Skip to main content
SpringerLink
Log in

On estimation of \(L_{r}\)-norms in Gaussian white noise models

  • Published:
Probability Theory and Related Fields Aims and scope Submit manuscript

Abstract

We provide a complete picture of asymptotically minimax estimation of \(L_r\)-norms (for any \(r\ge 1\)) of the mean in Gaussian white noise model over Nikolskii–Besov spaces. In this regard, we complement the work of Lepski et al. (Probab Theory Relat Fields 113(2):221–253, 1999), who considered the cases of \(r=1\) (with poly-logarithmic gap between upper and lower bounds) and r even (with asymptotically sharp upper and lower bounds) over Hölder spaces. We additionally consider the case of asymptotically adaptive minimax estimation and demonstrate a difference between even and non-even r in terms of an investigator’s ability to produce asymptotically adaptive minimax estimators without paying a penalty.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bernstein, S.: Sur l’ordre de la meilleure approximation des fonctions continues par des polynomes de degré donné, volume 4. Hayez, imprimeur des académies royales, (1912)

  2. Besov, O.V., Ilyin, V.P.: Integral representations of functions and imbedding theorems 2, (1979)

  3. Bickel, P.J., Klaassen, C.A., Bickel, P.J., Ritov, Y., Klaassen, J., Wellner, J.A., Ritov, Y.: Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins University Press, Baltimore (1993)

    MATH  Google Scholar 

  4. Bickel, P.J., Ritov, Y.: Estimating integrated squared density derivatives: sharp best order of convergence estimates. Sankhyā: Indian J. Stat. Ser. A 50, 381–393 (1988)

    MathSciNet  MATH  Google Scholar 

  5. Birgé, L., Massart, P.: Estimation of integral functionals of a density. Ann. Stat. 23, 11–29 (1995)

    MathSciNet  MATH  Google Scholar 

  6. Boucheron, S., Lugosi, G., Massart, P.: Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press, Oxford (2013)

    MATH  Google Scholar 

  7. Cai, T.T., Low, M.G.: A note on nonparametric estimation of linear functionals. Ann. Stat. 31, 1140–1153 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Cai, T.T., Low, M.G.: Minimax estimation of linear functionals over nonconvex parameter spaces. Ann. Stat. 32, 552–576 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Cai, T.T., Low, M.G.: Nonquadratic estimators of a quadratic functional. Ann. Stat. 33, 2930–2956 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Cai, T.T., Low, M.G.: Testing composite hypotheses, hermite polynomials and optimal estimation of a nonsmooth functional. Ann. Stat. 39(2), 1012–1041 (2011)

    MathSciNet  MATH  Google Scholar 

  11. Carpentier, A.: Honest and adaptive confidence sets in \(L_p\). Electron. J. Stat. 7, 2875–2923 (2013)

    MathSciNet  MATH  Google Scholar 

  12. DeVore, R.A., Lorentz, G.G.: Constructive Approximation, vol. 303. Springer, Berlin (1993)

    MATH  Google Scholar 

  13. Ditzian, Z., Totik, V.: Moduli of Smoothness. Springer, Berlin (1987)

    MATH  Google Scholar 

  14. Donoho, D.L., Liu, R.C., MacGibbon, B.: Minimax risk over hyperrectangles, and implications. Ann. Stat. 18, 1416–1437 (1990)

    MathSciNet  MATH  Google Scholar 

  15. Donoho, D.L., Nussbaum, M.: Minimax quadratic estimation of a quadratic functional. J. Complex. 6(3), 290–323 (1990)

    MathSciNet  MATH  Google Scholar 

  16. Fan, J.: On the estimation of quadratic functionals. Ann. Stat. 19, 1273–1294 (1991)

    MathSciNet  MATH  Google Scholar 

  17. Giné, E., Nickl, R.: Mathematical Foundations of Infinite-Dimensional Statistical Models, vol. 40. Cambridge University Press, Cambridge (2015)

    MATH  Google Scholar 

  18. Hall, P., Marron, J.S.: Estimation of integrated squared density derivatives. Stat. Probab. Lett. 6(2), 109–115 (1987)

    MathSciNet  MATH  Google Scholar 

  19. Han, Y., Jiao, J., Weissman, T.: Minimax rate-optimal estimation of divergences between discrete distributions. (2016). Preprint arXiv:1605.09124

  20. Han, Y., Jiao, J., Weissman, T., Wu, Y.: Optimal rates of entropy estimation over Lipschitz balls. (2017). Preprint arXiv:1711.02141

  21. Härdle, W., Kerkyacharian, G., Picard, D., Tsybakov, A.: Wavelets, Approximation, and Statistical Applications, vol. 129. Springer, Berlin (2012)

    MATH  Google Scholar 

  22. Ibragimov, I., Khas’minskii, R.: Estimates of the signal, its derivatives and point of maximum for Gaussian distributions. Theory Probab. Appl. 25(4), 703–720 (1981)

    MATH  Google Scholar 

  23. Ingster, Y., Suslina, I.A.: Nonparametric Goodness-of-Fit Testing Under Gaussian Models, vol. 169. Springer, Berlin (2012)

    MATH  Google Scholar 

  24. Ingster, Y.I.: Minimax testing of nonparametric hypotheses on a distribution density in the \(L_p\) metrics. Theory Probab. Appl. 31(2), 333–337 (1987)

    MathSciNet  MATH  Google Scholar 

  25. Jiao, J., Han, Y., Weissman, T.: Minimax estimation of the \(L_1\) distance. In: 2016 IEEE International Symposium on Information Theory (ISIT), pp. 750–754. IEEE (2016)

