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Consider a linear model Y = Xβ+σz, where X has n rows and p columns and z ~ N(0, In). We assume both p and n are large, including the case of p » n. The unknown signal vector β is assumed to be sparse in the sense that only a small fraction of its components is nonzero. The goal is to identify such nonzero coordinates (i.e., variable selection). We are primarily interested in the regime where signals are both rare and weak so that successful variable selection is challenging but is still possible. We assume the Gram matrix G = X'X is sparse in the sense that each row has relatively few large entries (diagonals of G are normalized to 1). The sparsity of G naturally induces the sparsity of the so-called Graph of Strong Dependence (GOSD). The key insight is that there is an interesting interplay between the signal sparsity and graph sparsity: in a broad context, the signals decompose into many small-size components of GOSD that are disconnected to each other. We propose Graphlet Screening for variable selection. This is a two-step Screen and Clean procedure, where in the first step, we screen subgraphs of GOSD with sequential χ²-tests, and in the second step, we clean with penalized MLE. The main methodological innovation is to use GOSD to guide both the screening and cleaning processes. For any variable selection procedure ..., we measure its performance by the Hamming distance between the sign vectors of ... and β, and assess the optimality by the minimax Hamming distance. Compared with more stringent criteria such as exact support recovery or oracle property, which demand strong signals, the Hamming distance criterion is more appropriate for weak signals since it naturally allows a small fraction of errors. We show that in a broad class of situations, Graphlet Screening achieves the optimal rate of convergence in terms of the Hamming distance. Unlike Graphlet Screening, well-known procedures such as the L0/L¹-penalization methods do not utilize local graphic structure for variable selection, so they generally do not achieve the optimal rate of convergence, even in very simple settings and even if the tuning parameters are ideally set. The the presented algorithm is implemented as R-CRAN package ScreenClean and in matlab (available at http: //www. stat. cmu. edu/~jiashun/Research/sof tware/GS-matlab/). [ABSTRACT FROM AUTHOR]

We propose a new method of learning a sparse nonnegative-definite target matrix. Our primary example of the target matrix is the inverse of a population covariance or correlation matrix. The algorithm first estimates each column of the target matrix by the scaled Lasso and then adjusts the matrix estimator to be symmetric. The penalty level of the scaled Lasso for each column is completely determined by data via convex minimization, without using cross-validation. We prove that this scaled Lasso method guarantees the fastest proven rate of convergence in the spectrum norm under conditions of weaker form than those in the existing analyses of other l1 regularized algorithms, and has faster guaranteed rate of convergence when the ratio of the l1 and spectrum norms of the target inverse matrix diverges to infinity. A simulation study demonstrates the computational feasibility and superb performance of the proposed method. Our analysis also provides new performance bounds for the Lasso and scaled Lasso to guarantee higher concentration of the error at a smaller threshold level than previous analyses, and to allow the use of the union bound in column-by-column applications of the scaled Lasso without an adjustment of the penalty level. In addition, the least squares estimation after the scaled Lasso selection is considered and proven to guarantee performance bounds similar to that of the scaled Lasso. [ABSTRACT FROM AUTHOR]

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]

This article derives characterizations and computational algorithms for continuous general gradient descent trajectories in high-dimensional parameter spaces for statistical model selection, prediction, and classification. Examples include proportional gradient shrinkage as an extension of LASSO and LARS, threshold gradient descent with right-continuous variable selectors, threshold ridge regression, and many more with proper combinations of variable selectors and functional forms of a kernel. In all these problems, general gradient descent trajectories are continuous piecewise analytic vector-valued curves as solutions to matrix differential equations. We show the mono- tonicity and convergence of the proposed algorithms in the loss or negative likelihood functions. We prove that approximations of continuous solutions via infinite series expansions are computationally more efficient and accurate compared with discretization methods. We demonstrate the applicability of our algorithms through numerical experiments with real and simulated datasets. [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.

Let Xn for n≥1 be independent random variables with EXn = 0 and EXn2=1⁠. Set Sk,n=∑i1<…cnk/2}. We study critical values ck,p for k≥2 and p>0, such that ETk,c,mp<∞ for cck,p and all sufficiently large m. In particular, c1,1=c2,1=1, c3,1=2 and c4,1=3 under certain moment conditions on X1, when Xn are identically distributed. We also investigate perturbed stopping rules of the form Th,m=inf{n≥m:h(S1,n/n1/2)<ξn or >ζn} for continuous functions h and random variables ξn∼a and ζn∼b with a

