Anti-concentration and honest, adaptive confidence bands

Annals of Statistics

Volumn 42, Issue 5, 2014, Pages 1787-1818

  Chernozhukov, Victor ^a   Chetverikov, Denis ^b   Kato, Kengo ^c  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c UNIVERSITY OF TOKYO   (Japan)

Author keywords

Anti concentration of separable gaussian processes; Honest confidence bands; Lepski's method; Multiplier method; Non donsker empirical processes

Indexed keywords

EID: 84908254539     PISSN: 00905364     EISSN: 21688966     Source Type: Journal    
DOI: 10.1214/14-AOS1235     Document Type: Article

References (45)

1. Bickel, P. J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071-1095. MR0348906
2. Birgé, L. (2001). An alternative point of view on Lepski's method. In State of the Art in Probability and Statistics (Leiden, 1999). Institute of Mathematical Statistics Lecture Notes-Monograph Series 36 113-133. IMS, Beachwood, OH. MR1836557
3. Bissantz, N., Dümbgen, L., Holzmann, H. and Munk, A. (2007). Non-parametric confidence bands in deconvolution density estimation. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 483-506. MR2323764
4. Bull, A. D. (2012). Honest adaptive confidence bands and self-similar functions. Electron. J. Stat. 6 1490-1516. MR2988456
5. Bull, A. D. (2013). A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets. Constr. Approx. 37 295-309. MR3019781
6. Cai, T. T. and Low, M. G. (2004). An adaptation theory for nonparametric confidence intervals. Ann. Statist. 32 1805-1840. MR2102494
7. Chernozhukov, V., Chetverikov, D. and Kato, K. (2012). Gaussian approximation of suprema of empirical processes. Preprint. Available at arXiv:1212.6885v2.
8. Chernozhukov, V., Chetverikov, D. and Kato, K. (2014). Comparison and anticoncentration bounds for maxima of Gaussian random vectors. Probab. Theory Related Fields. To appear. Available at arXiv:1301.4807v3.
9. Chernozhukov, V., Chetverikov, D. and Kato, K. (2014). Supplement to "Anticoncentration and honest, adaptive confidence bands."
10. Chernozhukov, V., Lee, S. and Rosen, A. M. (2013). Intersection bounds: Estimation and inference. Econometrica 81 667-737. MR3043345
11. Chetverikov, D. (2012). Testing regression monotonicity in econometric models. Available at arXiv:1212.6757.
12. Claeskens, G. and Van Keilegom, I. (2003). Bootstrap confidence bands for regression curves and their derivatives. Ann. Statist. 31 1852-1884. MR2036392
13. Daubechies, I. (1992). Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics 61. SIAM, Philadelphia, PA. MR1162107
14. Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge Univ. Press, Cambridge. MR1720712
15. Giné, E. and Guillou, A. (2001). A law of the iterated logarithm for kernel density estimators in the presence of censoring. Ann. Inst. Henri Poincaré Probab. Stat. 37 503-522.
16. Giné, E. and Guillou, A. (2002). Rates of strong uniform consistency for multivariate kernel density estimators. Ann. Inst. Henri Poincaré Probab. Stat. 38 907-921. MR1955344
17. Giné, E., Güntürk, C. S. and Madych, W. R. (2011). On the periodized square of L2 cardinal splines. Exp. Math. 20 177-188. MR2821389
18. Giné, E., Koltchinskii, V. and Sakhanenko, L. (2004). Kernel density estimators: Convergence in distribution for weighted sup-norms. Probab. Theory Related Fields 130 167-198. MR2093761
19. Giné, E. and Nickl, R. (2009). An exponential inequality for the distribution function of the kernel density estimator, with applications to adaptive estimation. Probab. Theory Related Fields 143 569-596. MR2475673
20. Giné, E. and Nickl, R. (2010). Confidence bands in density estimation. Ann. Statist. 38 1122-1170. MR2604707
21. Giné, E. and Nickl, R. (2010). Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections. Bernoulli 16 1137-1163. MR2759172
22. Giné, E. and Zinn, J. (1984). Some limit theorems for empirical processes. Ann. Probab. 12 929-998. MR0757767
23. Giné, E. and Zinn, J. (1990). Bootstrapping general empirical measures. Ann. Probab. 18 851-869. MR1055437
24. Hall, P. (1991). On convergence rates of suprema. Probab. Theory Related Fields 89 447-455. MR1118558
25. Hall, P. and Horowitz, J. (2013). A simple bootstrap method for constructing nonparametric confidence bands for functions. Ann. Statist. 41 1892-1921. MR3127852
26. Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximation, and Statistical Applications. Springer, New York. MR1618204
27. Hoffmann, M. and Nickl, R. (2011). On adaptive inference and confidence bands. Ann. Statist. 39 2383-2409. MR2906872
28. Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrsch. Verw. Gebiete 32 111-131. MR0375412
29. Ledoux, M. and Talagrand, M. (1988). Un critère sur les petites boules dans le théorème limite central. Probab. Theory Related Fields 77 29-47. MR0921817
30. LepskiѤ, O. V. (1991). Asymptotically minimax adaptive estimation. I. Upper bounds. Optimally adaptive estimates. Theory Probab. Appl. 36 682-697.
31. Li, K.-C. (1989). Honest confidence regions for nonparametric regression. Ann. Statist. 17 1001-1008. MR1015135
32. Lounici, K. and Nickl, R. (2011). Global uniform risk bounds for wavelet deconvolution estimators. Ann. Statist. 39 201-231. MR2797844
33. Low, M. G. (1997). On nonparametric confidence intervals. Ann. Statist. 25 2547-2554. MR1604412
34. Massart, P. (2000). About the constants in Talagrand's concentration inequalities for empirical processes. Ann. Probab. 28 863-884. MR1782276
35. Massart, P. (2007). Concentration Inequalities and Model Selection. Springer, Berlin. MR2319879
36. Picard, D. and Tribouley, K. (2000). Adaptive confidence interval for pointwise curve estimation. Ann. Statist. 28 298-335. MR1762913
37. Præstgaard, J. and Wellner, J. A. (1993). Exchangeably weighted bootstraps of the general empirical process. Ann. Probab. 21 2053-2086. MR1245301
38. Rio, E. (1994). Local invariance principles and their application to density estimation. Probab. Theory Related Fields 98 21-45. MR1254823
39. Rudelson, M. and Vershynin, R. (2009). Smallest singular value of a random rectangular matrix. Comm. Pure Appl. Math. 62 1707-1739. MR2569075
40. Smirnov, N. V. (1950). On the construction of confidence regions for the density of distribution of random variables. Doklady Akad. Nauk SSSR (N. S.) 74 189-191. MR0037494
41. Talagrand, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126 505-563. MR1419006
42. Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer, New York. MR2724359
43. van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York. MR1385671
44. Wasserman, L. (2006). All of Nonparametric Statistics. Springer, New York. MR2172729
45. Xia, Y. (1998). Bias-corrected confidence bands in nonparametric regression. J. R. Stat. Soc. Ser. B Stat. Methodol. 60 797-811. MR1649488