  26. Jiao, J., Venkat, K., Han, Y., Weissman, T.: Minimax estimation of functionals of discrete distributions. IEEE Trans. Inf. Theory 61(5), 2835–2885 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Jiao, J., Venkat, K., Han, Y., Weissman, T.: Maximum likelihood estimation of functionals of discrete distributions. IEEE Trans. Inf. Theory 63(10), 6774–6798 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Kerkyacharian, G., Picard, D.: Estimating nonquadratic functionals of a density using haar wavelets. Ann. Stat. 24(2), 485–507 (1996)

    MathSciNet  MATH  Google Scholar 

  29. Korostelev, A.P., Tsybakov, A.B.: Minimax Theory of Image Reconstruction, vol. 82. Springer, Berlin (2012)

    MATH  Google Scholar 

  30. Korostelevi, A.: On estimation accuracy for nonsmooth functionals of regression. Theory Probab. Appl. 35(4), 784–787 (1991)

    Google Scholar 

  31. Laurent, B.: Efficient estimation of integral functionals of a density. Ann. Stat. 24(2), 659–681 (1996)

    MathSciNet  MATH  Google Scholar 

  32. Lepski, O.: On a problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35(3), 454–466 (1991)

    MathSciNet  Google Scholar 

  33. Lepski, O.: Asymptotically minimax adaptive estimation. I: Upper bounds. Optimally adaptive estimates. Theory Probab. Appl. 36(4), 682–697 (1992)

    MathSciNet  Google Scholar 

  34. Lepski, O.: Asymptotically minimax adaptive estimation. II. Schemes without optimal adaptation: Adaptive estimators. Theory Probab. Appl. 37(3), 433–448 (1993)

    MathSciNet  Google Scholar 

  35. Lepski, O., Nemirovski, A., Spokoiny, V.: On estimation of the \(L_r\) norm of a regression function. Probab. Theory Relat. Fields 113(2), 221–253 (1999)

    MATH  Google Scholar 

  36. Lepski, O.V.: On problems of adaptive estimation in white Gaussian noise. Top. Nonparam. Estim. 12, 87–106 (1992)

    MathSciNet  Google Scholar 

  37. Mukherjee, R., Newey, W.K., Robins, J.M.: Semiparametric efficient empirical higher order influence function estimators. (2017). Preprint arXiv:1705.07577

  38. Nemirovski, A.: Topics in non-parametric. Ecole d’Eté de Probabilités de Saint-Flour 28, 85 (2000)

    MathSciNet  MATH  Google Scholar 

  39. Nikol’skii, S.M.: Approximation of Functions of Several Variables and Embedding Theorems, vol. 205. Springer, Berlin (2012)

    Google Scholar 

  40. Qazi, M., Rahman, Q.: Some coefficient estimates for polynomials on the unit interval. Serdica Math. J. 33, 449–474 (2007)

    MathSciNet  MATH  Google Scholar 

  41. Robins, J., Li, L., Mukherjee, R., Tchetgen, E.T., van der Vaart, A.: Higher order estimating equations for high-dimensional models. Ann. Stat. (To Appear) (2016)

  42. Robins, J., Li, L., Tchetgen, E., van der Vaart, A.: Higher order influence functions and minimax estimation of nonlinear functionals. In: Probability and Statistics: Essays in Honor of David A. Freedman, pp. 335–421. Institute of Mathematical Statistics (2008)

  43. Spokoiny, V.: Adaptive and spatially hypothesis testing of a nonparametric hypothesis. Math. Methods Stat. 7, 245–273 (1998)

    MATH  Google Scholar 

  44. Tchetgen, E., Li, L., Robins, J., van der Vaart, A.: Minimax estimation of the integral of a power of a density. Stat. Probab. Lett. 78(18), 3307–3311 (2008)

    MathSciNet  MATH  Google Scholar 

  45. Tsybakov, A.: Introduction to Nonparametric Estimation. Springer, Berlin (2008)

    MATH  Google Scholar 

  46. Valiant, G., Valiant, P.: The power of linear estimators. In: IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS), 2011 , pp. 403–412. IEEE (2011)

  47. Van der Vaart, A.W.: Asymptotic Statistics, vol. 3. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  48. Varga, R.S., Karpenter, A.D.: On a conjecture of S. Bernstein in approximation theory. Sbornik: Math. 57(2), 547–560 (1987)

    MathSciNet  Google Scholar 

  49. Wu, Y., Yang, P.: Minimax rates of entropy estimation on large alphabets via best polynomial approximation. IEEE Trans. Inf. Theory 62(6), 3702–3720 (2016)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank Tsachy Weissman for the tremendous support and very helpful discussions. We are also grateful to an anonymous referee for detailed and helpful comments on improving this paper.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Stanford University, 350 Serra Mall, Stanford, CA, 94305, USA

    Yanjun Han

  2. Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, 257M Cory Hall, Berkeley, CA, 94720, USA

    Jiantao Jiao

  3. Department of Biostatistics, Harvard T. H. Chan School of Public Health, Harvard University, 655 Huntington Avenue, Boston, MA, 02115, USA

    Rajarshi Mukherjee

Authors
  1. Yanjun Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiantao Jiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rajarshi Mukherjee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yanjun Han.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Yanjun Han and Jiantao Jiao: Supported partially by the NSF Center for Science of Information under Grant CCF-0939370. Jiantao Jiao was partially supported by NSF Grant IIS-1901252.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, Y., Jiao, J. & Mukherjee, R. On estimation of \(L_{r}\)-norms in Gaussian white noise models. Probab. Theory Relat. Fields 177, 1243–1294 (2020). https://doi.org/10.1007/s00440-020-00982-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-020-00982-x

Keywords

Mathematics Subject Classification

Search

Navigation