Journal of Statistical Theory and Practice

, Volume 2, Issue 3, pp 453–463 | Cite as

Wavelet Based Estimation of the Derivatives of a Density for a Negatively Associated Process

  • Yogendra P. ChaubeyEmail author
  • Hassan Doosti
  • B. L. S. Prakasa Rao


Here we adopt the method of estimation for the derivatives of a probability density function based on wavelets discussed in Prakasa Rao (1996) to the case of negatively associated random variables. An upper bound on Lp-loss for the resulting estimator is given which extends such a result for the integrated mean square error (IMSE) given in Prakasa Rao (1996). Also, considering the case of derivative of order zero, the results given by Kerkyacharian and Picard (1992), Tribouley (1995) and Leblanc (1996) are obtained as special cases.

AMS Subject Classification

62G05 62G07 


Negative dependence Multiresolution analysis Besov space Wavelets Nonparametric estimation 


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  1. Alam, K., Saxena, K.M.L., 1981. Positive dependence in multivariate distribution. Commun. Statist-Theor. Meth., A10, 1183–1196.MathSciNetzbMATHGoogle Scholar
  2. Antoniadis, A., Grégoire, G., McKeague, I., 1994. Wavelet methods for curve estimation. Journal of the American Statistical Association, 89, 1340–1353.MathSciNetCrossRefGoogle Scholar
  3. Chaubey, Y. P., Doosti, H., Prakasa Rao, B.L.S.P., 2006. Wavelet based estimation of the derivatives of a density with associated variables. Int. J. Pure and App. Math., 27, 97–106.MathSciNetzbMATHGoogle Scholar
  4. Daubechies, I., 1992. Ten Lectures on Wavelets, CBMS-NSF regional conferences series in applied mathematics. SIAM, Philadelphia.CrossRefGoogle Scholar
  5. Daubechies, I., 1988. Orthogonal bases of compactly supported wavelets. Communication in Pure and Applied Mathematics, 41, 909–996.CrossRefGoogle Scholar
  6. De Barra, G., 1974. Introduction to Measure Theory, Von Nostrand Reinhold Company Ltd., New York.zbMATHGoogle Scholar
  7. Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., Picard, D., 1996. Density estimation by wavelet thresholding. The Annals of Statistics, 2, 508–539.MathSciNetzbMATHGoogle Scholar
  8. Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., Picard, D., 1995. Wavelet shrinkage: asymptopia (with discussion). Journal of Royal Statistical Society, Ser. B 57, 301–370.MathSciNetzbMATHGoogle Scholar
  9. Doukhan, P. and Leon, J.R., 1990. Une note sur la deviation quadratique destimateurs de densites par projections orthogonales. C.R. Acad. Sci. Paris, t310, serie 1, 425–430.zbMATHGoogle Scholar
  10. Doosti, H., Fakoor, V., Chaubey, Y.P., 2006. Wavelet linear density estimation for negative associated sequences. Journal of the Indian Statistical Association, 44, 127–135.Google Scholar
  11. Essary, J.D., Proschan, F., Walkup, D.W., 1967. Association of random variables with applications. Ann. Math. Statist., 44, 1466–1474.MathSciNetCrossRefGoogle Scholar
  12. Härdle, W., Kerkyacharian, G., Picard, D., Tsybakov, A., 1998. Wavelets, Approximations, and Statistical Applications, Lecture Notes in Statistics, 129, Springer-Verlag, Berlin.CrossRefGoogle Scholar
  13. Joag-Dev, K., Proschan, F., 1983. Negative association of random variables with applications. The Annals of Statistics, 11, 286–295.MathSciNetCrossRefGoogle Scholar
  14. Kerkyacharian, G., Picard, D., 1992. Density estimation in Besov spaces. Statistics and Probability Letters, 13, 15–24.MathSciNetCrossRefGoogle Scholar
  15. Leblanc, F., 1996. Wavelet linear density estimator for a discrete-time stochastic process: L p-losses. Statistics and Probability Letters, 27, 71–84.MathSciNetCrossRefGoogle Scholar
  16. Masry, E., 1994. Probability density estimation from dependent observations using wavelets orthonormal bases. Statistics and Probability Letters, 21, 181–194.MathSciNetCrossRefGoogle Scholar
  17. Meyer, Y., 1990. Ondelettes et Operateurs, Hermann, Paris.zbMATHGoogle Scholar
  18. Prakasa Rao, B.L.S., 1996. Nonparametric estimation of the derivatives of a density by the method of wavelets. Bull. Inform. Cyb., 28, 91–100.MathSciNetzbMATHGoogle Scholar
  19. Prakasa Rao, B.L.S., 1999a. Estimation of the integrated squared density derivative by wavelets. Bull. Inform. Cyb., 31, 47–65.MathSciNetzbMATHGoogle Scholar
  20. Prakasa Rao, B.L.S., 1999b. Nonparametric functional estimation: An overview. In Asymptotics, Nonparametrics and Time Series, Ed. Subir Ghosh, Marcel Dekker Inc., New York, 461–509.zbMATHGoogle Scholar
  21. Prakasa Rao, B.L.S., 2003. Wavelet linear density estimation for associated sequences. Journal of the Indian Statistical Association, 41, 369–379.MathSciNetGoogle Scholar
  22. Roussas, G., 1996. Positive and negative dependence with some statistical applications. In Research Development in Probability and Statistics, Ed. Brunner, E. and Denker, M., VSP, Utrecht, Holland.Google Scholar
  23. Shao, Q.M., 2000. A comparison theorem on moment inequalities between negatively associated and independent random variables. J. Theoretical. Probab., 13, 343–356.MathSciNetCrossRefGoogle Scholar
  24. Triebel, H., 1992. Theory of Function Spaces II, Birkhauser Verlag, Berlin.CrossRefGoogle Scholar
  25. Tribouley, K., 1995. Density estimation by cross-validation with wavelet method. Statistica Neerlandica, 45, 41–62.MathSciNetCrossRefGoogle Scholar
  26. Vidakovic, B., 1999. Statistical Modeling by Wavelets, Wiley, New York.CrossRefGoogle Scholar
  27. Walter, G., Ghorai, J., 1992. Advantages and disadvantages of density estimation with wavelets, In Proceedings of the 24th Symp. on the Interface, Ed. H. Joseph Newton, Interface FNA, VA 24: 234–343.Google Scholar

Copyright information

© Grace Scientific Publishing 2008

Authors and Affiliations

  • Yogendra P. Chaubey
    • 1
    Email author
  • Hassan Doosti
    • 2
  • B. L. S. Prakasa Rao
    • 3
  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontréalCanada
  2. 2.Statistical Research and Training Center (SRTC), TehranIran & Department of Statistics, Ferdowsi UniversityMashhadIran
  3. 3.Department of Mathematics and StatisticsUniversity of HyderabadHyderabadIndia

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