Uniform asymptotic inference and the bootstrap after model selection. (English) Zbl 1392.62210

Summary: Recently, R. J. Tibshirani et al. [“Exact post-selection inference for sequential regression procedures”, J. Am. Stat. Assoc. 111, No. 514, 600–620 (2016; doi:10.1080/01621459.2015.1108848)] 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 [loc. cit.] 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 [loc. cit.] does not enjoy uniform validity in a high-dimensional setting, when the dimension \(d\) is allowed grow.


62J05 Linear regression; mixed models
62F05 Asymptotic properties of parametric tests
62F35 Robustness and adaptive procedures (parametric inference)
62J07 Ridge regression; shrinkage estimators (Lasso)


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