Semiparametric varying-coefficient partially linear models with auxiliary covariates

In this paper, we consider a semiparametric varyingcoefficient partially linear model when some of the covariates are only measured on a selected validation set whereas auxiliary variables are observed for all study subjects. The semiparametric profile-likelihood procedure for estimating parametric and nonparametric component which incorporates information from auxiliary covariates is proposed. The resulting estimators are consistent regardless of the specification of the relationship between the covariates and the surrogate variables. Moreover, the proposed estimators are asymptotically more efficient than the validation-set-only estimators. Asymptotic properties of the proposed estimators are established. The finite sample performance is investigated and compared with alternative methods via simulation studies. The simulated results demonstrate that the asymptotic approximations of the proposed estimators are adequate for practice. We use a Boston Housing dataset to illustrate the performance of the proposed method in practice.