The Annals of Statistics

Adaptive wavelet estimator for nonparametric density deconvolution

Marianna Pensky and Brani Vidakovic

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Abstract

The problem of estimating a density $g$ based on a sample $X_1, X_2,\dots, X_n$ from $p = q*g$ is considered. Linear and nonlinear wavelet estimators based on Meyer-type wavelets are constructed. The estimators are asymptotically optimal and adaptive if $g$ belongs to the Sobolev space $H^{\alpha}$ . Moreover, the estimators considered in this paper adjust automatically to the situation when $g$ is supersmooth.

Article information

Source
Ann. Statist. Volume 27, Number 6 (1999), 2033-2053.

Dates
First available in Project Euclid: 4 April 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1017939249

Mathematical Reviews number (MathSciNet)
MR1765627

Digital Object Identifier
doi:10.1214/aos/1017939249

Zentralblatt MATH identifier
0962.62030

Subjects
Primary: 62G05: Estimation
Secondary: 62G07: Density estimation

Keywords
Mixing distribution wavelet transformation Sobolev space Meyer wavelet

Citation

Pensky, Marianna; Vidakovic, Brani. Adaptive wavelet estimator for nonparametric density deconvolution. The Annals of Statistics 27 (1999), no. 6, 2033--2053. doi:10.1214/aos/1017939249. http://projecteuclid.org/euclid.aos/1017939249.


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References

  • Abramovich, F. and Silverman, B. W. (1998). Wavelet decomposition approaches to statistical inverse problems. Biometrika 85 115-129.
  • Antoniadis, A., Gr´egoire, G. and McKeague, I. W. (1994). Wavelet method for curve estimation. J. Amer. Statist. Assoc. 89 1340-1353.
  • Carroll, R. J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184-1186.
  • Desouza, C. M. (1991). An empirical Bayes formulation of cohort models in cancer epidemiology. Statistics in Medicine 10 1241-1256.
  • Devroye, L. (1989). Consistent deconvolution in density estimation. Canad. J. Statist. 17 235- 239.
  • Diggle, P. J. and Hall, P. (1993). A Fourier approach to nonparametric deconvolution of a density estimate. J. Roy. Statist. Assoc. Ser. B 55 523-531.
  • Donoho, D. and Johnstone, I. (1995). Adapting to unknown smoothness via wavelet shrinkage. J. Amer. Statist. Assoc. 90 1200-1224.
  • Donoho, D., Johnstone, I., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508-539.
  • Efromovich, S. (1997). Density estimation for the case of supersmooth measurement error. J. Amer. Statist. Assoc. 92 526-535. Fan, J. (1991a). On the optimal rates of convergence for nonparametric deconvolution problem. Ann. Statist. 19 1257-1272. Fan, J. (1991b). Asymptoticnormality for deconvolution kernel density estimators. Sankhy ¯a Ser. A 53 97-110. Fan, J. (1991c). Global behavior of deconvolution kernel estimates. Statist. Sinica 1 541-551.
  • Fan, J. (1993). Adaptively local one-dimensional subproblems with application to a deconvolution problem. Ann. Statist. 21 600-610.
  • Hall, P., Kerkyacharian, G. and Picard, D. (1998). Block threshold rules for curve estimation using kernel and wavelet methods. Ann. Statist. 26 922-942.
  • Hall, P. and Patil, P. (1995). Formulae for mean integrated squared error of nonlinear waveletbased density estimators. Ann. Statist. 23 905-928.
  • Hall, P., Penev, S., Kerkyacharian, G. and Picard, D. (1997). Numerical performance of block thresholded wavelet estimators. Statistics Comput. 7 115-124.
  • Hern´andez, E. and Weiss, G. (1996). A First Course on Wavelets. CRC Press, Boca Raton, FL.
  • Kerkyacharian, G. and Picard, D. (1992). Density estimation in Besov spaces. Statist. Probab. Lett. 13 15-24.
  • Liu, M. C. and Taylor, R. L. (1989). A consistent nonparametric density estimator for the deconvolution problem. Canad. J. Statist. 17 427-438.
  • Louis, T. A. (1991). Using empirical Bayes methods in biopharmaceutical research. Statistics in Medicine 10 811-827.
  • Masry, E. (1991). Multivariate probability density deconvolution for stationary random processes. IEEE Trans. Inform. Theory 37 1105-1115. Masry, E. (1993a). Strong consistency and rates for deconvolution of multivariate densities of stationary processes. Stochastic Process. Appl. 47 53-74. Masry, E. (1993b). Asymptoticnormality for deconvolution estimators of multivariate densities of stationary processes. J. Multivariate Anal. 44 47-68.
  • Masry, E. (1994). Probability density estimation from dependent observations using wavelet orthonormal bases. Statist. Probab. Lett. 21 181-194.
  • Penskaya, M. (1985). Projection estimators of the density of an a priori distribution and of functionals of it. Theory Probab. Math. Statist. 31 113-124.
  • Pensky, M. (1999). Estimation of a smooth density function using Meyer-type wavelets. Statist. Decisions 17 111-123
  • Piterbarg, V. and Penskaya, M. (1993). On asymptoticdistribution of integrated squared error of an estimate of a component of a convolution. Math. Methods Statist. 2 30-41.
  • Stefansky, L. A. (1990). Rates of convergence of some estimators in a class of deconvolution problems. Statist. Probab. Lett. 9 229-235.
  • Stefanski, L. and Carrol, R. J. (1990). Deconvoluting kernel density estimators. Statistics 21 169-184.
  • Taylor, R. L. and Zhang, H. M. (1990). On a strongly consistent non-parametric density estimator for deconvolution problem. Comm. Statist. Theory Methods 19 3325-3342.
  • Talagrand M. (1994). Sharper bounds for empirical processes. Ann. Probab. 22 28-76.
  • Walter, G. G. (1981). Orthogonal series estimators of the prior distribution. Sankhy¯a Ser. A 43 228-245.
  • Walter, G. G. (1994). Wavelets and Other Orthogonal Systems with Applications. CRC Press, Boca Raton, FL.
  • Zayed, A. I. and Walter, G. G. (1996). Characterization of analytic functions in terms of their wavelet coefficients. Complex Variables 29 265-276.
  • Zhang, C. H. (1990). Fourier methods for estimating mixing densities and distributions. Ann. Statist. 18 806-831.
  • Zhang, C. H. (1992). On deconvolution using time of flight information in positron emission tomography. Statist. Sinica 2 553-575.