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Nonasymptotic analysis of semiparametric regression models with high-dimensional parametric coefficients. (English) Zbl 1392.62201
Author’s abstract: We consider a two-step projection based Lasso procedure for estimating a partially linear regression model where the number of coefficients in the linear component can exceed the sample size and these coefficients belong to the $$l_{q}$$-“balls” for $$q\in[0,1]$$. Our theoretical results regarding the properties of the estimators are nonasymptotic. In particular, we establish a new nonasymptotic “oracle” result: Although the error of the nonparametric projection per se (with respect to the prediction norm) has the scaling $$t_{n}$$ in the first step, it only contributes a scaling $$t_{n}^{2}$$ in the $$l_{2}$$-error of the second-step estimator for the linear coefficients. This new “oracle” result holds for a large family of nonparametric least squares procedures and regularized nonparametric least squares procedures for the first-step estimation and the driver behind it lies in the projection strategy. We specialize our analysis to the estimation of a semiparametric sample selection model and provide a simple method with theoretical guarantees for choosing the regularization parameter in practice.

##### MSC:
 62J02 General nonlinear regression 62N01 Censored data models 62N02 Estimation in survival analysis and censored data 62G08 Nonparametric regression and quantile regression 62J12 Generalized linear models (logistic models)
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