High-dimensional inference in misspecified linear models

By Peter Buhlmann, Sara A. van de Geer. 2015

68 citations recorded by the source corpus.

We consider high-dimensional inference when the assumed linear model is misspecified. We describe some correct interpretations and corresponding sufficient assumptions for valid asymptotic inference of the model parameters, which still have a useful meaning when the model is misspecified. We largely focus on the de-sparsified Lasso procedure but we also indicate some implications for (multiple) sample splitting techniques. In view of available methods and software, our results contribute to robustness considerations with respect to model misspecification.

Subfields: Computational Mechanics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9078052af8ba7be7ceb673a3c053b63d16bd3959

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In Computational Mechanics

In Statistics and Probability

Citations in the corpus

Breaking the curse of dimensionality in regression (2017, background): “Zhang and Cheng (2016) (ZC) and Dezeure et al. (2016) define the simultaneous tests by designing appropriate bootstrap procedures.”
Models as Approximations I: Consequences Illustrated with Linear Regression (2014, background): “(9) It is important to distinguish between population and sample properties: The vectors δ, and η are not orthogonal to the regressor columns Xj in the sample.”
A unifying approach for doubly-robust $\ell_1$ regularized estimation of causal contrasts (2019, background): “This paper was motivated by a spate of recent papers (Farrell (2015), Avagyan and Vansteelandt (2017), Tan (2018), Chernozhukov et al.”
Linear Hypothesis Testing in Dense High-Dimensional Linear Models (2016, background): “…0 = o ( √ n/ log p ) ; see Zhang and Zhang (2014); Belloni et al. (2014); Van de Geer et al. (2014); Javanmard and Montanari (2014a); Ning and Liu (2014); Javanmard and Montanari (2015); Mitra and Zhang (2014); Bühlmann and van de Geer (2015); Belloni et al. (2015); Chernozhukov et al. (2015).”
Bootstrapping and sample splitting for high-dimensional, assumption-lean inference (2016, background): “While this is a larger literature than can be addressed completely here, it includes early work on model selection Hurvich and Tsai (1990) and model averaging interpretations Hjort and Claeskens (2003); the impossibility results of Leeb and Pötscher (2008), Buja et al. (2015) on random X and model m”
Models as Approximations - A Conspiracy of Random Regressors and Model Deviations Against Classical Inference in Regression (2015, background): “• The best population linear approximation to Y is l( ~ X) = β′ ~ X whose coefficients β = β(P ) are given by β(P ) := argminβ∈IRp+1 E[(Y −β ′ ~ X)(2)] = E[ ~ X ~ X ′]−1E[ ~ XY ] (5) = argminβ∈IRp+1 E[(μ( ~ X)−β′ ~ X)(2)] = E[ ~ X ~ X ′]−1E[ ~ Xμ( ~ X) ] (6) The right-most expressions in (5) and (6)”