On the grouped LSE under an errors-in-variables model

In the error-in-variables regression model, some or all of the covariates are observed with error. Standard approaches to this problem include maximum likelihood estimation, which requires some of the parameters to be known or be estimated from a separate experiment, moment estimation, which requires at least as many instrumental variables as there are variables observed with error, and the method of grouping initiated by Wald. The present method is kin to the grouping approach but here we allow the number of groups and the number of observations per group to tend to infinity. Without errors in variables the resulting estimator is asymptotically equivalent to the ordinary least-squares estimator. The present method is also related to the instrumental variables approach in the sense that information from an instrumental variable can be used in forming the groups; however, the present method is more generally applicable. When one of the covariates in the regression model is observed without error, the grouping can be done according to the values of this covariate and thus it requires no extraneous information.