An Adaptive Resampling Test for Detecting the Presence of Significant Predictors

REFERENCES

• Andrews, D. (2000), “Inconsistency of the Bootstrap When a Parameter is on the Boundary of the Parameter Space,” Econometrica, 68, 399–405.
   Web of Science ®Google Scholar
• Arias-Castro, E., Candès, E.J., and Plan, Y. (2011), “Global Testing Under Sparse Alternatives: ANOVA, Multiple Comparisons and the Higher Criticism,” The Annals of Statistics, 39, 2533–2556.
   Web of Science ®Google Scholar
• Belloni, A., Chernozhukov, V., and Hansen, C. (2014), “Inference on Treatment Effects After Selection Amongst High-Dimensional Controls,” Review of Economic Studies, 81, 608–650.
   Web of Science ®Google Scholar
• Benjamini, Y., and Hochberg, Y. (1995), “Controlling the False Discovery Rate: A practical and Powerful Approach to Multiple Testing,” Journal of the Royal Statistical Society, Series B, 57, 289–300.
   Web of Science ®Google Scholar
• Berk, R., Brown, L.D., Buja, A., Zhang, K., and Zhao, L. (2013), “Valid Post-Selection Inference,” The Annals of Statistics, 41, 802–837.
   Web of Science ®Google Scholar
• Breiman, L. (1992), “The Little Bootstrap and Other Methods for Dimensionality Selection in Regression: X-Fixed Prediction Error,” Journal of the American Statistical Association, 87, 738–754.
   Web of Science ®Google Scholar
• Bühlmann, P. (2013), “Statistical Significance in High-Dimensional Linear Models,” Bernoulli, 19, 1212–1242.
   Web of Science ®Google Scholar
• Cai, T.T., and Jiang, T. (2012), “Phase Transition in Limiting Distributions of Coherence of High-dimensional Random Matrices,” Journal of Multivariate Analysis, 107, 24–39.
   Web of Science ®Google Scholar
• (2008), “Comprehensive Genomic Characterization Defines Human Glioblastoma Genes and Core Pathways,” Nature, 455, 1061–1068.
   PubMed Web of Science ®Google Scholar
• Chatterjee, A., and Lahiri, S.N. (2011), “Bootstrapping Lasso Estimators,” Journal of the American Statistical Association, 106, 608–625.
   Web of Science ®Google Scholar
• Cheng, X. (2008), “Robust Confidence Intervals in Nonlinear Regression Under Weak Identification,” unpublished manuscript, Department of Economics, University of Pennsylvania. Available at https://economics.sas.upenn.edu/events/robust-confidence-intervals-nonlinear-regression-under-weak-identification
   Google Scholar
• ——— (2015), “Robust Inference in Nonlinear Models With Mixed Identification Strength,” Journal of Econometrics.
   Web of Science ®Google Scholar
• Davies, R.B. (1977), “Hypothesis Testing When a Nuisance Parameter is Present Only under the Alternative,” Biometrika, 64, 247–254.
   Web of Science ®Google Scholar
• Donoho, D., and Jin, J. (2004), “Higher Criticism for Detecting Sparse Heterogeneous Mixtures,” The Annals of Statistics, 32, 962–994.
   Web of Science ®Google Scholar
• ——— (2015), “Higher Criticism for Large-Scale Inference, Especially for Rare and Weak Effects,” Statistical Science, 30, 1–25.
   Web of Science ®Google Scholar
• Dudoit, S., Shaffer, J.P., and Boldrick, J.C. (2003), “Multiple Hypothesis Testing in Microarray Experiments,” Statistical Science, 18, 71–103.
   Web of Science ®Google Scholar
• Dudoit, S., and van der Laan, M.J. (2008), Multiple Testing Procedures With Applications to Genomics, New York: Springer.
   Google Scholar
• Efron, B. (2006), “Large-Scale Simultaneous Hypothesis Testing: The Choice of a Null Hypothesis,” Journal of American Statistical Association, 99, 96–104.
   Web of Science ®Google Scholar
• ——— (2010), Large-Scale Inference: Empirical Bayes Methods for Estimation, Testing, and Prediction, Cambridge, UK: Cambridge University Press.
   Google Scholar
• Efron, B., and Tibshirani, R.J. (1993), An Introduction to the Bootstrap (Monographs on Statistics & Applied Probability), Boca Raton, FL: Chapman & Hall/CRC.
   Google Scholar
• Fan, J., and Li, R. (2001), “Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties,” Journal of the American Statistical Association, 96, 1348–1360.
