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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.

##### MSC:
 62J05 Linear regression; mixed models 62F05 Asymptotic properties of parametric tests 62F35 Robustness and adaptive procedures (parametric inference) 62J07 Ridge regression; shrinkage estimators (Lasso)
covTest
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##### References:
 [1] Bachoc, F., Leeb, H. and Potscher, B. (2014). Valid confidence intervals for post-model-selection predictors. Available at arXiv:1412.4605. · Zbl 1419.62164 [2] Berk, R., Brown, L., Buja, A., Zhang, K. and Zhao, L. (2013). Valid post-selection inference. Ann. Statist.41 802-837. · Zbl 1267.62080 [3] Choi, Y., Taylor, J. and Tibshirani, R. (2014). Selecting the number of principal components: Estimation of the true rank of a noisy matrix. Available at arXiv:1410.8260. · Zbl 1394.62073 [4] Donoho, D. L. (1988). One-sided inference about functionals of a density. Ann. Statist.16 1390-1420. · Zbl 0665.62040 [5] Fithian, W., Sun, D. and Taylor, J. (2014). Optimal inference after model selection. Available at arXiv:1410.2597. [6] Hyun, S., G’Sell, M. and Tibshirani, R. J. (2016). Exact post-selection inference for changepoint detection and other generalized lasso problems. Available at arXiv:1606.03552. [7] Kasy, M. (2015). Uniformity and the delta method. Unpublished manuscript. · Zbl 1420.62491 [8] Lee, J. and Taylor, J. (2014). Exact post model selection inference for marginal screening. Adv. Neural Inf. Process. Syst.27 136-144. [9] Lee, J. D., Sun, D. L., Sun, Y. and Taylor, J. E. (2016). Exact post-selection inference, with application to the lasso. Ann. Statist.44 907-927. · Zbl 1341.62061 [10] Leeb, H. and Pötscher, B. M. (2003). The finite-sample distribution of post-model-selection estimators and uniform versus nonuniform approximations. Econometric Theory 19 100-142. · Zbl 1032.62011 [11] Leeb, H. and Pötscher, B. M. (2006). Can one estimate the conditional distribution of post-model-selection estimators? Ann. Statist.34 2554-2591. · Zbl 1106.62029 [12] Leeb, H. and Pötscher, B. M. (2008). Can one estimate the unconditional distribution of post-model-selection estimators? Econometric Theory 24 338-376. · Zbl 1284.62152 [13] Lockhart, R., Taylor, J., Tibshirani, R. J. and Tibshirani, R. (2014). A significance test for the lasso. Ann. Statist.42 413-468. · Zbl 1305.62254 [14] Loftus, J. and Taylor, J. (2014). A significance test for forward stepwise model selection. Available at arXiv:1405.3920. [15] O’Hagan, A. and Leonard, T. (1976). Bayes estimation subject to uncertainty about parameter constraints. Biometrika 63 201-203. · Zbl 0326.62025 [16] Reid, S., Taylor, J. and Tibshirani, R. (2017). Post-selection point and interval estimation of signal sizes in Gaussian samples. Canad. J. Statist.45 128-148. [17] Taylor, J. E., Loftus, J. R. and Tibshirani, R. J. (2016). Inference in adaptive regression via the Kac-Rice formula. Ann. Statist.44 743-770. · Zbl 1337.62304 [18] Tian, X. and Taylor, J. (2017). Asymptotics of selective inference. Scand. J. Stat.44 480-499. · Zbl 1422.62252 [19] Tibshirani, R. J., Taylor, J., Lockhart, R. and Tibshirani, R. (2016). Exact post-selection inference for sequential regression procedures. J. Amer. Statist. Assoc.111 600-620. [20] Tibshirani, R. J., Rinaldo, A., Tibshirani, R. and Wasserman, L. (2018). Supplement to “Uniform asymptotic inference and the bootstrap after model selection.” DOI:10.1214/17-AOS1584SUPP. · Zbl 1392.62210 [21] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge Univ. Press, Cambridge. [22] Wasserman, L. (2014). Discussion: “A significance test for the lasso” [MR3210970]. Ann. Statist.42 501-508. · Zbl 1305.62257
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