zbMATH — the first resource for mathematics

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.

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)
Full Text: DOI