, the regional radon averages, cannot be measured exactly due to the strong dependency of radon on geological conditions and various other environmental sources of influence. The distribution of the unobserved true radon measure is modelled as a finite mixture of normals