The Annals of Statistics

Adaptive nonparametric confidence sets

James Robins and Aad van der Vaart

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Abstract

We construct honest confidence regions for a Hilbert space-valued parameter in various statistical models. The confidence sets can be centered at arbitrary adaptive estimators, and have diameter which adapts optimally to a given selection of models. The latter adaptation is necessarily limited in scope. We review the notion of adaptive confidence regions, and relate the optimal rates of the diameter of adaptive confidence regions to the minimax rates for testing and estimation. Applications include the finite normal mean model, the white noise model, density estimation and regression with random design.

Article information

Source
Ann. Statist., Volume 34, Number 1 (2006), 229-253.

Dates
First available in Project Euclid: 2 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.aos/1146576262

Digital Object Identifier
doi:10.1214/009053605000000877

Mathematical Reviews number (MathSciNet)
MR2275241

Zentralblatt MATH identifier
1091.62039

Subjects
Primary: 62G15: Tolerance and confidence regions 62G20: Asymptotic properties 62F25: Tolerance and confidence regions

Keywords
Adaptation white noise model density estimation regression testing rate

Citation

Robins, James; van der Vaart, Aad. Adaptive nonparametric confidence sets. Ann. Statist. 34 (2006), no. 1, 229--253. doi:10.1214/009053605000000877. https://projecteuclid.org/euclid.aos/1146576262


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References

  • Baraud, Y. (2004). Confidence balls in Gaussian regression. Ann. Statist. 32 528--551.
  • Barron, A., Birgé, L. and Massart, P. (1999). Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 301--413.
  • Barron, A. R. and Cover, T. M. (1991). Minimum complexity density estimation. IEEE Trans. Inform. Theory 37 1034--1054.
  • Beran, R. (2000). REACT scatterplot smoothers: Superefficiency through basis economy. J. Amer. Statist. Assoc. 95 155--171.
  • Beran, R. and Dümbgen, L. (1998). Modulation of estimators and confidence sets. Ann. Statist. 26 1826--1856.
  • Bickel, P. J. and Ritov, Y. (1988). Estimating integrated squared density derivatives: Sharp best order of convergence estimates. Sankhyā Ser. A 50 381--393.
  • Birgé, L. (2002). Discussion of ``Random rates in anisotropic regression,'' by M. Hoffmann and O. Lepski. Ann. Statist. 30 359--363.
  • Birgé, L. and Massart, P. (2001). Gaussian model selection. J. Eur. Math. Soc. 3 203--268.
  • Bretagnolle, J. and Huber, C. (1979). Estimation des densités: Risque minimax. Z. Wahrsch. Verw. Gebiete 47 119--137.
  • Brown, L. D., Carter, A. V., Low, M. G. and Zhang, C.-H. (2004). Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift. Ann. Statist. 32 2074--2097.
  • Cai, T. and Low, M. (2006). Adaptive confidence balls. Ann. Statist. 34 202--228.
  • Donoho, D. L. and Johnstone, I. M. (1995). Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 1200--1224.
  • Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation via wavelet shrinkage. Biometrika 81 425--455.
  • Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: Asymptopia? (with discussion). J. Roy. Statist. Soc. Ser. B 57 301--369.
  • Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508--539.
  • Efromovich, S. Yu. and Pinsker, M. S. (1984). Learning algorithm for nonparametric filtering. Autom. Remote Control 11 1434--1440.
  • Fan, J. (1991). On the estimation of quadratic functionals. Ann. Statist. 19 1273--1294.
  • Genovese, C. R. and Wasserman, L. (2005). Confidence sets for nonparametric wavelet regression. Ann. Statist. 33 698--729.
  • Golubev, G. K. (1987). Adaptive asymptotically minimax estimates of smooth signals. Problems Inform. Transmission 23 57--67.
  • Hoffmann, M. and Lepski, O. (2002). Random rates in anisotropic regression (with discussion). Ann. Statist. 30 325--396.
  • Ibragimov, I. A. and Khasminskii, R. Z. (1980). Asymptotic properties of some nonparametric estimates in Gaussian white noise. Proc. Third Summer School on Probab. Theory and Math. Statist. Varna 1978.
  • Ibragimov, I. A. and Khasminskii, R. Z. (1981). Statistical Estimation. Asymptotic Theory. Springer, Berlin.
  • Ingster, Yu. I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives. I, II, III. Math. Methods Statist. 2 85--114, 171--189, 249--268.,
  • Ingster, Yu. I. and Suslina, I. A. (2003). Nonparametric Goodness-of-Fit Testing Under Gaussian Models. Lecture Notes in Statist. 168. Springer, New York.
  • Juditsky, A. and Lambert-Lacroix, S. (2003). Nonparametric confidence set estimation. Math. Methods Statist. 12 410--428.
  • Laurent, B. (1996). Efficient estimation of integral functionals of a density. Ann. Statist. 24 659--681.
  • Laurent, B. (1997). Estimation of integral functionals of a density and its derivatives. Bernoulli 3 181--211.
  • Laurent, B. and Massart, P. (2000). Adaptive estimation of a quadratic functional by model selection. Ann. Statist. 28 1302--1338.
  • Lepskii, O. (1990). On a problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 454--466.
  • Lepskii, O. (1991). Asymptotically minimax adaptive estimation. I. Upper bounds. Optimally adaptive estimates. Theory Probab. Appl. 36 682--697.
  • Lepskii, O. (1992). Asymptotically minimax adaptive estimation. II. Schemes without optimal adaptation. Adaptive estimates. Theory Probab. Appl. 37 433--448.
  • Li, K.-C. (1989). Honest confidence regions for nonparametric regression. Ann. Statist. 17 1001--1008.
  • Mikosch, T. (1993). A weak invariance principle for weighted $U$-statistics with varying kernels. J. Multivariate Anal. 47 82--102.
  • Nussbaum, M. (1985). Spline smoothing in regression models and asymptotic efficiency in $L_2$. Ann. Statist. 13 984--997.
  • Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399--2430.
  • Pinsker, M. (1980). Optimal filtering of square-integrable signals in Gaussian noise. Problems Inform. Transmission 16 120--133.
  • Stone, C. J. (1984). An asymptotically optimal window selection rule for kernel density estimates. Ann. Statist. 12 1285--1297.
  • Tsybakov, A. (2004). Introduction à l'estimation non-paramétrique. Springer, Berlin.
  • van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Univ. Press.