Asymptotic properties of bridge estimators in sparse high-dimensional regression models (0804.0693v1)

Abstract: We study the asymptotic properties of bridge estimators in sparse, high-dimensional, linear regression models when the number of covariates may increase to infinity with the sample size. We are particularly interested in the use of bridge estimators to distinguish between covariates whose coefficients are zero and covariates whose coefficients are nonzero. We show that under appropriate conditions, bridge estimators correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic distribution that they would have if the zero coefficients were known in advance. Thus, bridge estimators have an oracle property in the sense of Fan and Li [J. Amer. Statist. Assoc. 96 (2001) 1348--1360] and Fan and Peng [Ann. Statist. 32 (2004) 928--961]. In general, the oracle property holds only if the number of covariates is smaller than the sample size. However, under a partial orthogonality condition in which the covariates of the zero coefficients are uncorrelated or weakly correlated with the covariates of nonzero coefficients, we show that marginal bridge estimators can correctly distinguish between covariates with nonzero and zero coefficients with probability converging to one even when the number of covariates is greater than the sample size.

• The paper proves that bridge estimators possess the oracle property under certain conditions in high-dimensional sparse regression models, allowing correct identification of non-zero coefficients.
• A significant contribution is extending the analysis to the challenging scenario where the number of covariates exceeds the sample size, relevant for complex datasets.
• Simulation studies support the theoretical findings, demonstrating the practical effectiveness of bridge estimators in identifying relevant variables and outperforming other methods.

Asymptotic Properties of Bridge Estimators in Sparse High-Dimensional Regression Models

The paper by Huang, Horowitz, and Ma addresses the asymptotic behavior of bridge estimators in the context of high-dimensional sparse regression models. Within this framework, the number of covariates may increase with the sample size, presenting challenges for parameter estimation and variable selection. This work focuses on the ability of bridge estimators to effectively discern between covariates with zero and nonzero coefficients, even when the ratio of the covariates to the sample size is large.

Key Contributions and Results

1. Oracle Property: The authors prove that under certain conditions, bridge estimators exhibit the oracle property. This means they can correctly identify nonzero covariate coefficients with a probability approaching one, and the estimators themselves become asymptotically normal, with the same distribution they would have if the identity of the true nonzero coefficients were known in advance.
2. High-Dimensional Framework: A significant contribution of this paper is extending the investigation of bridge estimators to cases where the number of covariates exceeds the number of samples. Previous work primarily focused on finite-dimensional parameter settings, whereas Huang et al. address scenarios more typical of contemporary high-dimensional data analyses, such as those encountered in genetic data and other complex datasets.
3. Partial Orthogonality Condition: The paper introduces a partial orthogonality condition where covariates with zero coefficients are either uncorrelated or weakly correlated with those having nonzero coefficients. This condition is particularly relevant as it allows the algorithm to handle cases where the number of covariates exceeds the sample size, a common situation in genomics and other areas.
4. Simulation Studies: The numerical experiments demonstrate the practicality of the theoretical results. They show that the bridge estimator can reliably identify nonzero coefficients and performs well in terms of prediction and estimation errors compared to other methods such as LASSO and elastic-net.

Implications and Future Directions

Theoretical developments such as those presented in this paper are crucial for advancing statistical learning methods applied to high-dimensional data. Practically, demonstrating that bridge estimators maintain the oracle property under broad circumstances encourages their usage in real-world applications where model interpretability and efficient computation are necessary.

The paper opens avenues for refining the conditions under which the bridge estimator performs optimally, potentially leading to adaptive algorithms that adjust the penalty parameter dynamically based on data characteristics. Further investigations might explore the integration of adaptive techniques for selecting hyperparameters, such as the penalty factor, to further enhance performance beyond the current fixed setting.

Conclusion

Overall, the discussion of bridge estimators in sparse, high-dimensional settings marks a valuable step forward in regression modeling. By proving the oracle property in broader contexts and incorporating practical simulation verifications, Huang, Horowitz, and Ma significantly contribute to the toolkit available for statisticians grappling with complex, multidimensional datasets. Their results lay the groundwork for further exploration into more flexible regression frameworks adaptable to various high-dimensional data challenges.