The purpose of this paper is two fold. First, we investigate estimation for varying coefficient partially linear models in which covariates in the nonparametric part are measured with errors. As there would be some spurious covariates in the linear part, a penalized profile least squares estimation is suggested with the assistance from smoothly clipped absolute deviation penalty. However, the estimator is often biased due to the existence of measurement errors, a bias correction is proposed such that the estimation consistency with the oracle property is proved. Second, based on the estimator, a test statistic is constructed to check a linear hypothesis of the parameters and its asymptotic properties are studied. We prove that the existence of measurement errors causes intractability of the limiting null distribution that requires a Monte Carlo approximation and the absence of the errors can lead to a chi-square limit. Furthermore, confidence regions of the parameter of interest can also be constructed. Simulation studies and a real data example are conducted to examine the performance of our estimators and test statistic.
diverging number of parameters varying coefficient partially linear model penalized likelihood SCAD variable selection
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This work was supported by National Natural Science Foundation of China (Grant Nos. 11401006, 11671299 and 11671042), a grant from the University Grants Council of Hong Kong, the China Postdoctoral Science Foundation (Grant No. 2017M611083), and the National Statistical Science Research Program of China (Grant No. 2015LY55). The authors thank two referees for their insightful comments.
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