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Theory and Methods

Adaptive Huber Regression

, ORCID Icon &
Pages 254-265 | Received 11 Aug 2017, Accepted 24 Oct 2018, Published online: 22 Apr 2019

References

  • Alquier, P., Cottet, V., and Lecué, G. (2017), “Estimation Bounds and Sharp Oracle Inequalities of Regularized Procedures With Lipschitz Loss Functions,” arXiv no. 1702.01402.
  • Belloni, A., and Chernozhukov, V. (2011), “ℓ1 -Penalized Quantile Regression in High-Dimensional Sparse Models,” The Annals of Statistics, 39, 82–130. DOI: 10.1214/10-AOS827.
  • Bellec, P. C., Lecué, G., and Tsybakov, A. B. (2018), “Slope Meets Lasso: Improved Oracle Bounds and Optimality,” The Annals of Statistics, 46, 3603–3642. DOI: 10.1214/17-AOS1670.
  • Bickel, P. J., Ritov, Y., and Tsybakov, A. B. (2009), “Simultaneous Analysis of Lasso and Dantzig Selector,” The Annals of Statistics, 37, 1705–1732. DOI: 10.1214/08-AOS620.
  • Bogdan, M., van den Berg, E., Sabatti, C., Su, W., and Candès, E. J. (2015), “SLOPE—Adaptive Variable Selection via Convex Optimization,” The Annals of Applied Statistics, 9, 1103–1140. DOI: 10.1214/15-AOAS842.
  • Brownlees, C., Joly, E., and Lugosi, G. (2015), “Empirical Risk Minimization for Heavy-Tailed Losses,” The Annals of Statistics, 43, 2507–2536. DOI: 10.1214/15-AOS1350.
  • Bühlmann, P., and van de Geer, S. (2011), Statistics for High-Dimensional Data: Methods, Theory and Applications, Heidelberg: Springer.
  • Catoni, O. (2012), “Challenging the Empirical Mean and Empirical Variance: A Deviation Study,” Annales de I’Institut Henri Poincaré—Probabilités et Statistiques, 48, 1148–1185. DOI: 10.1214/11-AIHP454.
  • Catoni, O. (2016), “PAC-Bayesian Bounds for the Gram Matrix and Least Squares Regression With a Random Design,” arXiv no. 1603.05229.
  • Chen, M., Gao, C., and Ren, Z. (2018), “Robust Covariance and Scatter Matrix Estimation Under Huber’s Contamination Model,” The Annals of Statistics, 46, 1932–1960. DOI: 10.1214/17-AOS1607.
  • Cont, R. (2001), “Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues,” Quantitative Finance, 1, 223–236. DOI: 10.1080/713665670.
  • Delaigle, A., Hall, P., and Jin, J. (2011), “Robustness and Accuracy of Methods for High Dimensional Data Analysis Based on Student’s t-Statistic,” Journal of the Royal Statistical Society, Series B, 73, 283–301. DOI: 10.1111/j.1467-9868.2010.00761.x.
  • Devroye, L., Lerasle, M., Lugosi, G., and Oliveira, R. I. (2016), “Sub-Gaussian Mean Estimators,” The Annals of Statistics, 44, 2695–2725. DOI: 10.1214/16-AOS1440.
  • Efron, B., Hastie, T., Johnstone, I., and Tibshirani, R. (2004), “Least Angle Regression,” The Annals of Statistics, 32, 407–499. DOI: 10.1214/009053604000000067.
  • Eklund, A., Nichols, T., and Knutsson, H. (2016), “Cluster Failure: Why fMRI Inferences for Spatial Extent Have Inflated False-Positive Rates,” Proceedings of the National Academy of Sciences of the United States of America, 113, 7900–7905. DOI: 10.1073/pnas.1602413113.
  • Fan, J., Fan, Y., and Barut, E. (2014), “Adaptive Robust Variable Selection,” The Annals of Statistics, 42, 324–351. DOI: 10.1214/13-AOS1191.
  • Fan, J., Li, Q., and Wang, Y. (2017), “Estimation of High Dimensional Mean Regression in the Absence of Symmetry and Light Tail Assumptions,” Journal of the Royal Statistical Society, Series B, 79, 247–265. DOI: 10.1111/rssb.12166.
  • 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. DOI: 10.1198/016214501753382273.
  • Fan, J., Liu, H., Sun, Q., and Zhang, T. (2018), “I-LAMM for Sparse Learning: Simultaneous Control of Algorithmic Complexity and Statistical Error,” The Annals of Statistics, 96, 1348–1360.
  • Fan, J., Wang, W., and Zhu, Z. (2016), “A Shrinkage Principle for Heavy-Tailed Data: High-Dimensional Robust Low-Rank Matrix Recovery,” arXiv no. 1603.08315.
