Statistics and Its Interface

Volume 11 (2018)

Number 4

Semiparametric varying-coefficient partially linear models with auxiliary covariates

Pages: 587 – 602



Xiaojing Wang (Department of Statistics, University of Connecticut, Storrs, Ct., U.S.A.)

Yong Zhou (Faculty of Economics and Managment, Institute of Statistics and Interdisciplinary Sciences, East China Normal University, Shanghai, China)

Yang Liu (Department of Statistics, University of Connecticut, Storrs, Ct., U.S.A.)


In this paper, we consider a semiparametric varying-coefficient 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.


incomplete data, validated samples, profile-likelihood, surrogate variables

2010 Mathematics Subject Classification

Primary 62G05. Secondary 62G20.

Zhou’s work was supported by the State Key Program of National Natural Science Foundation of China (NSFC) (71331006), the State Key Program in the Major Research Plan of National Natural Science Foundation of China (91546202).

Received 20 July 2016

Published 19 September 2018