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Covariate-adjusted partially linear regression models. (English) Zbl 1284.62422
Summary: Motivated by covariate-adjusted regression (CAR) proposed by D. Şentürk and H.-G. Müller [Biometrika 92, No. 1, 75–89 (2005; Zbl 1068.62082)] and an application problem, in this article we introduce and investigate a covariate-adjusted partially linear regression model (CAPLM), in which both response and predictor vector can only be observed after being distorted by some multiplicative factors, and an additional variable such as age or period is taken into account. Although our model seems to be a special case of covariate-adjusted varying coefficient model (CAVCM) given by D. Şentürk [Biostatistics 7, No. 2, 235–251 (2006; Zbl 1169.62360)], the data types of CAPLM and CAVCM are basically different and then the methods for inferring the two models are different. In this article, the estimate method motivated by X. Cui et al. [Ann. Stat. 37, No. 4, 1839–1870 (2009; Zbl 1168.62035)] is employed to infer the new model. Furthermore, under some mild conditions, the asymptotic normality of estimator for the parametric component is obtained. Combined with the consistent estimate of asymptotic covariance, we obtain confidence intervals for the regression coefficients. Also, some simulations and a real data analysis are made to illustrate the new model and methods.

MSC:
62J05 Linear regression; mixed models
62F12 Asymptotic properties of parametric estimators
62G05 Nonparametric estimation
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