Estimation of Linear Functionals in High-Dimensional Linear Models: From Sparsity to Nonsparsity

Abstract

ABSTRACT High-dimensional linear models are commonly used in practice. In many applications, one is interested in linear transformations β ⊤ x of regression coefficients β ∈ R p , where x is a specific point and is not required to be identically distributed as the training data. One common approach is the plug-in technique which first estimates β , then plugs the estimator in the linear transformation for prediction. Despite its popularity, estimation of β can be difficult for high-dimensional problems. Commonly used assumptions in the literature include that the signal of coefficients β is sparse and predictors are weakly correlated. These assumptions, however, may not be easily verified, and can be violated in practice. When β is non-sparse or predictors are strongly correlated, estimation of β can be very difficult. In this article, we propose a novel pointwise estimator for linear transformations of β . This new estimator greatly relaxes the common assumptions for high-dimensional problems, and is adaptive to the degree of sparsity of β and strength of correlations among the predictors. In particular, β can be sparse or nonsparse and predictors can be strongly or weakly correlated. The proposed method is simple for implementation. Numerical and theoretical results demonstrate the competitive advantages of the proposed method for a wide range of problems. Supplementary materials for this article are available online.