Uniform asymptotic inference and the bootstrap after model selection

By Robert Tibshirani, Alessandro Rinaldo, Larry A. Wasserman. Annals of Statistics, 2015

113 citations recorded by the source corpus.

Recently, Tibshirani et al. (2016) proposed a method for making inferences about parameters defined by model selection, in a typical regression setting with normally distributed errors. Here, we study the large sample properties of this method, without assuming normality. We prove that the test statistic of Tibshirani et al. (2016) is asymptotically valid, as the number of samples n grows and the dimension d of the regression problem stays fixed. Our asymptotic result holds uniformly over a wide class of nonnormal error distributions. We also propose an efficient bootstrap version of this test that is provably (asymptotically) conservative, and in practice, often delivers shorter intervals than those from the original normality-based approach. Finally, we prove that the test statistic of Tibshirani et al. (2016) does not enjoy uniform validity in a high-dimensional setting, when the dimension d is allowed grow.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1829b812d1c0ee6b6d02aaafe44c1fefaad4c7f9

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Citations in the corpus

Uniformly valid confidence intervals post-model-selection (2016, methodology): “One needs to mention here that Tibshirani et al. (2015) also discuss a bootstrap version of their “TG” intervals.”
Valid confidence intervals for post-model-selection predictors (2014, background): “alternative to the PoSI approach, that also focuses on (n) M^ as the quantity of interest, is developed in Fithian et al. (2015), Lee et al. (2015), Lee and Taylor (2014), Tian and Taylor (2015), and Tibshirani et al. (2015). In these papers, condence sets for (n) M^ are considered that have a guara”
Splitting strategies for post-selection inference (2021, methodology): “Verifying that E′ is a union of convex sets can still be difficult problem in general, but it has been shown to be true for some widely used selection rules such as forward stepwise regression, the lasso, and least angle regression, all with fixed tuning parameters (Tibshirani et al., 2018).”
Automatic Database Management System Tuning Through Large-scale Machine Learning (2017, background): “There is also both practical and theoretical work backing its effectiveness as a consistent feature selection algorithm [56, 57, 64, 9].”
Exact post-selection inference for the generalized lasso path (2018, methodology): “Asymptotics for non-Gaussian error distributions are presented in Tian and Taylor [33], Tibshirani et al. [36].”
More powerful post-selection inference, with application to the Lasso (2018, background): “In the regime where p is fixed and n is growing, these assumptions can be dispensed with: • Tibshirani et al. (2014), Tian & Taylor (2017) show that the truncated normal pivot underlying the TZMs method holds asymptotically for a wide class of (non-normal) error distributions.”