We provide a principled way for investigators to analyze randomized experiments when the number of covariates is large. Investigators often use linear multivariate regression to analyze randomized experiments instead of simply reporting the difference of means between treatment and control groups. Their aim is to reduce the variance of the estimated treatment effect by adjusting for covariates. If there are a large number of covariates relative to the number of observations, regression may perform poorly because of overfitting. In such cases, the least absolute shrinkage and selection operator (Lasso) may be helpful. We study the resulting Lasso-based treatment effect estimator under the Neyman-Rubin model of randomized experiments. We present theoretical conditions that guarantee that the estimator is more efficient than the simple difference-of-means estimator, and we provide a conservative estimator of the asymptotic variance, which can yield tighter confidence intervals than the difference-of means estimator. Simulation and data examples show that Lasso based adjustment can be advantageous even when the number of covariates is less than the number of observations. Specifically, a variant using Lasso for selection and ordinary least squares (OLS) for estimation performs particularly well, and it chooses a smoothing parameter based on combined performance of Lasso and OLS. [ABSTRACT FROM AUTHOR]

As the nonparametric generalization of the one-way analysis of variance model, the Kruskal-Wallis test applies when the goal is to test the difference between multiple samples and the underlying population distributions are nonnormal or unknown. Although the Kruskal-Wallis test has been widely used for data analysis, power and sample size methods for this test have been investigated to a much lesser extent. This article proposes new power and sample size calculation methods for the Kruskal-Wallis test based on the pilot study in either a completely nonparametric model or a semiparametric location model. No assumption is made on the shape of the underlying population distributions. Simulation results show that, in terms of sample size calculation for the Kruskal-Wallis test, the proposed methods are more reliable and preferable to some more traditional methods. A mouse peritoneal cavity study is used to demonstrate the application of the methods. [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]

Rankboost has been shown to be an effective algorithm for combining ranks. However, its ability to generalize well and not overfit is directly related to the choice of weak learner, in the sense that regularization of the rank function is due to the regularization properties of its weak learners. We present a regularization property called consistency in preference and confidence that mathematically translates into monotonic concavity, and describe a new weak ranking learner (MWGR) that generates ranking functions with this property. In experiments combining ranks from multiple face recognition algorithms and an experiment combining text information retrieval systems, rank functions using MWGR proved superior to binary weak learners. [ABSTRACT FROM AUTHOR]

A basic question in analyzing cDNA microarray data is normalization, the purpose of which is to remove systematic bias in the observed expression values by establishing a normalization curve across the whole dynamic range. A proper normalization procedure ensures that the normalized intensity ratios provide meaningful measures of relative expression levels. We propose a two-way semilinear model (TW-SLM) for normalization and analysis of microarray data. This method does not make the usual assumptions underlying some of the existing methods. For example, it does not assume that the percentage of differentially expressed genes is small or that there is symmetry in the expression levels of up-regulated and down-regulated genes, as required in the lowess normalization method. The TW-SLM also naturally incorporates uncertainty due to normalization into significance analysis of microarrays. We use a semiparametric approach based on polynomial splines in the TW-SLM to estimate the normalization curves and the normalized expression values. We study the theoretical properties of the proposed estimator in the TW-SLM, including the finite-sample distributional properties of the estimated gene effects and the rate of convergence of the estimated normalization curves when the number of genes under study is large. We also conduct simulation studies to evaluate the TW-SLM method and illustrate the proposed method using a published microarray dataset. [ABSTRACT FROM AUTHOR]

In many tumour growth inhibition studies, tumour sizes are recorded over a period of time, forming a growth curve for each experimental animal. A frequently asked question in these studies is whether or not different treatment agents/inhibitors result in different growth rates. To compare tumour growth rates under two treatments, either the t‐test or the Wilcoxon–Mann–Whitney test could be used under suitable assumptions, provided that a one‐dimensional variable, such as the area under the curve, is used to summarise a growth curve. Such tests may not be valid in the presence of informative heterogeneous censoring, in that a subject's continued participation in the experiment may depend on the tumour size and/or the level of toxicity of the treatment given. We propose test statistics that naturally correct the bias caused by the censorship and retain high efficiency. They are easy to construct and are nonparametric in nature, making no assumption on the distributions of the growth curves. The method is illustrated with an example from a tumour growth inhibition study on mice. Simulation results are also reported showing that the method performs well with moderate sample sizes. [ABSTRACT FROM PUBLISHER]

A simple fluorescent probe able to selectively and sensitively detect cysteine (Cys) with an excellent dose-dependent relationship between fluorescence intensity and concentration of Cys from 0 to 100 mMhas been designed and synthesised. [ABSTRACT FROM AUTHOR]

In multiple regression problems when covariates can be naturally grouped, it is important to carry out feature selection at the group and within-group individual variable levels simultaneously. The existing methods, including the lasso and group lasso, are designed for either variable selection or group selection, but not for both. We propose a group bridge approach that is capable of simultaneous selection at both the group and within-group individual variable levels. The proposed approach is a penalized regularization method that uses a specially designed group bridge penalty. It has the oracle group selection property, in that it can correctly select important groups with probability converging to one. In contrast, the group lasso and group least angle regression methods in general do not possess such an oracle property in group selection. Simulation studies indicate that the group bridge has superior performance in group and individual variable selection relative to several existing methods. [ABSTRACT FROM AUTHOR]