The Annals of Statistics

Perturbation bootstrap in adaptive Lasso

Debraj Das, Karl Gregory, and S. N. Lahiri

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The Adaptive Lasso (Alasso) was proposed by Zou [J. Amer. Statist. Assoc. 101 (2006) 1418–1429] as a modification of the Lasso for the purpose of simultaneous variable selection and estimation of the parameters in a linear regression model. Zou [J. Amer. Statist. Assoc. 101 (2006) 1418–1429] established that the Alasso estimator is variable-selection consistent as well as asymptotically Normal in the indices corresponding to the nonzero regression coefficients in certain fixed-dimensional settings. In an influential paper, Minnier, Tian and Cai [J. Amer. Statist. Assoc. 106 (2011) 1371–1382] proposed a perturbation bootstrap method and established its distributional consistency for the Alasso estimator in the fixed-dimensional setting. In this paper, however, we show that this (naive) perturbation bootstrap fails to achieve second-order correctness in approximating the distribution of the Alasso estimator. We propose a modification to the perturbation bootstrap objective function and show that a suitably Studentized version of our modified perturbation bootstrap Alasso estimator achieves second-order correctness even when the dimension of the model is allowed to grow to infinity with the sample size. As a consequence, inferences based on the modified perturbation bootstrap will be more accurate than the inferences based on the oracle Normal approximation. We give simulation studies demonstrating good finite-sample properties of our modified perturbation bootstrap method as well as an illustration of our method on a real data set.

Article information

Ann. Statist., Volume 47, Number 4 (2019), 2080-2116.

Received: November 2016
Revised: February 2018
First available in Project Euclid: 21 May 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J07: Ridge regression; shrinkage estimators
Secondary: 62G09: Resampling methods 62E20: Asymptotic distribution theory

Alasso naive perturbation bootstrap modified perturbation bootstrap second-order correctness oracle


Das, Debraj; Gregory, Karl; Lahiri, S. N. Perturbation bootstrap in adaptive Lasso. Ann. Statist. 47 (2019), no. 4, 2080--2116. doi:10.1214/18-AOS1741.

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  • Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion. Ann. Statist. 6 434–451.
  • Bhattacharya, R. N. and Ranga Rao, R. (1986). Normal Approximation and Asymptotic Expansions. John Wiley & Sons, New York.
  • Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge Univ. Press, Cambridge.
  • Bühlmann, P., Kalisch, M. and Meier, L. (2014). High-dimensional statistics with a view towards applications in biology. Annu. Rev. Statist. Appl. 1 255–278.
  • Camponovo, L. (2015). On the validity of the pairs bootstrap for Lasso estimators. Biometrika 102 981–987.
  • Caner, M. and Fan, Q. (2010). The adaptive Lasso method for instrumental variable selection. Working paper, North Carolina State Univ., Raleigh, NC.
  • Chatterjee, S. and Bose, A. (2005). Generalized bootstrap for estimating equations. Ann. Statist. 33 414–436.
  • Chatterjee, A. and Lahiri, S. N. (2010). Asymptotic properties of the residual bootstrap for Lasso estimators. Proc. Amer. Math. Soc. 138 4497–4509.
  • Chatterjee, A. and Lahiri, S. N. (2011). Bootstrapping Lasso estimators. J. Amer. Statist. Assoc. 106 608–625.
  • Chatterjee, A. and Lahiri, S. N. (2013). Rates of convergence of the adaptive LASSO estimators to the oracle distribution and higher order refinements by the bootstrap. Ann. Statist. 41 1232–1259.
  • Das, D., Gregory, K. and Lahiri, S. N. (2019). Supplement to “Perturbation bootstrap in adaptive Lasso.” DOI:10.1214/18-AOS1741SUPP.
  • Das, D. and Lahiri, S. N. (2019). Second order correctness of perturbation bootstrap M-estimator of multiple linear regression parameter. Bernoulli 25 654–682.
  • Dezeure, R., Bühlmann, P. and Zhang, C.-H. (2017). High-dimensional simultaneous inference with the bootstrap. TEST 26 685–719. Available at arXiv:1606.03940.
  • Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc. 96 1348–1360.
  • Friedman, J., Hastie, T. and Tibshirani, R. (2010). Regularization paths for generalized linear models via coordinate descent. J. Stat. Softw. 33 1–22.
  • Fuk, D. H. and Nagaev, S. V. (1971). Probabilistic inequalities for sums of independent random variables. Teor. Veroyatn. Primen. 16 660–675.
  • Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
  • Hall, P., Lee, Y. K., Park, B. U. and Paul, D. (2009). Tie-respecting bootstrap methods for estimating distributions of sets and functions of eigenvalues. Bernoulli 15 380–401.
  • Jin, Z., Ying, Z. and Wei, L. J. (2001). A simple resampling method by perturbing the minimand. Biometrika 88 381–390.
  • Knight, K. and Fu, W. (2000). Asymptotics for Lasso-type estimators. Ann. Statist. 28 1356–1378.
  • Minnier, J., Tian, L. and Cai, T. (2011). A perturbation method for inference on regularized regression estimates. J. Amer. Statist. Assoc. 106 1371–1382.
  • Tibshirani, R. (1996). Regression shrinkage and selection via the Lasso. J. Roy. Statist. Soc. Ser. B 58 267–288.
  • Turnbull, H. W. (1930). A matrix form of Taylor’s theorem. Proc. Edinb. Math. Soc. (2) 33 33–54.
  • van de Geer, S., Bühlmann, P., Ritov, Y. and Dezeure, R. (2014). On asymptotically optimal confidence regions and tests for high-dimensional models. Ann. Statist. 42 1166–1202.
  • Wang, X. and Song, L. (2011). Adaptive Lasso variable selection for the accelerated failure models. Comm. Statist. Theory Methods 40 4372–4386.
  • Zhang, C.-H. and Zhang, S. S. (2014). Confidence intervals for low dimensional parameters in high dimensional linear models. J. Roy. Statist. Soc. Ser. B 76 217–242.
  • Zhou, Q. M., Song, P. X.-K. and Thompson, M. E. (2012). Information ratio test for model misspecification in quasi-likelihood inference. J. Amer. Statist. Assoc. 107 205–213.
  • Zou, H. (2006). The adaptive Lasso and its oracle properties. J. Amer. Statist. Assoc. 101 1418–1429.

Supplemental materials

  • Supplement to “Perturbation bootstrap in adaptive Lasso”. Details of the proofs and additional simulation studies are provided.