Group bound: confidence intervals for groups of variables in sparse high dimensional regression without assumptions on the design

By Nicolai Meinshausen. 2013

53 citations recorded by the source corpus.

It is in general challenging to provide confidence intervals for individual variables in high dimensional regression without making strict or unverifiable assumptions on the design matrix. We show here that a ‘group bound’ confidence interval can be derived without making any assumptions on the design matrix. The lower bound for the regression coefficient of individual variables can be derived via linear programming. The idea also generalizes naturally to groups of variables, where we can derive a one‐sided confidence interval for the joint effect of a group. Although the confidence intervals of individual variables are by the nature of the problem often very wide, it is shown to be possible to detect the contribution of groups of highly correlated predictor variables even when no variable individually shows a significant effect. The assumptions that are necessary to detect the effect of groups of variables are shown to be weaker than the weakest known assumptions that are necessary to detect the effect of individual variables.

Subfields: Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/fcbc7d5fec519559a5eeddf9511aad36f7b7bb5f

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In Management Science and Operations Research

In Statistics and Probability

Citations in the corpus

Goodness‐of‐fit tests for high dimensional linear models (2015, methodology): “…2009; Meinshausen et al., 2009; Bühlmann, 2013; Zhang and Zhang, 2014; Lockhart et al., 2014; van de Geer et al., 2014; Javanmard and Montanari, 2014; Meinshausen, 2015; Ning and Liu, 2017; Voorman et al., 2014; Zhou, 2015)—see Dezeure et al. (2015) for an overview of some of these methods.”
High-dimensional inference in misspecified linear models (2015, methodology): “Most of the recent contributions for this task discuss some methods and approaches for high-dimensional linear models (Bühlmann, 2013; Zhang and Zhang, 2014; van de Geer et al., 2014; Javanmard and Montanari, 2014; Meinshausen, 2015; Foygel Barber and Candès, 2014), but generalized linear models (Me”
A modern maximum-likelihood theory for high-dimensional logistic regression (2018, background): “In recent times, multiple approaches aimed to provide stronger simultaneous inference procedures for the ultra high-dimensional sparse linear model, see [36, 70, 73, 111].”
Simultaneous Inference for High-Dimensional Linear Models (2016, methodology): “…is obtained by inverting the KKT condition, where Θ is a suitable approximation for the inverse of the Gram matrix Σ := X T X /n. b b In what follows, we consider the approximate inverse b Θ given by Lasso for the nodewise regression on the design matrix X ; see Meinshausen and B¨uhlmann (2006).”
Bootstrapping and sample splitting for high-dimensional, assumption-lean inference (2016, background): “Some partial exceptions exist: Bühlmann and van de Geer (2015) relaxes the requirement of a correctly-specified linear model, while Meinshausen (2015) removes the design assumptions.”