Linear Hypothesis Testing in Dense High-Dimensional Linear Models

By Yinchu Zhu, Jelena Bradic. 2016

88 citations recorded by the source corpus.

ABSTRACT We propose a methodology for testing linear hypothesis in high-dimensional linear models. The proposed test does not impose any restriction on the size of the model, that is, model sparsity or the loading vector representing the hypothesis. Providing asymptotically valid methods for testing general linear functions of the regression parameters in high-dimensions is extremely challenging—especially without making restrictive or unverifiable assumptions on the number of nonzero elements. We propose to test the moment conditions related to the newly designed restructured regression, where the inputs are transformed and augmented features. These new features incorporate the structure of the null hypothesis directly. The test statistics are constructed in such a way that lack of sparsity in the original model parameter does not present a problem for the theoretical justification of our procedures. We establish asymptotically exact control on Type I error without imposing any sparsity assumptions on model parameter or the vector representing the linear hypothesis. Our method is also shown to achieve certain optimality in detecting deviations from the null hypothesis. We demonstrate the favorable finite-sample performance of the proposed methods, via a number of numerical and a real data example. Supplementary materials for this article are available online.

Subfields: Artificial Intelligence, Computational Mechanics, Statistics and Probability

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Citations in the corpus

Double/De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers (2018, background): “…(2017); Chernozhukov et al. (2015); Neykov et al. (2018); Ren et al. (2015); Jankova and Van De Geer (2015, 2016, 2018); Bradic and Kolar (2017); Zhu and Bradic (2017, 2018)], which focuses on coefficients in high-dimensional linear and generalized linear regression models, without considering…”
Approximate residual balancing: debiased inference of average treatment effects in high dimensions (2016, methodology): “…lasso regression adjustments allow for efficient average treatment effect estimation in randomized trials even when k n log.p/, whereas the method of Zhu and Bradic (2016) for estimating dense contrasts ξβc does not rely on sparsity of βc and instead places assumptions on the joint distribution of…”
High-dimensional econometrics and regularized GMM (2018, background): “Finally, [34], [127], [128], and [72] consider different approaches which allow testing hypotheses about functionals that may involve the entire high-dimensional parameter vector within different high-dimensional contexts.”
Learning L2 Continuous Regression Functionals via Regularized Riesz Representers (2018, background): “egression to be sparse or linear. We do this via regularized RR learning that can also be interpreted as learning of balancing weights or inverse propensity scores, as further discussed in Section 4. Zhu and Bradic (2017) showed that it is possible to attain root-n consistency for the coeﬃcients of ”
A modern maximum-likelihood theory for high-dimensional logistic regression (2018, background): “Imposing some additional conditions on the design matrix, [115] was able to remove sparsity requirements entirely.”
Targeted Undersmoothing. (2017, methodology): “Both [58] and [59] construct hypothesis tests for objects similar to those considered in our paper via `1-projections of coefficient estimates to the set of coefficients consistent with the null.”
Robust Estimation of Causal Effects via High-Dimensional Covariate Balancing Propensity Score. (2018, background): “…high-dimensional inference on regression models; see Zhang & Zhang (2014), Javanmard &70 Montanari (2014), Van de Geer et al. (2014), Belloni et al. (2016), Ning & Liu (2017); Ning et al. (2017), Cai & Guo (2017), Neykov et al. (2018) Zhu & Bradic (2018), and Dukes et al. (2018), among many others.”
Double/de-biased machine learning using regularized Riesz representers (2018, methodology): “…conditions used in Javanmard and Montanari (2015) in the case of inference on a single coefﬁcient in Gaussian sparse linear regression models, and those in Zhu and Bradic (2017a) in the case of point testing general linear hypotheses on regression coefﬁcients in linear Gaussian regression models.”
Breaking the curse of dimensionality in regression (2017, methodology): “Estimators (9) are inspired by the recent work of Zhu and Bradic (2016a) for example.”