Locally efficient estimators for semiparametric models with measurement error

We derive constructive locally efficient estimators in semiparametric measurement error models. The setting is one in which the likelihood function depends on variables measured with and without error, where the variables measured without error can be modeled nonparametrically. The algorithm is based on backfitting. We show that if one adopts a parametric model for the latent variable measured with error and if this model is correct, then the estimator is semiparametric efficient; if the latent variable model is misspecified, then our methods lead to a consistent and asymptotically normal estimator. Our method further produces an estimator of the nonparametric function that achieves the standard bias and variance property. We extend the methodology to allow estimation of parameters in the measurement error model by additional data in the form of replicates or instrumental variables. The methods are illustrated through a simulation study and a data example, where the putative latent variable distribution is a shifted lognormal, but concerns about the effects of misspecification of this assumption and the linear assumption of another covariate demand a more model-robust approach. A special case of wide interest is the partial linear measurement error model. If one assumes that the model error and the measurement error are both normally distributed, then our estimator has a closed form. When a normal model for the unobservable variable is also posited, our estimator becomes consistent and asymptotically normally distributed for the general partially linear measurement error model, even without any of the normality assumptions under which the estimator is originally derived. We show that the method in fact reduces to a same estimator as that of Liang et al., thus demonstrating a previously unknown optimality property of their method.