Different Nonlinear Regression Models with Incorrectly Observed Covariates

Abstract

We present quasi-likelihood models for di erent regression problems when one of the explanatory variables is measured with heteroscedastic error. In order to derive models for the observed data the conditional mean and variance functions of the regression models are only expressed through functions of the observable covariates. The latent c o variable is treated as a random variable that follows a normal distribution. Furthermore it is assumed that enough additional information is provided to estimate the individual measurement error variances, e.g. through replicated measurements of the fallible predictor variable. The discussion includes the polynomial regression model as well as the probit and logit model for binary data, the Poisson model for count data and ordinal regression models.

Some Basic Results on the Extension of Quasi-likelihood Based Measurement Error Correction to Multivariate and Flexible Structural Models

“…Thamerus (1998A, 1998B) and Augustin (2000) worked with simpler versions of the model used here letting some of the main aspects of the arguments given below already shine up. For modeling the distribution of the latent variable, Thamerus (1998B) and Augustin (2000) do only allow for a single normal distribution, but not for mixtures. Even more important, all three papers just quoted had to concentrate on the case where only one dimension, X i 1] say, of the covariate vector is measured with error.…”

Consistent estimation and testing in heteroscedastic polynomial errors-in-variables models

“…Gimenez and Bolfarine (1997) however, provide rigorous proofs for consistency and asymptotic normality for the estimators obtained by solving estimating equations that follows by using the corrected score approach (Stefanski, 1989;Nakamura, 1990) in functional situations. In the structural model, Thamerus (1998) concerns with heteroscedastic measurement errors for all kind of nonlinear models.…”