The Annals of Statistics

Locally adaptive confidence bands

Tim Patschkowski and Angelika Rohde

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We develop honest and locally adaptive confidence bands for probability densities. They provide substantially improved confidence statements in case of inhomogeneous smoothness, and are easily implemented and visualized. The article contributes conceptual work on locally adaptive inference as a straightforward modification of the global setting imposes severe obstacles for statistical purposes. Among others, we introduce a statistical notion of local Hölder regularity and prove a correspondingly strong version of local adaptivity. We substantially relax the straightforward localization of the self-similarity condition in order not to rule out prototypical densities. The set of densities permanently excluded from the consideration is shown to be pathological in a mathematically rigorous sense. On a technical level, the crucial component for the verification of honesty is the identification of an asymptotically least favorable stationary case by means of Slepian’s comparison inequality.

Article information

Source
Ann. Statist., Volume 47, Number 1 (2019), 349-381.

Dates
Received: August 2017
Revised: January 2018
First available in Project Euclid: 30 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.aos/1543568591

Digital Object Identifier
doi:10.1214/18-AOS1690

Mathematical Reviews number (MathSciNet)
MR3909936

Zentralblatt MATH identifier
07036204

Subjects
Primary: 62G07: Density estimation 62G15: Tolerance and confidence regions

Keywords
Local adaptivity confidence bands in density estimation

Citation

Patschkowski, Tim; Rohde, Angelika. Locally adaptive confidence bands. Ann. Statist. 47 (2019), no. 1, 349--381. doi:10.1214/18-AOS1690. https://projecteuclid.org/euclid.aos/1543568591


Export citation

References

  • Baraud, Y. (2004). Confidence balls in Gaussian regression. Ann. Statist. 32 528–551.
  • Bickel, P. J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071–1095.
  • Bull, A. D. (2012). Honest adaptive confidence bands and self-similar functions. Electron. J. Stat. 6 1490–1516.
  • Bull, A. D. and Nickl, R. (2013). Adaptive confidence sets in $L^{2}$. Probab. Theory Related Fields 156 889–919.
  • Cai, T. T. and Low, M. G. (2004). An adaptation theory for nonparametric confidence intervals. Ann. Statist. 32 1805–1840.
  • Cai, T. T. and Low, M. G. (2006). Adaptive confidence balls. Ann. Statist. 34 202–228.
  • Carpentier, A. (2013). Honest and adaptive confidence sets in $L_{p}$. Electron. J. Stat. 7 2875–2923.
  • Chernozhukov, V., Chetverikov, D. and Kato, K. (2014a). Anti-concentration and honest, adaptive confidence bands. Ann. Statist. 42 1787–1818.
  • Chernozhukov, V., Chetverikov, D. and Kato, K. (2014b). Gaussian approximation of suprema of empirical processes. Ann. Statist. 42 1564–1597.
  • Davies, P. L., Kovac, A. and Meise, M. (2009). Nonparametric regression, confidence regions and regularization. Ann. Statist. 37 2597–2625.
  • Dümbgen, L. (1998). New goodness-of-fit tests and their application to nonparametric confidence sets. Ann. Statist. 26 288–314.
  • Dümbgen, L. (2003). Optimal confidence bands for shape-restricted curves. Bernoulli 9 423–449.
  • Genovese, C. R. and Wasserman, L. (2005). Confidence sets for nonparametric wavelet regression. Ann. Statist. 33 698–729.
  • Genovese, C. and Wasserman, L. (2008). Adaptive confidence bands. Ann. Statist. 36 875–905.
  • Giné, E. and Nickl, R. (2010). Confidence bands in density estimation. Ann. Statist. 38 1122–1170.
  • Hengartner, N. W. and Stark, P. B. (1995). Finite-sample confidence envelopes for shape-restricted densities. Ann. Statist. 23 525–550.
  • Hoffmann, M. and Nickl, R. (2011). On adaptive inference and confidence bands. Ann. Statist. 39 2383–2409.
  • Juditsky, A. and Lambert-Lacroix, S. (2003). Nonparametric confidence set estimation. Math. Methods Statist. 12 410–428.
  • Kerkyacharian, G., Nickl, R. and Picard, D. (2012). Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds. Probab. Theory Related Fields 153 363–404.
  • Kueh, A. (2012). Locally adaptive density estimation on the unit sphere using needlets. Constr. Approx. 36 433–458.
  • Lepskiĭ, O. V. (1990). A problem of adaptive estimation in Gaussian white noise. Teor. Veroyatn. Primen. 35 459–470.
  • Low, M. G. (1997). On nonparametric confidence intervals. Ann. Statist. 25 2547–2554.
  • Nickl, R. and Szabó, B. (2016). A sharp adaptive confidence ball for self-similar functions. Stochastic Process. Appl. 126 3913–3934.
  • Patschkowski, T. and Rohde, A. (2019). Supplement to “Locally adaptive confidence bands.” DOI:10.1214/18-AOS1690SUPP.
  • Picard, D. and Tribouley, K. (2000). Adaptive confidence interval for pointwise curve estimation. Ann. Statist. 28 298–335.
  • Robins, J. and van der Vaart, A. (2006). Adaptive nonparametric confidence sets. Ann. Statist. 34 229–253.
  • Seuret, S. and Véhel, J. L. (2002). The local Hölder function of a continuous function. Appl. Comput. Harmon. Anal. 13 263–276.
  • Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer, New York. Revised and extended from the 2004 French original, translated by Vladimir Zaiats.

Supplemental materials

  • Supplement to “Locally adaptive confidence bands”. Supplement A is organized as follows. Section A.1 develops connections between the Weierstraß function and the Admissibility Condition 3.5. Further notation and auxiliary results from empirical process theory are provided in Section A.2, whereas Section A.3 provides a simulation study together with an algorithm for the calculation of the locally adaptive confidence band. Section A.4 presents the remaining proofs of the results of Section 3. We proceed with the proofs of the results of Section 4 in Section A.5. Auxiliary results are stated and proved in Section A.6.