## The Annals of Statistics

### Rates of convergence of the Adaptive LASSO estimators to the Oracle distribution and higher order refinements by the bootstrap

#### Abstract

Zou [J. Amer. Statist. Assoc. 101 (2006) 1418–1429] proposed the Adaptive LASSO (ALASSO) method for simultaneous variable selection and estimation of the regression parameters, and established its oracle property. In this paper, we investigate the rate of convergence of the ALASSO estimator to the oracle distribution when the dimension of the regression parameters may grow to infinity with the sample size. It is shown that the rate critically depends on the choices of the penalty parameter and the initial estimator, among other factors, and that confidence intervals (CIs) based on the oracle limit law often have poor coverage accuracy. As an alternative, we consider the residual bootstrap method for the ALASSO estimators that has been recently shown to be consistent; cf. Chatterjee and Lahiri [J. Amer. Statist. Assoc. 106 (2011a) 608–625]. We show that the bootstrap applied to a suitable studentized version of the ALASSO estimator achieves second-order correctness, even when the dimension of the regression parameters is unbounded. Results from a moderately large simulation study show marked improvement in coverage accuracy for the bootstrap CIs over the oracle based CIs.

#### Article information

Source
Ann. Statist., Volume 41, Number 3 (2013), 1232-1259.

Dates
First available in Project Euclid: 13 June 2013

https://projecteuclid.org/euclid.aos/1371150899

Digital Object Identifier
doi:10.1214/13-AOS1106

Mathematical Reviews number (MathSciNet)
MR3113809

Zentralblatt MATH identifier
1293.62153

#### Citation

Chatterjee, A.; Lahiri, S. N. Rates of convergence of the Adaptive LASSO estimators to the Oracle distribution and higher order refinements by the bootstrap. Ann. Statist. 41 (2013), no. 3, 1232--1259. doi:10.1214/13-AOS1106. https://projecteuclid.org/euclid.aos/1371150899

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#### Supplemental materials

• Supplementary material: Supplement to “Rates of convergence of the Adaptive LASSO estimators to the Oracle distribution and higher order refinements by the bootstrap”. Detailed proofs of all results.