On asymptotically optimal confidence regions and tests for high-dimensional models

By Sara A. van de Geer, Peter Buhlmann, Yaacov Ritov, Ruben Dezeure. 2013

1,244 citations recorded by the source corpus.

We propose a general method for constructing confidence intervals and statistical tests for single or low-dimensional components of a large parameter vector in a high-dimensional model. It can be easily adjusted for multiplicity taking dependence among tests into account. For linear models, our method is essentially the same as in Zhang and Zhang [J. R. Stat. Soc. Ser. B Stat. Methodol. 76 (2014) 217-242]: we analyze its asymptotic properties and establish its asymptotic optimality in terms of semiparametric efficiency. Our method naturally extends to generalized linear models with convex loss functions. We develop the corresponding theory which includes a careful analysis for Gaussian, sub-Gaussian and bounded correlated designs.

Subfields: Statistics and Probability, Statistical Learning Theory

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Citations in the corpus

Double/De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers (2018, background): “…estimation and inference [Zhang and Zhang (2014); Belloni et al. (2011, 2014, 2015); Javanmard and Montanari (2014a,b, 2018); Van de Geer et al. (2014); Ning and Liu (2017); Chernozhukov et al. (2015); Neykov et al. (2018); Ren et al. (2015); Jankova and Van De Geer (2015, 2016,…”
Distributed Multi-Task Learning (2015, methodology): “Here, we adopt the approach proposed in [Javanmard and Montanari, 2014], as it leads to weakest assumption on the model in (16): each machine uses a matrix M t = ( ˆ m tj ) =1 with rows: where e j is the vector with j -th component equal to 1 and 0 otherwise and ˆΣ t = n − 1 X Tt X t .”
Confidence intervals for high-dimensional linear regression: Minimax rates and adaptivity (2015, methodology): “11) is similar to the ones based on the de-biased estimators introduced in [16, 25, 27].”
Group bound: confidence intervals for groups of variables in sparse high dimensional regression without assumptions on the design (2013, methodology): “The Ridge effect estimator proposed in Bühlmann [2012] (“R”) and the Lasso-based test of individual variables of van de Geer et al. [2013] (“L”).”
Double machine learning for treatment and causal parameters (2016, methodology): “This second approach represents a generalization of the approach of [31, 52, 57] considered for the case of Lasso-type methods without cross-fitting.”
A Unified Theory of Confidence Regions and Testing for High-Dimensional Estimating Equations (2015, methodology): “The conditions in Corollary 1 agree with the existing conditions in Zhang and Zhang (2014); van de Geer et al. (2014).”
A Projection-based Conditional Dependence Measure with Applications to High-dimensional Undirected Graphical Models. (2015, methodology): “It is worth mentioning that an alternative approach to reduce the estimation bias is the de-biased lasso method (Zhang and Zhang, 2014; Van de Geer et al., 2014).”
Convergence rates of least squares regression estimators with heavy-tailed errors (2017, background): “Previous work guaranteeing rate-optimality of the Lasso estimator typically assumes that both X and ξ are sub-Gaussian; see [14, 51, 62].”
Targeted Undersmoothing. (2017, methodology): “This approach is adopted in the high-dimensional linear model setting in [13], [49] and [56]”
Handbook of Big Data (2016, background): “Analyses of big data are not immune to the problems of small data.”
High-dimensional econometrics and regularized GMM (2018, methodology): “[109] suggest using Bonferroni-Holm in conjunction with the de-sparsified lasso using a limiting distribution derived under homoscedastic Gaussian errors.”