Distribution-Free Prediction Sets

REFERENCES

• Aichison , J. and Dunsmore , I. R. 1975 . Statistical Prediction Analysis , Cambridge : Cambridge University Press .
   Google Scholar
• Audibert , J. and Tsybakov , A. 2007 . “Fast Learning for Plug-in Classifiers,” . The Annals of Statistics , 35 : 608 – 633 .
   Web of Science ®Google Scholar
• Baillo , A. 2003 . “Total Error in a Plug-in Estimator of Level Sets,” . Statistics & Probability Letters , 65 : 411 – 417 .
   Web of Science ®Google Scholar
• Baillo , A. , Cuestas-Alberto , J. and Cuevas , A. 2001 . “Convergence Rates in Nonparametric Estimation of Level Sets,” . Statistics & Probability Letters , 53 : 27 – 35 .
   Web of Science ®Google Scholar
• Cadre , B. 2006 . “Kernel Estimation of Density Level Sets,” . Journal of Multivariate Analysis , 97 : 999 – 1023 .
   Web of Science ®Google Scholar
• Cadre , B. , Pelletier , B. and Pudlo , P. 2009 . “Clustering by Estimation of Density Level Sets at a Fixed Probability,” . manuscript. Available at http://hal.archives-ouvertes.fr/docs/00/39/74/37/PDF/tlevel.pdf
   Google Scholar
• Chatterjee , S. K. and Patra , N. K. 1980 . “Asymptotically Minimal Multivariate Tolerance Sets,” . Calcutta Statistical Association Bulletin , 29 : 73 – 93 .
   Google Scholar
• Di Bucchianico , A. , Einmahl , J. H. and Mushkudiani , N. A. 2001 . “Smallest Nonparametric Tolerance Regions,” . The Annals of Statistics , 29 : 1320 – 1343 .
   Web of Science ®Google Scholar
• Fraser , D. A. S. and Guttman , I. 1956 . “Tolerance Regions,” . The Annals of Mathematical Statistics , 27 : 162 – 179 .
   Google Scholar
• Giné , E. and Guillou , A. 2002 . “Rates of Strong Uniform Consistency for Multivariate Kernel Density Estimators,” . Annales de l’Institut Henri Poincare (B) Probability and Statistics , 38 : 907 – 921 .
   Web of Science ®Google Scholar
• Guttman , I. 1970 . Statistical Tolerance Regions: Classical and Bayesian , London : Griffin .
   Google Scholar
• Hartigan , J. 1975 . Clustering Algorithms , New York : Wiley .
   Google Scholar
• Hyndman , R. 1996 . “Computing and Graphing Highest Density Regions,” . The American Statistician , 50 : 120 – 125 .
   Web of Science ®Google Scholar
• Li , J. and Liu , R. 2008 . “Multivariate Spacings Based on Data Depth: I. Construction of Nonparametric Multivariate Tolerance Regions,” . The Annals of Statistics , 36 : 1299 – 1323 .
   Web of Science ®Google Scholar
• Loader , C. 1999 . “Bandwidth Selection: Classical or Plug-in?,” . The Annals of Statistics , 27 : 415 – 438 .
   Web of Science ®Google Scholar
• Mammen , E. , Miranda , M. D. M. , Nielsen , J. P. and Sperlich , S. 2011 . “Do-Validation for Kernel Density Estimation,” . Journal of the American Statistical Association , 106 : 651 – 660 .
   Web of Science ®Google Scholar
• Park , C. , Huang , J. Z. and Ding , Y. 2010 . “A Computable Plug-in Estimator of Minimum Volume Sets for Novelty Detection,” . Operations Research , 58 : 1469 – 1480 .
   Web of Science ®Google Scholar
• Polonik , W. 1995 . “Measuring Mass Concentrations and Estimating Density Contour Clusters–An Excess Mass Approach,” . The Annals of Statistics , 23 : 855 – 881 .
   Web of Science ®Google Scholar
• Polonik , W. 1997 . “Minimum Volume Sets and Generalized Quantile Processes,” . Stochastic Processes and Their Applications , 69 : 1 – 24 .
   Web of Science ®Google Scholar
• Prakasa Rao , B. 1983 . Nonparametric Functional Estimation , New York : Academic Press .
