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Parallelism, uniqueness, and large-sample asymptotics for the Dantzig selector.

Lee Dicker1, Xihong Lin

  • 1Department of Statistics and Biostatistics, Rutgers University, 501 Hill Center, 110 Frelinghuysen Road, Piscataway, NJ 08854Department of Biostatistics, Harvard School of Public Health, 655 Huntington Avenue, Boston, MA 02115.

The Canadian Journal of Statistics = Revue Canadienne De Statistique
|April 17, 2013
PubMed
Summary
This summary is machine-generated.

We identified a geometric condition ensuring unique Dantzig selector estimators in linear regression. This condition, related to predictor parallelism, holds with probability 1 for continuous distributions, aiding variable selection and estimation.

Keywords:
LassoRegularized regressionVariable selection and estimation

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Area of Science:

  • Statistics
  • Machine Learning
  • Optimization

Background:

  • The Dantzig selector is a key ℓ1-regularization technique for variable selection and estimation in linear regression.
  • Ensuring the uniqueness of estimators is crucial for reliable statistical inference.

Purpose of the Study:

  • To establish a weak geometric condition that guarantees the uniqueness of Dantzig selector estimators.
  • To investigate the necessity of this condition and extend it to Lasso estimators.
  • To derive large sample asymptotic properties of the Dantzig selector.

Main Methods:

  • Introducing a geometric condition based on predictor parallelism.
  • Analyzing the probability of this condition holding for predictors from continuous distributions.
  • Utilizing continuity arguments and uniqueness results to establish asymptotic properties.

Main Results:

  • A novel, weak geometric condition ensuring the uniqueness of Dantzig selector estimators is presented.
  • This condition is shown to hold with probability 1 for predictors drawn from continuous distributions.
  • Large sample asymptotics, including almost sure convergence and asymptotic distribution, are derived, revealing a generally non-normal limiting distribution.

Conclusions:

  • The identified geometric condition is fundamental for the uniqueness of Dantzig selector estimators.
  • The study provides a theoretical foundation for the reliability of the Dantzig selector in variable selection and estimation.
  • Asymptotic results are established, offering insights into the behavior of the Dantzig selector for large sample sizes.