   Web of Science ®Google Scholar
• ——— (2006), “Statistical Challenges With High Dimensionality: Feature Selection in Knowledge Discovery,” in Proceedings of the International Congress of Mathematicians (Vol. 110), eds. M. Sanz-Sole, J. Soria, J. L. Varona, and J. Verdera, Zurich: European Mathematical Society, pp. 595–622.
   Google Scholar
• Fan, J., and Lv, J. (2008), “Sure Independence Screening for Ultra-High Dimensional Feature Space” (with discussion), Journal of the Royal Statistical Society, Series B, 70, 849–911.
   PubMed Web of Science ®Google Scholar
• Genovese, C., Jin, J., Wasserman, L., and Yao, Z. (2012), “A Comparison of the Lasso and Marginal Regression,” Journal of Machine Learning Research, 13, 2107–2143.
   Web of Science ®Google Scholar
• (2014), “Genotype-Phenotype Datasets,” Stanford University, available at http://hivdb.stanford.edu/pages/genopheno.dataset.html
   Google Scholar
• Huang, J., Ma, S., and Zhang, C.-H. (2008), “Adaptive Lasso for High-Dimensional Regression Models,” Statistica Sinica, 18, 1603–1618.
   Web of Science ®Google Scholar
• Ingster, Y.I., Tsybakov, A.B., and Verzelen, N. (2010), “Detection Boundary in Sparse Regression,” Electronic Journal of Statistics, 4, 1476–1526.
   Web of Science ®Google Scholar
• Johnson, R.A., and Wichern, D.W. (2007), Applied Multivariate Statistical Analysis (6th ed.), Upper Saddle River, NJ: Prentice Hall.
   Google Scholar
• Laber, E., Lizotte, D., Qian, M., and Murphy, S.A. (2014), “Dynamic Treatment Regimes: Technical Challenges and Applications,” Electronic Journal of Statistics, 8, 1225–1272.
   PubMed Web of Science ®Google Scholar
• Laber, E., and Murphy, S.A. (2011), “Adaptive Confidence Intervals for the Test Error in Classification” (with discussion), Journal of the American Statistical Association, 106, 904–913.
   PubMed Web of Science ®Google Scholar
• ——— (2015), “Adaptive Inference After Model Selection,” under review.
   Google Scholar
• Leeb, H., and Pötscher, B.M. (2006), “Can One Estimate the Conditional Distribution of Post-Model-Selection Estimators? The Annals of Statistics, 34, 2554–2591.
   Web of Science ®Google Scholar
• Lockhart, R., Taylor, J., Tibshirani, R.J., and Tibshirani, R. (2014), “A Significance Test for the Lasso,” The Annals of Statistics, 42, 413–468.
   PubMed Web of Science ®Google Scholar
• McCloskey, A. (2012), “Bonferroni-Based Size-Correction for Nonstandard Testing Problems,” Working Paper. Available at http://www.econ.brown.edu/fac/adam_mccloskey/Research_files/McCloskey_BBCV.pdf
   Google Scholar
• Meinshausen, N., Meier, L., and Bühlmann, P. (2009), “P-Values for High-Dimensional Regression,” Journal of the American Statistical Association, 104, 1671–1681.
   Web of Science ®Google Scholar
• Ning, Y., and Liu, H. (2015), “A General Theory of Hypothesis Tests and Confidence Regions for Sparse High Dimensional Models,” available at http://arxiv.org/abs/1412.8765
   Google Scholar
• Rhee, S.-Y., Gonzales, M.J., Kantor, R., Betts, B.J., Ravela, J., and Shafer, R.W. (2003), “Human Immunodeficiency Virus Reverse Transcriptase and Protease Sequence Database,” Nucleic Acids Research, 31, 298–303.
   PubMed Web of Science ®Google Scholar
• Samworth, R. (2003), “A Note on Methods of Restoring Consistency to the Bootstrap,” Biometrika, 90, 985–990.
   Web of Science ®Google Scholar
• van der Vaart, A.W. (1998), Asymptotic Statistics, Cambridge, UK: Cambridge University Press.
   Google Scholar
• Zhang, C.-H., and Zhang, S.S. (2014), “Confidence Intervals for Low Dimensional Parameters in High Dimensional Linear Models,” Journal of the Royal Statistical Society, Series B, 76, 217–242.
   Web of Science ®Google Scholar
• Zou, H., and Hastie, T. (2005), “Regularization and Variable Selection via the Elastic Net,” Journal of the Royal Statistical Society, Series B, 67, 301–320.
   Google Scholar