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Theory and Methods

Gaining Outlier Resistance With Progressive Quantiles: Fast Algorithms and Theoretical Studies

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Pages 1282-1295 | Received 15 Oct 2019, Accepted 03 Nov 2020, Published online: 14 Jan 2021

References

  • Alfons, A., Croux, C., and Gelper, S. (2013), “Sparse Least Trimmed Squares Regression for Analyzing High-Dimensional Large Data Sets,” The Annals of Applied Statistics, 7, 226–248. DOI: 10.1214/12-AOAS575.
  • Avella-Medina, M. (2017), “Influence Functions for Penalized M-Estimators,” Bernoulli, 23, 3178–3196. DOI: 10.3150/16-BEJ841.
  • Avella-Medina, M., and Ronchetti, E. (2018), “Robust and Consistent Variable Selection in High-Dimensional Generalized Linear Models,” Biometrika, 105, 31–44. DOI: 10.1093/biomet/asx070.
  • Belloni, A., and Chernozhukov, V. (2011), “l1-Penalized Quantile Regression in High-Dimensional Sparse Models,” The Annals of Statistics, 39, 82–130.
  • Bianco, A. M., and Yohai, V. J. (1996), “Robust Estimation in the Logistic Regression Model,” in Robust Statistics, Data Analysis, and Computer Intensive Methods, ed. H. Rieder, New York: Springer, pp. 17–34.
  • Boyd, S., and Vandenberghe, L. (2004), Convex Optimization, Cambridge: Cambridge University Press.
  • Bregman, L. M. (1967), “The Relaxation Method of Finding the Common Point of Convex Sets and Its Application to the Solution of Problems in Convex Programming,” USSR Computational Mathematics and Mathematical Physics, 7, 200–217. DOI: 10.1016/0041-5553(67)90040-7.
  • Bühlmann, P., and Yu, B. (2003), “Boosting With the L2 Loss: Regression and Classification,” Journal of the American Statistical Association, 98, 324–339. DOI: 10.1198/016214503000125.
  • Candes, E. J., and Tao, T. (2005), “Decoding by Linear Programming,” IEEE Transactions on Information Theory, 51, 4203–4215. DOI: 10.1109/TIT.2005.858979.
  • Cantoni, E., and Ronchetti, E. (2001), “Robust Inference for Generalized Linear Models,” Journal of the American Statistical Association, 96, 1022–1030. DOI: 10.1198/016214501753209004.
  • Chapelle, O. (2007), “Training a Support Vector Machine in the Primal,” Neural Computation, 19, 1155–1178. DOI: 10.1162/neco.2007.19.5.1155.
  • Chinchor, N. (1992), “MUC-4 Evaluation Metrics,” in Proceedings of the 4th Conference on Message Understanding, pp. 22–29. DOI: 10.3115/1072064.1072067.
  • Croux, C., and Haesbroeck, G. (2003), “Implementing the Bianco and Yohai Estimator for Logistic Regression,” Computational Statistics & Data Analysis, 44, 273–295.
  • Freue, G. V. C., Kepplinger, D., Salibián-Barrera, M., and Smucler, E. (2019), “Robust Elastic Net Estimators for Variable Selection and Identification of Proteomic Biomarkers,” The Annals of Applied Statistics, 13, 2065–2090. DOI: 10.1214/19-AOAS1269.
  • Hadi, A. S., and Luceño, A. (1997), “Maximum Trimmed Likelihood Estimators: A Unified Approach, Examples, and Algorithms,” Computational Statistics & Data Analysis, 25, 251–272.
  • Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., and Stahel, W. A. (2011), Robust Statistics: The Approach based on Influence Functions (Vol. 1), Hoboken, NJ: Wiley.
  • Hastie, T., Tibshirani, R., and Wainwright, M. (2015), Statistical Learning With Sparsity: The Lasso and Generalizations, Boca Raton, FL: Chapman and Hall/CRC.
  • Huber, P., and Ronchetti, E. (2009), Robust Statistics, Hoboken, NJ: Wiley.
  • Hunter, D. R., and Lange, K. (2004), “A Tutorial on MM Algorithms,” The American Statistician, 58, 30–37. DOI: 10.1198/0003130042836.
  • Khan, J. A., Van Aelst, S., and Zamar, R. H. (2007), “Robust Linear Model Selection Based on Least Angle Regression,” Journal of the American Statistical Association, 102, 1289–1299. DOI: 10.1198/016214507000000950.
  • Kurnaz, F. S., Hoffmann, I., and Filzmoser, P. (2018), “Robust and Sparse Estimation Methods for High-Dimensional Linear and Logistic Regression,” Chemometrics and Intelligent Laboratory Systems, 172, 211–222. DOI: 10.1016/j.chemolab.2017.11.017.
  • Little, M. A., McSharry, P. E., Roberts, S. J., Costello, D. A., and Moroz, I. M. (2007), “Exploiting Nonlinear Recurrence and Fractal Scaling Properties for Voice Disorder Detection,” Biomedical Engineering Online, 6, 23. DOI: 10.1186/1475-925X-6-23.
