## The Annals of Statistics

- Ann. Statist.
- Volume 45, Number 5 (2017), 2274-2298.

### Nonasymptotic analysis of semiparametric regression models with high-dimensional parametric coefficients

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 45, Number 5 (2017), 2274-2298.

**Dates**

Received: October 2015

Revised: November 2016

First available in Project Euclid: 31 October 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1509436835

**Digital Object Identifier**

doi:10.1214/16-AOS1528

**Mathematical Reviews number (MathSciNet)**

MR3718169

**Zentralblatt MATH identifier**

06821126

**Subjects**

Primary: 62J02: General nonlinear regression

Secondary: 62N01: Censored data models 62N02: Estimation 62G08: Nonparametric regression 62J12: Generalized linear models

**Keywords**

High-dimensional statistics Lasso nonasymptotic analysis partially linear models sample selection

#### Citation

Zhu, Ying. Nonasymptotic analysis of semiparametric regression models with high-dimensional parametric coefficients. Ann. Statist. 45 (2017), no. 5, 2274--2298. doi:10.1214/16-AOS1528. https://projecteuclid.org/euclid.aos/1509436835

#### Supplemental materials

- Supplementary materials for “Nonasymptotic analysis of semiparametric regression models with high-dimensional parametric coefficients”. This supplement contains two Appendices. Appendix A provides the proofs for the main results and Appendix S provides the remaining technical lemmas and proofs.Digital Object Identifier: doi:10.1214/16-AOS1528SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.