The root–unroot algorithm for density estimation as implemented via wavelet block thresholding

  • Lawrence Brown
  • Tony Cai
  • Ren Zhang
  • Linda Zhao
  • Harrison Zhou
Article

Abstract

We propose and implement a density estimation procedure which begins by turning density estimation into a nonparametric regression problem. This regression problem is created by binning the original observations into many small size bins, and by then applying a suitable form of root transformation to the binned data counts. In principle many common nonparametric regression estimators could then be applied to the transformed data. We propose use of a wavelet block thresholding estimator in this paper. Finally, the estimated regression function is un-rooted by squaring and normalizing. The density estimation procedure achieves simultaneously three objectives: computational efficiency, adaptivity, and spatial adaptivity. A numerical example and a practical data example are discussed to illustrate and explain the use of this procedure. Theoretically it is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the Besov classes. The estimator also automatically adapts to the local smoothness of the underlying function, and attains the local adaptive minimax rate for estimating functions at a point. There are three key steps in the technical argument: Poissonization, quantile coupling, and oracle risk bound for block thresholding in the non-Gaussian setting. Some of the technical results may be of independent interest.

Keywords

Adaptation Block thresholding Coupling inequality Density estimation Nonparametric regression Root–unroot transform Wavelets 

Mathematics Subject Classification (2000)

