## Adaptive Lasso for sparse highdimensional regression (2006)

Venue: | University of Iowa |

Citations: | 36 - 4 self |

### BibTeX

@TECHREPORT{Huang06adaptivelasso,

author = {Jian Huang and Shuangge Ma and Cun-hui Zhang},

title = {Adaptive Lasso for sparse highdimensional regression},

institution = {University of Iowa},

year = {2006}

}

### Years of Citing Articles

### OpenURL

### Abstract

Summary. We study the asymptotic properties of adaptive LASSO estimators in sparse, high-dimensional, linear regression models when the number of covariates may increase with the sample size. We consider variable selection using the adaptive LASSO, where the L1 norms in the penalty are re-weighted by data-dependent weights. We show that, if a reasonable initial estimator is available, then under appropriate conditions, adaptive LASSO correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic dis-tribution that they would have if the zero coefficients were known in advance. Thus, the adaptive LASSO has an oracle property in the sense of Fan and Li (2001) and Fan and Peng (2004). In addition, under a partial orthogonality condition in which the covariates with zero coefficients are weakly correlated with the covariates with nonzero coefficients, univariate regression can be used to obtain the initial estimator. With this initial estimator, adaptive LASSO has the oracle property even when the number of covariates is greater than the sample size. Key Words and phrases. Penalized regression, high-dimensional data, variable selection, asymptotic normality, oracle property, zero-consistency. Short title. Sparse high-dimensional regression