  • Friedline, J. A., Garrett, S. H., Somji, S., Todd, J. H., and Sens, D. A. (1998), “Differential Expression of the MT-1E Gene in Estrogen-Receptor-Positive and -Negative Human Breast Cancer Cell Lines,” The American Journal of Pathology, 152, 23–27.
  • Giulini, I. (2017), “Robust PCA and Pairs of Projections in a Hilbert Space,” Electronic Journal of Statistics, 11, 3903–3926. DOI: 10.1214/17-EJS1343.
  • Hastie, T., Tibshirani, R., and Wainwright, M. J. (2015), Statistical Learning With Sparsity: The Lasso and Generalizations, Boca Raton, FL: CRC Press.
  • He, X., and Shao, Q.-M. (1996), “A General Bahadur Representation of M-Estimators and Its Application to Linear Regression With Nonstochastic Designs,” The Annals of Statistics, 24, 2608–2630. DOI: 10.1214/aos/1032181172.
  • He, X., and Shao, Q.-M. (2000), “On Parameters of Increasing Dimensions,” Journal of Multivariate Analysis, 73, 120–135. DOI: 10.1006/jmva.1999.1873.
  • Huber, P. J. (1964), “Robust Estimation of a Location Parameter,” The Annals of Mathematical Statistics, 35, 73–101. DOI: 10.1214/aoms/1177703732.
  • Huber, P. J. (1973), “Robust Regression: Asymptotics, Conjectures and Monte Carlo,” The Annals of Statistics, 1, 799–821. DOI: 10.1214/aos/1176342503.
  • Koenker, R. (2005), Quantile Regression, New York: Cambridge University Press.
  • Kretzschmar, M. (2000), “Transforming Growth Factor-β and Breast Cancer: Transforming Growth Factor-β/Smad Signaling Defects and Cancer,” Breast Cancer Research, 2, 107–115.
  • Landi, A., Vermeire, J., Iannucci, V., Vanderstraeten, H., Naessens, E., Bentahir, M., and Verhasselt, B. (2014), “Genome-Wide shRNA Screening Identifies Host Factors Involved in Early Endocytic Events for HIV-1-Induced CD4 Down-Regulation,” Retrovirology, 11, 118–129. DOI: 10.1186/PREACCEPT-3552320321395097.
  • Lepski, O. V. (1991), “Asymptotically Minimax Adaptive Estimation. I. Upper Bounds. Optimally Adaptive Estimates,” IEEE Transactions on Information Theory, 36, 682–697.
  • Liu, R. Y. (1990), “On a Notion of Data Depth Based on Random Simplices,” The Annals of Statistics, 18, 405–414. DOI: 10.1214/aos/1176347507.
  • Liu, R. Y., Parelius, J. M., and Singh, K. (1999), “Multivariate Analysis by Data Depth: Descriptive Statistics, Graphics and Inference” (with discussion and a rejoinder by Liu and Singh), The Annals of Statistics, 27, 783–858.
  • Loh, P., and Wainwright, M. J. (2015), “Regularized M-Estimators With Nonconvexity: Statistical and Algorithmic Theory for Local Optima,” Journal of Machine Learning Research, 16, 559–616.
  • Mammen, E. (1989), “Asymptotics With Increasing Dimension for Robust Regression With Applications to the Bootstrap,” The Annals of Statistics, 17, 382–400. DOI: 10.1214/aos/1176347023.
  • Minsker, S. (2018), “Sub-Gaussian Estimators of the Mean of a Random Matrix With Heavy-Tailed Entries,” The Annals of Statistics, 46, 2871–2903. DOI: 10.1214/17-AOS1642.
  • Mizera, I. (2002), “On Depth and Deep Points: A Calculus,” The Annals of Statistics, 30, 1681–1736. DOI: 10.1214/aos/1043351254.
  • Mizera, I., and Müller, C. H. (2004), “Location-Scale Depth,” Journal of the American Statistical Association, 99, 949–966. DOI: 10.1198/016214504000001312.
  • Nakata, B., Takashima, T., Ogawa, Y., Ishikawa, T., and Hirakawa, K. (2004), “Serum CYFRA 21-1 (Cytokeratin-19 Fragments) Is a Useful Tumour Marker for Detecting Disease Relapse and Assessing Treatment Efficacy in Breast Cancer,” British Journal of Cancer, 91, 873–878. DOI: 10.1038/sj.bjc.6602074.
  • Oh, J. H., Yang, J. O., Hahn, Y., Kim, M. R., Byun, S. S., Jeon, Y. J., Kim, J. M., Song, K. S., Noh, S. M., Kim, S., and Yoo, H. S. (2005), “Transcriptome Analysis of Human Gastric Cancer,” Mammalian Genome, 16, 942–954. DOI: 10.1007/s00335-005-0075-2.
