  Measurement error data or errors-in-variable data have been collected in many
studies. Natural criterion functions are often unavailable for general
functional measurement error models due to the lack of information on the
distribution of the unobservable covariates. Typically, the parameter
estimation is via solving estimating equations. In addition, the construction
of such estimating equations routinely requires solving integral equations,
hence the computation is often much more intensive compared with ordinary
regression models. Because of these difficulties, traditional best subset
variable selection procedures are not applicable, and in the measurement error
model context, variable selection remains an unsolved issue. In this paper, we
develop a framework for variable selection in measurement error models via
penalized estimating equations. We first propose a class of selection
procedures for general parametric measurement error models and for general
semi-parametric measurement error models, and study the asymptotic properties
of the proposed procedures. Then, under certain regularity conditions and with
a properly chosen regularization parameter, we demonstrate that the proposed
procedure performs as well as an oracle procedure. We assess the finite sample
performance via Monte Carlo simulation studies and illustrate the proposed
methodology through the empirical analysis of a familiar data set.