Primary: 62G99 Secondary: 62F12 62F35 62M99 

References

  1. 1.
    Anscombe F.J.: The transformation of Poisson, binomial and negative binomial data. Biometrika 35, 246–254 (1948)MATHMathSciNetGoogle Scholar
  2. 2.
    Bar-Lev S.K., Enis P.: On the construction of classes of variance stabilizing transformations. Statist. Probab. Lett. 10, 95–100 (1990)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bartlett M.S.: The square root transformation in analysis of variance. J. Roy. Statist. Soc. Suppl. 3, 68–78 (1936)CrossRefGoogle Scholar
  4. 4.
    Brown, L.D., Cai, T., Zhou, H.: Robust nonparametric estimation via wavelet median regression. Ann. Statist. (2008, to appear)Google Scholar
  5. 5.
    Brown L.D., Carter A.V., Low M.G., Zhang C.: Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift. Ann. Statist. 32, 2074–2097 (2004)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Brown L.D., Gans N., Mandelbaum A., Sakov A., Shen H., Zeltyn S., Zhao L.H.: Statistical analysis of a telephone call center: a queuing science perspective. J. Am. Statist. Assoc. 100, 36–50 (2005)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Brown L.D., Low M.G.: A constrained risk inequality with applications to nonparametric functional estimation. Ann. Statist. 24, 2524–2535 (1996)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cai T.: Adaptive wavelet estimation: a block thresholding and oracle inequality approach. Ann. Statist. 27, 898–924 (1999)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cai T.: On block thresholding in wavelet regression: adaptivity, block Size, and threshold level. Statistica Sinica 12, 1241–1273 (2002)MATHMathSciNetGoogle Scholar
  10. 10.
    Cai T., Low M.: Adaptive confidence balls. Ann. Statist. 34, 202–228 (2006)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Cai T., Silverman B.W.: Incorporating information on neighboring coefficients into wavelet estimation. Sankhya Ser. B 63, 127–148 (2001)MathSciNetGoogle Scholar
  12. 12.
    Chicken E., Cai T.: Block thresholding for density estimation: local and global adaptivity. J. Multivar. Anal. 95, 76–106 (2005)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Cohen A., Daubechies I., Jawerth B., Vial P.: Multiresolution analysis, wavelets, and fast algorithms on an interval. Comptes Rendus Acad. Sci. Paris (A). 316, 417–421 (1993)MATHMathSciNetGoogle Scholar
  14. 14.
    Daubechies I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)MATHGoogle Scholar
  15. 15.
    Daubechies I.: Two recent results on wavelets: wavelet bases for the interval, and biorthogonal wavelets diagonalizing the derivative operator. In: Schumaker, L.L., Webb, G. (eds) Recent Advances in Wavelet Analysis., pp. 237–258. Academic Press, New York (1994)Google Scholar
  16. 16.
    Donoho, D.L.: Nonlinear wavelet methods for recovery of signals, densities, and spectra from indirect and noisy data. In: Daubechies, I. (ed.) Different Perspectives on Wavelets. Proc. Symp. Appl. Math., vol. 47, pp. 173–205 (1993)Google Scholar
  17. 17.
    Donoho D.L., Johnstone I.M.: Minimax estimation via wavelet shrinkage. Ann. Statist. 26, 879–921 (1998)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Donoho D.L., Johnstone I.M., Kerkyacharian G., Picard D.: Density estimation by wavelet thresholding. Ann. Statist. 24, 508–539 (1996)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Efron B.: Transformation theory: how normal is a family of a distributions?. Ann. Statist. 10, 323–339 (1982)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Genovese C.R., Wasserman L.: Confidence sets for nonparametric wavelet regression. Ann. Statist. 33, 698–729 (2005)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Hall P., Kerkyacharian G., Picard D.: Block threshold rules for curve estimation using kernel and wavelet methods. Ann. Statist. 26, 922–942 (1998)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Hall P., Patil P.: Formulae for mean integrated squared error of nonlinear wavelet-based density estimators. Ann. Statist. 23, 905–928 (1995)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Hoyle M.H.: Transformations—an introduction and bibliography. Int. Stat. Rev. 41, 203–223 (1973)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Komlós J., Major P., Tusnády G.: An approximation of partial sums of independent rv’s, and the sample df. I. Z. Wahrsch. verw. Gebiete 32, 111–131 (1975)MATHCrossRefGoogle Scholar
  25. 25.
    Le Cam L.: On the information contained in additional observations. Ann. Statist. 2, 630–649 (1974)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Lepski O.V.: On a problem of adaptive estimation in white Gaussian noise. Theor. Probab. Appl. 35, 454–466 (1990)CrossRefGoogle Scholar
  27. 27.
    Low M.G., Zhou H.H.: A complement to Le Cam’s theorem. Ann. Statist. 35, 1146–1165 (2007)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Meyer Y.: Wavelets and Operators. Cambridge University Press, Cambridge (1992)MATHGoogle Scholar
  29. 29.
    Nussbaum M.: Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24, 2399–2430 (1996)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Runst T.: Mapping properties of non-linear operators in spaces of Triebel–Lizorkin and Besov type. Anal. Math. 12, 313–346 (1986)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Silverman B.W.: Density Estimation for Statistics and Data Analysis. Chapman and Hall, New York (1986)MATHGoogle Scholar
  32. 32.
    Strang G.: Wavelet and dilation equations: a brief introduction. SIAM Rev. 31, 614–627 (1992)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Triebel H.: Theory of Function Spaces II. Birkhäuser Verlag, Basel (1992)MATHGoogle Scholar
  34. 34.
    Weinberg J., Brown L.D., Stroud J.: Bayesian forecasting of an inhomogeneous Poisson process, with applications to call center data. J. Am. Statist. Assoc. 102, 1185–1198 (2007)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Zhang, R.: Nonparametric density estimation via wavelets. Ph.D. dissertation, Department of Statistics, University of Pennsylvania (2002)Google Scholar
  36. 36.
    Zhou, H.H.: A note on quantile coupling inequalities and their applications. Available from http://www.stat.yale.edu/hz68 (2006, submitted)

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Lawrence Brown
    • 1
  • Tony Cai
    • 1
  • Ren Zhang
    • 1
  • Linda Zhao
    • 1
  • Harrison Zhou
    • 2
  1. 1.The Wharton School, Department of StatisticsUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Yale UniversityNew HavenUSA

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