  • Portnoy, S. (1985), “Asymptotic Behavior of M Estimators of p Regression Parameters When p2/n Is Large; II. Normal Approximation,” The Annals of Statistics, 13, 1403–1417. DOI: 10.1214/aos/1176349744.
  • Purdom, E., and Holmes, S. P. (2005), “Error Distribution for Gene Expression Data,” Statistical Applications in Genetics and Molecular Biology, 4, 16.
  • Ross, D. T., Scherf, U., Eisen, M. B., Perou, C. M., Rees, C., Spellman, P., Iyer, W., Jeffrey, S. S., Van de Rijn, M., Pergamenschikov, A., Lee, J. C. F., Lashkari, D., Shalon, D., Myers, T. G., Weinstein, J. N., Botstein, D., and Brown, P. O. (2000), “Systematic Variation in Gene Expression Patterns in Human Cancer Cell Lines,” Nature Genetics, 24, 227–235. DOI: 10.1038/73432.
  • Shangguan, L., Ti, X., Krause, U., Hai, B., Zhao, Y., Yang, Z., and Liu, F. (2012), “Inhibition of TGF-β/Smad Signaling by BAMBI Blocks Differentiation of Human Mesenchymal Stem Cells to Carcinoma-Associated Fibroblasts and Abolishes Their Protumor Effects,” Stem Cells, 30, 2810–2819. DOI: 10.1002/stem.1251.
  • Shankavaram, U. T., Reinhold, W. C., Nishizuka, S., Major, S., Morita, D., Chary, K. K., Reimers, M. A., Scherf, U. Kahn, A., Dolginow, D., Cossman, J., Kaldjian, E. P., Scudiero, D. A., Petricoin, E., Liotta, L., Lee, J. K., and Weinstein, J. N. (2007), “Transcript and Protein Expression Profiles of the NCI-60 Cancer Cell Panel: An Integromic Microarray Study,” Molecular Cancer Therapeutics, 40, 2877–2909.
  • Shehata, M., Bièche, I., Boutros, R., Weidenhofer, J., Fanayan, S., Spalding, L., Zeps, N., Byth, K., Bright, R. K., Lidereau, R., and Byrne, J. A. (2008), “Nonredundant Functions for Tumor Protein D52-Like Proteins Support Specific Targeting of TPD52,” Clinical Cancer Research, 14, 5050–5060. DOI: 10.1158/1078-0432.CCR-07-4994.
  • Tibshirani, R. (1996), “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Series B, 58, 267–288. DOI: 10.1111/j.2517-6161.1996.tb02080.x.
  • Tukey, J. W. (1975), “Mathematics and the Picturing of Data,” in Proceedings of the International Congress of Mathematicians (Vol. 2), pp. 523–531.
  • Wainwright, M. J. (2009), “Sharp Thresholds for High-Dimensional and Noisy Sparsity Recovery Using ℓ1 -Constrained Quadratic Programming (Lasso),” IEEE Transactions on Information Theory, 55, 2183–2202. DOI: 10.1109/TIT.2009.2016018.
  • Wang, L. (2013), “The L1 Penalized LAD Estimator for High Dimensional Linear Regression,” Journal of Multivariate Analysis, 120, 135–151. DOI: 10.1016/j.jmva.2013.04.001.
  • Wang, L., Peng, B., and Li, R. (2015), “A High-Dimensional Nonparametric Multivariate Test for Mean Vector,” Journal of the American Statistical Association, 110, 1658–1669. DOI: 10.1080/01621459.2014.988215.
  • Wu, Y., Siadaty, M. S., Berens, M. E., Hampton, G. M., and Theodorescu, D. (2008), “Overlapping Gene Expression Profiles of Cell Migration and Tumor Invasion in Human Bladder Cancer Identify Metallothionein E1 and Nicotinamide N-Methyltransferase as Novel Regulators of Cell Migration,” Oncogene, 27, 6679–6689. DOI: 10.1038/onc.2008.264.
  • Yohai, V. J., and Maronna, R. A. (1979), “Asymptotic Behavior of M-Estimators for the Linear Model,” The Annals of Statistics, 7, 258–268. DOI: 10.1214/aos/1176344610.
  • Zheng, Q., Peng, L., and He, X. (2015), “Globally Adaptive Quantile Regression With Ultra-High Dimensional Data,” The Annals of Statistics, 43, 2225–2258. DOI: 10.1214/15-AOS1340.
  • Zhou, T., Li, Y., Yang, L., Liu, L., Ju, Y., and Li, C. (2017), “Silencing of ANXA3 Expression by RNA Interference Inhibits the Proliferation and Invasion of Breast Cancer Cells,” Oncology Reports, 37, 388–398. DOI: 10.3892/or.2016.5251.
  • Zuo, Y., and Serfling, R. (2000), “General Notions of Statistical Depth Function,” The Annals of Statistics, 28, 461–482. DOI: 10.1214/aos/1016218226.

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