  • Loh, P.-L. (2017), “Statistical Consistency and Asymptotic Normality for High-Dimensional Robust M-Estimators,” The Annals of Statistics, 45, 866–896.
  • Maronna, R. A., Martin, D. R., and Yohai, V. J. (2006), Robust Statistics: Theory and Methods (Vol. 1), New York: Wiley.
  • Müller, S., and Welsh, A. (2005), “Outlier Robust Model Selection in Linear Regression,” Journal of the American Statistical Association, 100, 1297–1310. DOI: 10.1198/016214505000000529.
  • Nair, V., and Hinton, G. E. (2010), “Rectified Linear Units Improve Restricted Boltzmann Machines,” in Proceedings of the 27th International Conference on Machine Learning (ICML-10), pp. 807–814.
  • Needell, D., and Tropp, J. A. (2009), “CoSaMP: Iterative Signal Recovery From Incomplete and Inaccurate Samples,” Applied and Computational Harmonic Analysis, 26, 301–321. DOI: 10.1016/j.acha.2008.07.002.
  • Öllerer, V., Croux, C., and Alfons, A. (2015), “The Influence Function of Penalized Regression Estimators,” Statistics, 49, 741–765. DOI: 10.1080/02331888.2014.922563.
  • Pregibon, D. (1982), “Resistant Fits for Some Commonly Used Logistic Models With Medical Applications,” Biometrics, 38, 485–498. DOI: 10.2307/2530463.
  • Qin, Y., Li, S., Li, Y., and Yu, Y. (2017), “Penalized Maximum Tangent Likelihood Estimation and Robust Variable Selection,” arXiv no. 1708.05439.
  • Ronchetti, E., Field, C., and Blanchard, W. (1997), “Robust Linear Model Selection by Cross-Validation,” Journal of the American Statistical Association, 92, 1017–1023. DOI: 10.1080/01621459.1997.10474057.
  • Rousseeuw, P. J. (1985), “Multivariate Estimation With High Breakdown Point,” Mathematical Statistics and Applications, 8, 283–297.
  • Rousseeuw, P. J., and Driessen, K. V. (1999), “Computing LTS Regression for Large Data Sets,” Technical Report, Institute of Mathematical Statistics Bulletin.
  • Rousseeuw, P. J., and Hubert, M. (1997), “Recent Developments in PROGRESS,” Lecture Notes-Monograph Series, 31, 201–214.
  • Rousseeuw, P. J., and Yohai, V. (1984), “Robust Regression by Means of S-Estimators,” in Robust and Nonlinear Time Series Analysis, eds. J. Franke, W. Härdle, and D. Martin, New York: Springer, pp. 256–272.
  • Sakar, B. E., Isenkul, M. E., Sakar, C. O., Sertbas, A., Gurgen, F., Delil, S., Apaydin, H., and Kursun, O. (2013), “Collection and Analysis of a Parkinson Speech Dataset With Multiple Types of Sound Recordings,” IEEE Journal of Biomedical and Health Informatics, 17, 828–834. DOI: 10.1109/JBHI.2013.2245674.
  • Salibian-Barrera, M., and Van Aelst, S. (2008), “Robust Model Selection Using Fast and Robust Bootstrap,” Computational Statistics & Data Analysis, 52, 5121–5135.
  • She, Y., and Chen, K. (2017), “Robust Reduced-Rank Regression,” Biometrika, 104, 633–647. DOI: 10.1093/biomet/asx032.
  • She, Y., and Owen, A. B. (2011), “Outlier Detection Using Nonconvex Penalized Regression,” Journal of the American Statistical Association, 106, 626–639. DOI: 10.1198/jasa.2011.tm10390.
  • She, Y., and Tran, H. (2019), “On Cross-Validation for Sparse Reduced Rank Regression,” Journal of the Royal Statistical Society, Series B, 81, 145–161. DOI: 10.1111/rssb.12295.
  • She, Y., Wang, J., Li, H., and Wu, D. (2013), “Group Iterative Spectrum Thresholding for Super-Resolution Sparse Spectral Selection,” IEEE Transactions on Signal Processing, 61, 6371–6386. DOI: 10.1109/TSP.2013.2281303.
  • She, Y., Wang, Z., and Jin, J. (2020), “Analysis of Generalized Bregman Surrogate Algorithms for Nonsmooth Nonconvex Statistical Learning,” The Annals of Statistics (to appear).
  • 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.
  • van de Geer, S. A., and Bühlmann, P. (2009), “On the Conditions Used to Prove Oracle Results for the Lasso,” Electronic Journal of Statistics, 3, 1360–1392. DOI: 10.1214/09-EJS506.
  • van der Vaart, A. W. (1998), Asymptotic Statistics, Cambridge: Cambridge University Press.
  • Vandev, D., and Neykov, N. (1998), “About Regression Estimators With High Breakdown Point,” Statistics: A Journal of Theoretical and Applied Statistics, 32, 111–129. DOI: 10.1080/02331889808802657.
  • Zhang, T. (2013), “Multi-Stage Convex Relaxation for Feature Selection,” Bernoulli, 19, 2277–2293. DOI: 10.3150/12-BEJ452.

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