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Estimation for a partial-linear single-index model. (English) Zbl 1181.62038
Summary: We study estimation of a partial-linear single-index model. A two-stage estimation procedure is proposed to estimate the link function for the single index and the parameters in the single index, as well as the parameters in the linear component of the model. Asymptotic normality is established for both parametric components.
For the index, a constrained estimating equation leads to an asymptotically more efficient estimator than existing estimators in the sense that it is of a smaller limiting variance. The estimator of the nonparametric link function achieves optimal convergence rates, and the structural error variance is obtained. In addition, the results facilitate the construction of confidence regions and hypothesis testing for the unknown parameters. A simulation study is performed and an application to a real dataset is illustrated. The extension to multiple indices is briefly sketched.

MSC:
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62G08 Nonparametric regression and quantile regression
62L12 Sequential estimation
62E20 Asymptotic distribution theory in statistics
62G15 Nonparametric tolerance and confidence regions
65C60 Computational problems in statistics (MSC2010)
Software:
SemiPar
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