Hierarchical Testing in the High-Dimensional Setting With Correlated Variables

REFERENCES

• Benjamini, Y., 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.
   Google Scholar
• Bühlmann, P. (2013), Statistical Significance in High-Dimensional Linear Models, Bernoulli, 19, 1212–1242.
   Web of Science ®Google Scholar
• Bühlmann, P., Kalisch, M., Meier, L. (2014), High-Dimensional Statistics With a View Toward Applications in Biology, Annual Review of Statistics and Its Application, 1, 255–278.
   Web of Science ®Google Scholar
• Bühlmann, P., Mandozzi, J. (2014), High-Dimensional Variable Screening and Bias in Subsequent Inference, With an Empirical Comparison, Computational Statistics, 29, 407–430.
   Web of Science ®Google Scholar
• Bühlmann, P., Rütimann, P., van de Geer, S., Zhang, C.-H. (2013), Correlated Variables in Regression: Clustering and Sparse Estimation ( with discussion), Journal of Statistical Planning and Inference, 143, 1835–1871.
   Web of Science ®Google Scholar
• Bühlmann, P., and van de Geer, S. (2011), Statistics for High-Dimensional Data: Methods, Theory and Applications, New York: Springer Verlag.
   Google Scholar
• Chatterjee, A., Lahiri, S.N. (2013), Rates of Convergence of the Adaptive LASSO Estimators to the Oracle Distribution and Higher Order Refinements by the Bootstrap, Annals of Statistics, 41, 1232–1259.
   Web of Science ®Google Scholar
• Conlon, E.M., Liu, X.S., Lieb, J.D., Liu, J.S. (2003), Integrating Regulatory Motif Discovery and Genome-Wide Expression Analysis, Proceedings of the National Academy of Sciences, 100, 3339–3344.
   PubMed Web of Science ®Google Scholar
• Dezeure, R., Bühlmann, P., Meier, L., Meinshausen, N. (2014), High-Dimensional Inference: Confidence Intervals, p-Values and R-Software hdi, arXiv:1408.4026v1.
   Google Scholar
• Goeman, J.J., Geer, S.A., Van Houwelingen, H.C. (2006), Testing Against a High Dimensional Alternative, Journal of the Royal Statistical Society, Series B, 68, 477–493.
   Google Scholar
• Javanmard, A., Montanari, A. (2014a), Confidence Intervals and Hypothesis Testing for High-Dimensional Regression, Journal of Machine Learning Research, 15, 2869–2909.
   Web of Science ®Google Scholar
• Javanmard, A., Montanari, A. (2014b), Hypothesis Testing in High-Dimensional Regression Under the Gaussian Random Design Model: Asymptotic Theory, IEEE Transactions on Information Theory, 60, 6522–6554.
   Web of Science ®Google Scholar
• Liu, H., Yu, B. (2013), Asymptotic Properties of Lasso+mLS and Lasso+Ridge in Sparse High-Dimensional Linear Regression, Electronic Journal of Statistics, 7, 3124–3169.
   Web of Science ®Google Scholar
• Meinshausen, N. (2008), Hierarchical Testing of Variable Importance, Biometrika, 95, 265–278.
   Web of Science ®Google Scholar
• ——— (2014), Group Bound: Confidence Intervals for Groups of Variables in Sparse High Dimensional Regression Without Assumptions on the Design, Journal of the Royal Statistical Society, Series B, arXiv:1309.3489v2,.
   Google Scholar
• Meinshausen, N., Meier, L., Bühlmann, P. (2009), P-Values for High-Dimensional Regression, Journal of the American Statistical Association, 104, 1671–1681.
   Web of Science ®Google Scholar
• Minnier, J., Tian, L., Cai, T. (2011), A Perturbation Method for Inference on Regularized Regression Estimates, Journal of the American Statistical Association, 106, 1371–1382.
   PubMed Web of Science ®Google Scholar
• Shaffer, J.P. (1986), Modified Sequentially Rejective Multiple Test Procedures, Journal of the American Statistical Association, 81, 826–831.
   Web of Science ®Google Scholar
• Tibshirani, R. (1996), Regression Shrinkage and Selection via the Lasso, Journal of the Royal Statistical Society, Series B, 58, 267–288.
   Google Scholar
• Geer, S., Bühlmann, P., Ritov, Y., Dezeure, R. (2014), On Asymptotically Optimal Confidence Regions and Tests for High-Dimensional Models, Annals of Statistics, 42, 1166–1202.
   Web of Science ®Google Scholar
• van de Geer, S., Bühlmann, P., Zhou, S. (2011), The Adaptive and the Thresholded Lasso for Potentially Misspecified Models (and a Lower Bound for the Lasso), Electronic Journal of Statistics, 5, 688–749.
   Web of Science ®Google Scholar
• Veer, L.J., Dai, H., van de Vijver, M.J., He, Y.D., Hart, A. A.M., Mao, M., Peterse, H.L., van der Kooy, K., Marton, M.J., Witteveen, A.T., Schreiber, G.J., Kerkhoven, R.M., Roberts, C., Linsley, P.S., Bernards, R., Friend, S.H. (2002), Gene Expression Profiling Predicts Clinical Outcome of Breast Cancer, Nature, 415, 530–536.
   PubMed Web of Science ®Google Scholar
• Wasserman, L., Roeder, K. (2009), High Dimensional Variable Selection, Annals of Statistics, 37, 2178–2201.
   PubMed Web of Science ®Google Scholar
• Zhang, C.-H., Huang, J. (2008), The Sparsity and Bias of the Lasso Selection in High-Dimensional Linear Regression, Annals of Statistics, 36, 1567–1594.
   Web of Science ®Google Scholar
• Zhang, C.-H., 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.
   Google Scholar