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New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models. (English) Zbl 1209.62074
Summary: The complexity of semiparametric models poses new challenges to statistical inference and model selection that frequently arise from real applications. We propose new estimation and variable selection procedures for the semiparametric varying-coefficient partially linear model. We first study quantile regression estimates for the nonparametric varying-coefficient functions and the parametric regression coefficients. To achieve nice efficiency properties, we further develop a semiparametric composite quantile regression procedure. We establish the asymptotic normality of the proposed estimators for both the parametric and nonparametric parts and show that the estimators achieve the best convergence rate. Moreover, we show that the proposed method is much more efficient than the least-squares-based method for many non-normal errors and that it only loses a small amount of efficiency for normal errors.
In addition, it is shown that the loss in efficiency is at most 11.1% for estimating varying coefficient functions and is no greater than 13.6% for estimating parametric components. To achieve sparsity with high-dimensional covariates, we propose adaptive penalization methods for variable selection in the semiparametric varying-coefficient partially linear model and prove that the methods possess the oracle property. Extensive Monte Carlo simulation studies are conducted to examine the finite-sample performance of the proposed procedures. Finally, we apply the new methods to analyze the plasma beta-carotene level data.

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62H12 Estimation in multivariate analysis
62F12 Asymptotic properties of parametric estimators
65C05 Monte Carlo methods
62G05 Nonparametric estimation
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