Probability Density Functions of the Empirical Wavelet Coefficients of Multidimensional Poisson Intensities

  • José Carlos Simon de Miranda
Part of the Contributions to Statistics book series (CONTRIB.STAT.)

We determine the probability density functions of the empirical wavelet coefficient estimator   =   dNin the wavelet series expansion p=    of non homogeneous multidimensional Poisson processes intensity functions.


Probability Density Function Poisson Process Point Process Haar Wavelet Wavelet Expansion 
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  1. [1]
    Timmermann, K. E. and Novak, R. D.: Multiscale Bayesian Estimation of Poisson Intensities. IEEE. 85-90 (1998).Google Scholar
  2. [2]
    Novak, R. D. and Kolaczyk, E. D.: A Multiscale MAP Estimation Method for Poisson Inverse Problems. IEEE 1682-1686 (1998).Google Scholar
  3. [3]
    Heikkinen, J. and Arjas, E.: Non-Parametric Bayesian Estimation of a Spatial Pois-son Intensity. Scand J Statist. 25 435-450 (1998).MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Novak, R. D. and Kolaczyk, E. D.: Multiscale Maximum Penalized Likelihood Esti-mators. IEEE. 156, (2002).Google Scholar
  5. [5]
    Müller, P. and Vidakovic, B.: Bayesian Inference with Wavelets: Density Estimation Journal of computational and Graphical Statistics. 7 (4), 456-468 (1998).Google Scholar
  6. [6]
    Barber, S., Nason, G. P. and Silverman, B. W.: Posterior Probability Intervals for Wavelet Thresholding, J. R. Statist. Soc. B. 64, part2, 189-205 (2002).MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Kolaczyk, E. D. and Novak, R. D.: Multiscale Likelihood Analysis and Complexity Penalized Estimation The Annals of Statistics. 32 (2), 2004, 500-527 (2004).MATHGoogle Scholar
  8. [8]
    Lam, W. M. and Wornell, G. W.: Multiscale Representation and Estimation of Frac-tal Point Processes IEEE Transactions on Signal Processing. 43 (11), 2606-2617 (1995).Google Scholar
  9. [9]
    Winter, A., Maître, H., Cambou, N. and Legrand, E.: Object Detection Using A Multiscale Probability Model IEEE. 269-272 (1996).Google Scholar
  10. [10]
    Figueiredo, M. A. T. and Novak, R. O.: Wavelet-Based Image Estimation: An Em-pirical Bayes Approach Using Je rey's Noninformative Prior. IEEE Transaction on Image Processing. 10 (9), September, 1322-1331 (2001).MATHCrossRefGoogle Scholar
  11. [11]
    Miranda, J.C.S. and Morettin, P.A.: Estimation of the Density of Point Processes on Rm via Wavelets, Technical Report -Department of Mathematics -IME-USP. No. 09, June, 2005.Google Scholar
  12. [12]
    Miranda, J.C.S.: Sobre a estimação da intensidade dos processos pontuais via on-daletas. São Paulo. 92 p. Tese de Doutorado. Instituto de Matemática e Estatística da Universidade de São Paulo. (2003).Google Scholar
  13. [13]
    Miranda, J.C.S. and Morettin, P.A.: On the Estimation of the Intensity of Point Processes on via Wavelets, Technical Report -Department of Statistics -IME-USP. 6,(2006).Google Scholar
  14. [14]
    De Miranda, J.C.S.: Adaptive Maximum Probability Estimation of Multidimensional Poisson Processes Intensity Function. Technical Report -Department of Mathematics -IME-USP. 01, March. (2006).Google Scholar
  15. [15]
    Meyer, Y.: Wavelets and Operators, Cambridge Studies in Advanced Mathematics, 37, April. (1993).Google Scholar
  16. [16]
    Daubechies, I.: Ten Lectures on Wavelets, Philadelphia, P.A. Society for Industrial and Applied Mathematics (CBMS -NSF Regional Conference Series in Applied Mathematics) 61 (1992).Google Scholar
  17. [17]
    Donoho, D. L., Johnstone, I. M.: Ideal Spatial Adaptation by Wavelet Shrinkage Biometrika. 81 (3), 425-455 (1994).MATHMathSciNetGoogle Scholar
  18. [18]
    Donoho, D. L., Johnstone, I. M. Kerkyacharian, G. and Picard, D. : Wavelet Shrink-age: Asymptopia. J. R. Statist. Soc. B. 57 (2), 301-369 (1995).MATHMathSciNetGoogle Scholar
  19. [19]
    Donoho, D. L., Johnstone, I. M. I. M., Kerkyacharian, G. and D. Picard: Density Es-timation by Wavelet Thresholding. The Annals of Statistics. 24 (2), 508-539 (1996).MATHCrossRefMathSciNetGoogle Scholar

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© Physica-Verlag Heidelberg 2008

Authors and Affiliations

  • José Carlos Simon de Miranda
    • 1
  1. 1.University of São PauloBrazil

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