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A central limit theorem for theL 2 error of positive wavelet density estimator

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Abstract

The asymptotic distribution of the integrated squared error of positive wavelet density estimator is derived. It is shown that three different cases arise depending on the smoothness of the unknown density. In each case the asymptotic distribution is shown to be normal. A Martingale central limit theorem is used to prove the results.

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References

  • Csörgő, S. and Horváth, L. (1988). Central limit theorems forL p -norms of density estimators,Zeitschrift Wahrsheinlichkeitstheorie und Verwandte Gebiete,80, 269–291.

    Google Scholar 

  • Daubechies, I. (1988). Orthogonal bases of compactly supported wavelets,Comm. Pure Appl. Math.,41, 909–996.

    MATH  MathSciNet  Google Scholar 

  • Delyon, B. and Judisky, A. (1996). On minimax wavelet estimators,Appl. Comput. Harmon. Anal.,3, 215–228.

    Article  MATH  MathSciNet  Google Scholar 

  • Donoho, B., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1996). Density estimation by wavelet thresholding,Ann. Statist.,24 (2), 508–539.

    Article  MATH  MathSciNet  Google Scholar 

  • Ghorai, J. (1980). Asymptotic normality of a quadratic measure of orthogonal series type density estimate,Ann. Inst. Statist. Math.,32, 341–350.

    Article  MATH  MathSciNet  Google Scholar 

  • Hall, P. (1984). Central limit theorem for integrated square error of multivariate nonparametric density estimators,J. Multivariate Anal.,14, 1–16.

    Article  MATH  MathSciNet  Google Scholar 

  • Hall, P. and Patil, P. (1995). Formulas for mean integrated error of nonlinear wavelet-based density estimators,Ann. Statist.,23, 905–928.

    MATH  MathSciNet  Google Scholar 

  • Horváth, L. (1991). OnL p -norms of multivariate density estimators,Ann. Statist.,19, 1933–1949.

    MATH  MathSciNet  Google Scholar 

  • Kerkyacharian, G. and Picard, D. (1992). Density estimation in Besov space,Statist. Probab. Lett.,13, 15–24.

    Article  MATH  MathSciNet  Google Scholar 

  • Masry, E. (1994). Probability density estimation from dependent observations using wavelet orthogonal basis,Statist. Probab. Lett.,21, 181–194.

    Article  MATH  MathSciNet  Google Scholar 

  • Rosenblatt, M. (1956). Remark on some nonparametric estimates of a density function,Ann. Math. Statist.,27, 832–837.

    MATH  MathSciNet  Google Scholar 

  • Walter, G. G. (1994).Wavelets and Other Orthogonal Systems with Application, CRC Press, Boca Ratan, Florida.

    Google Scholar 

  • Walter, G. G. and Shen, X. (1999). Continuous non-negative wavelets and their use in density estimation,Comm. Statist. Theory Methods,28, 1–17.

    MATH  MathSciNet  Google Scholar 

  • Wu, D. (1996). Asymptotic normality of the multiscale wavelet density estimator,Comm. Statist. Theory Methods,25 (9), 1957–1970.

    MATH  MathSciNet  Google Scholar 

  • Zhang, S. and Zheng, Z. (1999). On the asymptotic normality forL 2-error of wavelet density estimator with application,Comm. Statist. Theory Methods,23 (5), 1093–1104.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, The University of Wisconsin-Milwaukee, P. O. Box 413, 53201, Milwaukee, WI, U.S.A.

    J. K. Ghorai

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  1. J. K. Ghorai
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Ghorai, J.K. A central limit theorem for theL 2 error of positive wavelet density estimator. Ann Inst Stat Math 55, 619–637 (2003). https://doi.org/10.1007/BF02517811

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  • DOI: https://doi.org/10.1007/BF02517811

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