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In many applications, researchers often know a certain set of predictors is related to the response from some previous investigations and experiences. Based on the conditional information, we propose a conditional screening feature procedure via ranking conditional marginal empirical likelihood ratios. Due to the use of centralized variable, the proposed screening approach works well when there exist either or both hidden important variables and unimportant variables that are highly marginal correlated with the response. Moreover, the new method is demonstrated effective in scenarios with less restrictive distributional assumptions by inheriting the advantage of empirical likelihood approach and is computationally simple because it only needs to evaluate the conditional marginal empirical likelihood ratio at one point, without parameter estimation and iterative algorithm. The theoretical results reveal that the proposed procedure has sure screening properties. The merits of the procedure are illustrated by extensive numerical examples.

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The least squares fit in a linear regression is always unique even when the design matrix has rank deficiency. In this paper, we extend this classic result to linearly constrained generalized lasso. It is shown that under a mild condition, the fit can be represented as a projection onto a polytope and, hence, is unique no matter whether design matrix X has full column rank or not. Furthermore, a formula for the degrees of freedom is derived to characterize the effective number of parameters. It directly yields an unbiased estimate of degrees of freedom, which can be incorporated in an information criterion for model selection.

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