An R-package globaltest is available from http://www.bioconductor.org

We consider high-dimensional generalized linear models with Lipschitz loss functions, and prove a nonasymptotic oracle inequality for the empirical risk minimizer with Lasso penalty. The penalty is based on the coefficients in the linear predictor, after normalization with the empirical norm. The examples include logistic regression, density estimation and classification with hinge loss. Least squares regression is also discussed.Comment: Published in at http://dx.doi.org/10.1214/009053607000000929 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Oracle inequalities and variable selection properties for the Lasso in linear models have been established under a variety of different assumptions on the design matrix. We show in this paper how the different conditions and concepts relate to each other. The restricted eigenvalue condition (Bickel et al., 2009) or the slightly weaker compatibility condition (van de Geer, 2007) are sufficient for oracle results. We argue that both these conditions allow for a fairly general class of design matrices. Hence, optimality of the Lasso for prediction and estimation holds for more general situations than what it appears from coherence (Bunea et al, 2007b,c) or restricted isometry (Candes and Tao, 2005) assumptions.Comment: 33 pages, 1 figur

As the dimensionality of the alternative hypothesis increases, the power of classical tests tends to diminish quite rapidly. This is especially true for high dimensional data in which there are more parameters than observations. We discuss a score test on a hyperparameter in an empirical Bayesian model as an alternative to classical tests. It gives a general test statistic which can be used to test a point null hypothesis against a high dimensional alternative, even when the number of parameters exceeds the number of samples. This test will be shown to have optimal power on average in a neighbourhood of the null hypothesis, which makes it a proper generalization of the locally most powerful test to multiple dimensions. To illustrate this new locally most powerful test we investigate the case of testing the global null hypothesis in a linear regression model in more detail. The score test is shown to have significantly more power than the "F"-test whenever under the alternative the large variance principal components of the design matrix explain substantially more of the variance of the outcome than do the small variance principal components. The score test is also useful for detecting sparse alternatives in truly high dimensional data, where its power is comparable with the test based on the maximum absolute "t"-statistic. Copyright 2006 Royal Statistical Society.

This paper is a selective review of the regularization methods scattered in statistics literature. We introduce a general conceptual approach to regularization and fit most existing methods into it. We have tried to focus on the importance of regularization when dealing with today's high-dimensional objects: data and models. A wide range of examples are discussed, including nonparametric regression, boosting, covariance matrix estimation, principal component estimation, subsampling.