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.

We consider functional measurement error models, i.e., models where covariates are measured with classical error, and yet no distributional assumptions are made about the mismeasured variable. We propose and study a score-like local test and a series expansion based omnibus test in this context, where no likelihood function is available or calculated–that is, all the tests are proposed in the semiparametric model framework. We demonstrate that our tests have optimality properties and computational advantages similar to those of the classical score tests in the parametric model framework. The test procedures are applicable to several semiparametric extensions of measurement error models, including when the measurement error distribution is estimated nonparametrically as well as for generalized partially linear models. The performance of the local and omnibus tests is demonstrated through simulation studies and analysis of a nutrition data set.

Local and omnibus goodness-of-fit tests in classical measurement error models