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ADAPTIVE ROBUST VARIABLE SELECTION
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DOI：10.1214/13-AOS1191.png)
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Abstract
En 中文
Heavy-tailed high-dimensional data are commonly encountered in various scientific fields and pose great challenges to modern statistical analysis. A natural procedure to address this problem is to use penalized quantile regression with weighted L-1-penalty, called weighted robust Lasso (WR-Lasso), in which weights are introduced to ameliorate the bias problem induced by the L-1-penalty. In the ultra-high dimensional setting, where the dimensionality can grow exponentially with the sample size, we investigate the model selection oracle property and establish the asymptotic normality of the WR-Lasso. We show that only mild conditions on the model error distribution are needed. Our theoretical results also reveal that adaptive choice of the weight vector is essential for the WR-Lasso to enjoy these nice asymptotic properties. To make the WR-Lasso practically feasible, we propose a two-step procedure, called adaptive robust Lasso (AR-Lasso), in which the weight vector in the second step is constructed based on the L-1-penalized quantile regression estimate from the first step. This two-step procedure is justified theoretically to possess the oracle property and the asymptotic normality. Numerical studies demonstrate the favorable finite-sample performance of the AR-Lasso.
Keywords:
Adaptive weighted L-1
high dimensions
oracle properties
robust regularization
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