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The l1-penalized method, or the Lasso, has emerged as an important tool for the analysis of large data sets. Many important results have been obtained for the Lasso in linear regression which have led to a deeper understanding of high-dimensional statistical problems. In this article, we consider a class of weighted l1-penalized estimators for convex loss functions of a general form, including the generalized linear models. We study the estimation, prediction, selection and sparsity properties of the weighted l1-penalized estimator in sparse, high-dimensional settings where the number of predictors p can be much larger than the sample size n. Adaptive Lasso is considered as a special case. A multistage method is developed to approximate concave regularized estimation by applying an adaptive Lasso recursively. We provide prediction and estimation oracle inequalities for single- and multi-stage estimators, a general selection consistency theorem, and an upper bound for the dimension of the Lasso estimator. Important models including the linear regression, logistic regression and log-linear models are used throughout to illustrate the applications of the general results. [ABSTRACT FROM AUTHOR]

We consider the estimation of regression coefficients in a high-dimensional linear model. For regression coefficients in l r balls, we provide lower bounds for the minimax ℓq risk and minimax quantiles of the ℓq loss for all design matrices. Under an ℓ0 sparsity condition on a target coefficient vector, we sharpen and unify existing oracle inequalities for the Lasso and Dantzig selector. We derive oracle inequalities for target coefficient vectors with many small elements and smaller threshold levels than the universal threshold. These oracle inequalities provide sufficient conditions on the design matrix for the rate minimaxity of the Lasso and Dantzig selector for the l q risk and loss in ℓr balls, 0 ≤ r ≤ l ≤ q ≤ ∞. By allowing q = ∞, our risk bounds imply the variable selection consistency of threshold Lasso and Dantzig selectors. [ABSTRACT FROM AUTHOR]

Frequentist confidence intervals for population ranks and their statistical justifications are not well established, even though there is a great need for such procedures in practice. How do we assign confidence bounds for the ranks of health care facilities, schools, and financial institutions based on data that do not clearly separate the performance of different entities apart? The commonly used bootstrap-based frequentist confidence intervals and Bayesian intervals for population ranks may not achieve the intended coverage probability in the frequentist sense, especially in the presence of unknown ties or "near ties" among the populations to be ranked. Given random samples from k populations, we propose confidence bounds for population ranking parameters and develop rigorous frequentist theory and nonstandard bootstrap inference for population ranks, which allow ties and near ties. In the process, a notion of modified population rank is introduced that appears quite suitable for dealing with the population ranking problem. The proposed methodology and theoretical results are illustrated through simulations and a real dataset from a health research study involving 79 Veterans Health Administration (VHA) facilities. The results are extended to general risk adjustment models. [ABSTRACT FROM AUTHOR]

Presents the authors' comments on the paper "Semilinear High-Dimensional Model for Normalization of Microarray Data: A Theoretical Analysis and Partial Consistency," by Jianqing Fan, Heng Peng and Tao Huang, published in the September 2005 issue of the "Journal of the American Statistical Association." Appreciation of the authors for their significant contribution to the semiparametric estimation theory; Suggestions for a two-way semilinear model for normalization and analysis of cDNA microarray data; Comments on multiway semilinear models; Description of models for location normalization; Opinion on the relationship between the relationship between the multiway semilinear models and the semilinear in-slide model.

Confirms that the u,v method of estimation can provide asymptotically efficient estimators in an important special case. Concept of the method; Computational procedures; Sample applications.