   Google Scholar
• Rigollet , P. and Vert , R. 2009 . “Optimal Rates for Plug-in Estimators of Denslty Level Sets,” . Bernoulli , 14 : 1154 – 1178 .
   Google Scholar
• Rinaldo , A. and Wasserman , L. 2010 . “Generalized Density Clustering,” . The Annals of Statistics , 38 : 2678 – 2722 .
   Web of Science ®Google Scholar
• Samworth , R. J. and Wand , M. P. 2010 . “Asymptotics and Optimal Bandwidth Selection for Highest Density Region Estimation,” . The Annals of Statistics , 38 : 1767 – 1792 .
   Web of Science ®Google Scholar
• Scott , C. D. and Nowak , R. D. 2006 . “Learning Minimum Volume Sets,” . Journal of Machine Learning Research , 7 : 665 – 704 .
   Web of Science ®Google Scholar
• Shafer , G. and Vovk , V. 2008 . “A Tutorial on Conformal Prediction,” . Journal of Machine Learning Research , 9 : 371 – 421 .
   Web of Science ®Google Scholar
• Sricharan , K. and Hero , A. 2011 . “Efficient Anomaly Detection Using Bipartite k-NN Graphs,” . In Advances in Neural Information Processing Systems 24 Edited by: Shawe-Taylor , J. , Zemel , R. , Bartlett , P. , Pereira , F. and Weinberger , K. 478 – 486 . Red Hook, NY: Curran Associates, Inc.
   Google Scholar
• Steinwart , I. , Hush , D. and Scovel , C. 2005 . “A Classification Framework for Anomaly Detection,” . Journal of Machine Learning Research , 6 : 211 – 232 .
   Web of Science ®Google Scholar
• Tsybakov , A. 1997 . “On Nonparametric Estimation of Density Level Sets,” . The Annals of Statistics , 25 : 948 – 969 .
   Web of Science ®Google Scholar
• Tsybakov , A. 2009 . Introduction to Nonparametric Estimation , Berlin : Springer .
   Google Scholar
• Tukey , J. 1947 . “Nonparametric Estimation, II. Statistical Equivalent Blocks and Multivarate Tolerance Regions,” . The Annals of Mathematical Statistics , 18 : 529 – 539 .
   Google Scholar
• van der Vaart , A. W. and Wellner , J. A. 1996 . Weak Convergence and Empirical Processes , New York : Springer .
   Google Scholar
• Vovk , V. , Gammerman , A. and Shafer , G. 2005 . Algorithmic Learning in a Random World , New York : Springer .
   Google Scholar
• Vovk , V. , Nouretdinov , I. and Gammerman , A. 2009 . “On-Line Preditive Linear Regression,” . The Annals of Statistics , 37 : 1566 – 1590 .
   Web of Science ®Google Scholar
• Wald , A. 1943 . “An Extension of Wilks Method for Setting Tolerance Limits,” . The Annals of Mathematical Statistics , 14 : 45 – 55 .
   Google Scholar
• Walther , G. 1997 . “Granulometric Smoothing,” . The Annals of Statistics , 25 : 2273 – 2299 .
   Web of Science ®Google Scholar
• Wei , Y. 2008 . “An Approach to Multivariate Covariate-Dependent Quantile Contours With Application to Bivariate Conditional Growth Charts,” . Journal of the American Statistical Association , 103 : 397 – 409 .
   Web of Science ®Google Scholar
• Wilks , S. 1941 . “Determination of Sample Sizes for Setting Tolerance Limits,” . The Annals of Mathematical Statistics , 12 : 91 – 96 .
   Google Scholar
• Willett , R. and Nowak , R. 2007 . “Minimax Optimal Level-Set Estimation,” . IEEE Transactions on Image Processing , 16 : 2965 – 2979 .
   PubMed Web of Science ®Google Scholar
• Zhao , M. and Saligrama , V. 2009 . “Anomaly Detection With Score Functions Based on Nearest Neighbor Graphs,” . In Advances in Neural Information Processing Systems 22 Edited by: Bengio , Y. , Schuurmans , D. , Lafferty , J. , Williams , C. K. I. and Culotta , A. 2250 – 2258 . Red Hook, NY: Curran Associates, Inc.
   Google Scholar